Committee Feed
https://feeds.library.caltech.edu/people/Hou-T-Y/committee.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenSat, 13 Apr 2024 01:22:05 +0000Continuum dynamics of solid-solid phase transitions
https://resolver.caltech.edu/CaltechETD:etd-10222007-135103
Authors: {'items': [{'email': 'allanzhong@hotmail.com', 'id': 'Zhong-Xiaoguang-Allan', 'name': {'family': 'Zhong', 'given': 'Xiaoguang Allan'}, 'show_email': 'NO'}]}
Year: 1995
DOI: 10.7907/PW4M-9B73
<p>This work focuses on the applications in dynamics of recently developed continuum-mechanical models of solid-solid phase transitions. The dynamical problems considered here involve only one space coordinate, and attention is limited to hyperelastic materials that involve two phases. This investigation has two purposes. The first is to determine the predictions of the models in complicated situations. Secondly, the present study attempts to develop analytical and numerical approaches to problems that may be relevant to the interpretation and understanding of experiments involving phase transitions under dynamical conditions.</p>
<p>The first problem studied involves the study of a semi-infinite bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. The end of the bar is suddenly subject to a constant impact velocity that persists for a finite time and is then removed. Interaction between the phase boundary and the elastic waves generated by the impact and subsequent reflections are studied in detail, and the trajectory of the phase boundary is determined exactly. The second task addressed involves the development of a Riemann solver to be applied to the numerical solution of Riemann problems for two-phase elastic materials. Riemann problems for such materials involve complications not present in the corresponding problems that arise, for example, in classical gas dynamics. Finally, a finite-difference method of Godunov type is developed for the numerical treatment of boundary-initial-value problems arising in the model of Abeyaratne and Knowles. The method is applied to specific problems.</p>https://thesis.library.caltech.edu/id/eprint/4215Control of Uncertain Systems: State-Space Characterizations
https://resolver.caltech.edu/CaltechETD:etd-03022006-131646
Authors: {'items': [{'id': 'Lu-Wei-Min', 'name': {'family': 'Lu', 'given': 'Wei-Min'}, 'show_email': 'NO'}]}
Year: 1995
DOI: 10.7907/dnxg-nz58
<p>A central issue in control system design has been to deal with uncertainty and nonlinearity in the systems. In this dissertation, an integrated treatment for both uncertainty and nonlinearity is proposed. This dissertation consists of two relatively independent parts. The first part deals with uncertain linear systems, while the second part treats uncertain nonlinear systems.</p>
<p>In the first part, the problem of control synthesis of uncertain linear systems is considered. A linear fractional transformation (LFT) framework is proposed for robust control design of uncertain linear control systems with structured uncertainty. Linear parameter-varying systems whose coefficients depend on some time-invariant unknown parameters are treated in a general algebraic framework; both the stabilization and the H<sub>∞</sub>-control problems are considered. For uncertain linear systems under structured perturbations, robustness synthesis problems are characterized in terms of linear matrix inequalities (LMIs) in the LFT framework. A generalized PBH test is also used to characterize the robustness synthesis problems. Moreover, a separation principle for the control synthesis of uncertain linear systems is revealed. The machinery also streamlines a number of results concerning the analysis and synthesis of multidimensional systems.</p>
<p>In the second part, the problem of control synthesis for nonlinear systems is addressed; stabilization, L<sup>1</sup>-control, H<sub>∞</sub>-control, robustness analysis, and robustness synthesis problems for nonlinear systems are examined in detail. In particular, locally and globally stabilizing controller parameterizations for nonlinear systems are derived; the formulae generalize the celebrated Youla-parameterization for linear systems. Both nonlinear L<sup>1</sup>-control and nonlinear H<sub>∞</sub>-control are also considered for dealing with disturbance attenuation problems for nonlinear systems. The L<sup>1</sup>-performance and L<sup>1</sup>-control of nonlinear systems are characterized in terms of certain invariance sets of the state space; in addition, the relation between the L<sup>1</sup>-control of a continuous-time system and the ℓ<sup>1</sup>-control of the related Euler approximated discrete-time systems is established. A systematic treatment for H<sub>∞</sub>-control synthesis of nonlinear systems is provided; the nonlinear H<sub>∞</sub>-control problem is characterized in terms of Hamilton-Jacobi Inequalities (HJIs) and nonlinear matrix inequalities (NLMIs); a class of H<sub>∞</sub>-controllers are parameterized as a fractional transformation of contractive stable parameters. Finally, the problems of stability and performance robustness analysis and synthesis for uncertain nonlinear systems subject to structured perturbations with bounded L<sub>2</sub>-gains are introduced; they are characterized in terms of HJIs and NLMIs as well. Computational issues are also addressed; it is confirmed that the computation needed for robustness analysis and synthesis of nonlinear systems is of equivalent difficulty to that for checking Lyapunov stability.</p>https://thesis.library.caltech.edu/id/eprint/835Run-Up and Nonlinear Propagation of Oceanic Internal Waves and Their Interactions
https://resolver.caltech.edu/CaltechETD:etd-12192007-084353
Authors: {'items': [{'id': 'Lin-Duo-min', 'name': {'family': 'Lin', 'given': 'Duo-min'}, 'show_email': 'NO'}]}
Year: 1996
DOI: 10.7907/rgkn-4g56
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
<p>A weakly nonlinear and weakly dispersive oceanic internal long wave (ILW) model, in analogy with the generalized Boussinesq's (gB) model, is developed to investigate generation and propagation of internal waves (IWs) in a system of two-layer fluids. The ILW model can be further derived to give a bidirectional ILW model for facilitating calculations of head-on collisions of nonlinear internal solitary waves (ISWs). The important nonlinear features, such as phase shift of ISWs resulting from nonlinear collision encounters, are presented. The nonlinear processes of reflection and transmission of waves in channels with a slowly varying bottom are studied.</p>
<p>The terminal effects of IWs running up submerged sloping seabed are studied by the ILW model in considerable detail. Explicit solution of the nonlinear equations are obtained for several classes of wave forms, which are taken as the inner solutions and matched, when necessary for achieving uniformly valid results, with the outer solution based on linear theory for the outer region with waves in deep water. Based on the nonlinear analytic solution, two kinds of initial run-up problems can be solved analytically, and the breaking criteria and run-up law for IWs are obtained. The run-up of ISWs along the uniform beach is simulated by numerical computations using a moving boundary technique. The numerical results based on the ILW model are found in good agreement with the run-up law of ISWs when the amplitudes of the ISWs are small.</p>
<p>The ILW model differs from the corresponding KdV model in admitting bidirectional waves simultaneously and conserving mass. This model is applied to analyze the so-called critical depth problem of ISWs propagating across a critical station at which the depths of the two fluid layers are about equal so as to give rise to a critical point of the KdV equation. As the critical point is passed, the KdV model may predict a new upward facing ISW relative to a local mean interface is about to emerge from the effects of disintegrating original downward ISW. This phenomenon has never been observed in our laboratory. Numerical results are presented based on the present ILW model for ISWs climbing up a curved shelf and a sloping plane seabed. It is shown that in the transcritical region, the behaviour of the ISWs predicted by the ILW model depends on the relative importance of two dimensionless parameters, s<sub>w</sub>, the order of ISW wave slope, and s, the beach slope. For s >> s<sub>w</sub>, the wave profile of ISWs exhibits a smooth transition across the transcritical region; for s <s<sub>w</sub>, ISWs emerge with an oscillatory tail after passing across the critical point. Numerical simulations based on the ILW model are found in good agreement with laboratory observations.</p>
<p>Finally, conclusions are drawn from the results obtained in the present study based on the ILW model.</p>https://thesis.library.caltech.edu/id/eprint/5066I. Run-Up of Ocean Waves on Beaches II. Nonlinear Waves in a Fluid-Filled Elastic Tube
https://resolver.caltech.edu/CaltechETD:etd-01072008-105605
Authors: {'items': [{'email': 'jinzhang@hku.hk', 'id': 'Zhang-Jin-E', 'name': {'family': 'Zhang', 'given': 'Jin E.'}, 'show_email': 'NO'}]}
Year: 1996
DOI: 10.7907/TGA4-F552
<p>Part I.</p>
<p>This study considers the three-dimensional run-up of long waves on a horizontally uniform beach of vertically constant or variable slope which is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is first applied to obtain the fundamental solution for a uniform train of sinusoidal waves obliquely incident upon a uniform beach of variable downward slope without wave breaking. The linearly superposable solutions provide a basis for subsequent comparative studies when the nonlinear and dispersive effects are taken into account, both separately and jointly, thus providing a comprehensive prospect of the extents of influences due to these physical effects. These comparative results seem to be new.</p>
<p>By linear theory for waves at nearly grazing incidence, run-up is significant only for the waves in a set of eigenmodes being trapped within the beach at resonance with the exterior ocean waves. Fourier synthesis is employed to analyze a solitary wave and a train of cnoidal waves obliquely incident upon a sloping beach, with the nonlinear and dispersive effects neglected at this stage. Comparison is made between the present theory and the ray theory to ascertain a criterion of validity for the classical ray theory. The wave-induced longshore current is evaluated by finding the Stokes drift of the fluid particles carried by the momentum of the waves obliquely incident upon a sloping beach. Currents of significant velocities are produced by waves at incidence angles about 45° and by grazing waves trapped on the beach. Also explored are the effects of the variable downward slope and curvature of a uniform beach on three-dimensional run-up and reflection of long waves.</p>
<p>When the nonlinear effects are taken into account, the exact governing equations for determining a moving inviscid waterline are introduced here based on the local Lagrangian coordinates. A special numerical scheme has been developed for efficient evaluation of these governing equations. The scheme is shown to have a very high accuracy by comparison with some exact solutions of the shallow water equations. The maximum run-up of a solitary wave predicted by the shallow water equations depends on the initial location of the solitary wave and is not unique in value because the wave becomes increasingly more steepened given longer time to travel in the absence of the dispersive effects; it is in general larger than that predicted by the linear long-wave theory. The farther the initial solitary wave of the KdV form is imposed from the beach, the larger the maximum run-up it will reach.</p>
<p>The dispersive effects are also very important in two-dimensional run-ups in its role of keeping the nonlinear effects balanced at equilibrium, so that the run-ups predicted by the generalized Boussinesq model (Wu 1979) always yield unique values for run-up of a given initial solitary wave, regardless of its initial position. The result for the gB model is slightly larger than the wave run-up predicted by linear long-wave theory. The dispersive effects tend to reduce the wave run-up either for linear system or for nonlinear system.</p>
<p>A three-dimensional process of wave run-up upon a vertical wall has also been studied.</p>
<p>Part II.</p>
<p>This part is a study of nonlinear waves in a fluid-filled elastic tube, whose wall material satisfies the stress-strain law given by the kinetic theory of rubber. The results of this study have extended the scope of this subject, which has been limited to dealing with unidirectional solitary waves only (Olsen and Shapiro 1967), by establishing an exact theory for bidirectional solitons of arbitrary shape. This class of solitons has several remarkable characteristics. These solitons may have arbitrary shape and arbitrary polarity (upward or downward), and all propagate with the same phase velocity. The last feature of wave velocity renders the interactions impossible between unidirectional waves. However, the present new theory shows that bidirectional waves can have head-on collision through which our exact solution leaves each wave a specific phase shift as a permanent mark of the waves having made the nonlinear encounter. The system is at least tri-Hamiltonian and integrable. An iteration scheme has been developed to integrate the system. The system is distinguished by the fact that any local initial disturbance released from a state of rest will become two solitons traveling to the opposite direction, and shocks do not form if initial value is continuous.</p>https://thesis.library.caltech.edu/id/eprint/56Two-Dimensional Steady Bow Waves in Water of Finite Depth
https://resolver.caltech.edu/CaltechETD:etd-01222008-091958
Authors: {'items': [{'id': 'Kao-John', 'name': {'family': 'Kao', 'given': 'John'}, 'show_email': 'NO'}]}
Year: 1998
DOI: 10.7907/tp0w-sd61
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
<p>In this study, the two-dimensional steady bow flow in water of arbitrary finite depth has been investigated. The two-dimensional bow is assumed to consist of an inclined flat plate connected downstream to a horizontal semi-infinite draft plate. The bottom of the channel is assumed to be a horizontal plate; the fluid is assumed to be invicid , incompressible; and the flow irrotational. For the angle of incidence α (held by the bow plate) lying between 0° and 60°, the local flow analysis near the stagnation point shows that the angle lying between the free surface and the inclined plate, β, must always be equal to 120°, otherwise no solution can exist. Moreover, we further find that the local flow solution does not exist if α > 60°, and that on the inclined plate there exists a negative pressure region adjacent to the stagnation point for α < 30°. Singularities at the stagnation point and the upstream infinity are found to have multiple branch-point singularities of irrational orders.</p>
<p>A fully nonlinear theoretical model has been developed in this study for evaluating the incompressible irrotational flow satisfying the free-surface conditions and two constraint equations. To solve the bow flow problem, successive conformal mappings are first used to transform the flow domain into the interior of a unit semi-circle in which the unknowns can be represented as the coefficients of an infinite series. A total error function equivalent to satisfying the Bernoulli equation is defined and solved by minimizing the error function and applying the method of Lagrange's multiplier. Smooth solutions with monotonic free surface profiles have been found and presented here for the range of 35° < α < 60°, a draft Froude number Fr<sub>d</sub> less then 0.5, and a water-depth Froude number Fr<sub>h</sub> less than 0.4.</p>
<p>The dependence of the solution on these key parameters is examined. As α decreases for fixed Fr<sub>d</sub> and Fr<sub>h</sub>, the free surface falls off more steeply from the stagnation point. Similarly, as Fr<sub>d</sub> increases, the free surface falls off quickly from the stagnation point, but for decreasing Fr<sub>h</sub> it descends rather slowly towards the upstream level. As Fr<sub>h</sub> decreases further, difficulties cannot be surmounted in finding an exact asymptotic water level at upstream infinity, which may imply difficulties in finding solutions for water of infinite depth. Our results may be useful in designing the optimum bow shape.</p>https://thesis.library.caltech.edu/id/eprint/277The Multiscale Finite Element Method (MsFEM) and Its Applications
https://resolver.caltech.edu/CaltechETD:etd-11102005-090314
Authors: {'items': [{'email': 'efendiev@math.tamu.edu', 'id': 'Efendiev-Yalchin-R', 'name': {'family': 'Efendiev', 'given': 'Yalchin R.'}, 'orcid': '0000-0001-9626-303X', 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/2QJN-2S06
<p>Multiscale problems occur in many scientific and engineering disciplines, in petroleum engineering, material science, etc. These problems are characterized by the great deal of spatial and time scales which make it difficult to analyze theoretically or solve numerically. On the other hand, the large scale features of the solutions are often of main interest. Thus, it is desirable to have a numerical method that can capture the effect of small scales on large scales without resolving the small scale details.</p>
<p>In the first part of this work we analyze the multiscale finite element method (MsFEM) introduced in [28] for elliptic problems with oscillatory coefficients. The idea behind MsFEM is to capture the small scale information through the base functions constructed in elements that are larger than the small scale of the problem. This is achieved by solving for the finite element base functions from the leading order of homogeneous elliptic equation. We analyze MsFEM for different situations both analytically and numerically. We also investigate the origin of the resonance errors associated with the method and discuss the ways to improve them.</p>
<p>In the second part we discuss flow based upscaling of absolute permeability which is an important step in the practical simulations of flow through heterogeneous formations. The central idea is to compute the upscaled, grid-block permeability from fine scale solutions of the flow equation. It is well known that the grid block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of grid blocks. We analyze the effects of the boundary conditions and grid block sizes on the computed grid block absolute permeabilities. Moreover, we employ the ideas developed in the analysis of MsFEM to improve the computed values of absolute permeability.</p>
<p>The last part of the work is the application of MsFEM as well as upscaling of absolute permeability on upscaling of two-phase flow. In this part we consider coarse models using MsFEM. We demonstrate the efficiency of these models for practical problems. Moreover, we show that these models improve the existing approaches.</p>https://thesis.library.caltech.edu/id/eprint/4487Studies on Nonlinear Dispersive Water Waves
https://resolver.caltech.edu/CaltechETD:etd-08152006-140314
Authors: {'items': [{'email': 'wendong.qu@pimco.com', 'id': 'Qu-Wendong', 'name': {'family': 'Qu', 'given': 'Wendong'}, 'show_email': 'NO'}]}
Year: 2000
DOI: 10.7907/19B4-2N21
This study investigates the phenomena of evolution of two-dimensional, fully nonlinear, fully dispersive, incompressible and irrotational waves in water of uniform depth in single and in double layers. The study is based on an exact fully nonlinear and fully dispersive (FNFD) wave model developed by Wu (1997, 1999a). This FNFD wave model is first based on two exact equations involving three variables all pertaining to their values at the water surface. Closure of the system of model equations is accomplished either in differential form, by attaining a series expansion of the velocity potential, or in integral form by adopting a boundary integral equation for the velocity field.
A reductive perturbation method for deriving asymptotic theory for higher-order solitary waves is developed using the differential closure equation of the FNFD wave theory. Using this method, we have found the leading 15th-order solitary wave solutions. The solution is found to be an asymptotic solution which starts to diverge from the 12th-order so that the 11th-order solution appears to provide the best approximation to the fully nonlinear solitary waves, with a great accuracy for waves of small to moderately large amplitudes.
Two numerical methods for calculating unsteady fully nonlinear waves, namely, the FNFD method and the Point-vortex method, are developed and applied to compute evolutions of fully nonlinear solitary waves. The FNFD method, which is based on the integral closure equation of Wu's theory, can provide good performance on computation of solitary waves of very large amplitude. The Point-vortex method using the Lagrange markers is very efficient for computation of waves of small to moderate amplitudes, but has intrinsic difficulties in computing waves of large amplitudes. These two numerical methods are applied to carry out a comparative study of interactions between solitary waves.
Capillary-gravity solitary waves are investigated both theoretically and numerically. The theoretical study based on the reductive perturbation method provides asymptotic theories for higher-order capillary-gravity solitary waves. A stable numerical method (FNFD) for computing exact solutions for unsteady capillary-gravity solitary waves is developed based on the FNFD wave theory. The results of the higher-order asymptotic theories compare extremely well with those given by the FNFD method for waves of small to moderate amplitudes.
A numerical method for computing unsteady fully nonlinear interfacial waves in two-layer fluid systems is developed based on the FNFD model. The subcritical and supercritical cases can be clearly distinguished by this method, especially for waves of amplitudes approaching the maximum attainable for the fully nonlinear theory.https://thesis.library.caltech.edu/id/eprint/3134Hard vs. soft bounds in probablilistic robustness analysis and generalized source coding and optimal web layout design
https://resolver.caltech.edu/CaltechETD:etd-05042006-131410
Authors: {'items': [{'id': 'Zhu-Xiaoy', 'name': {'family': 'Zhu', 'given': 'Xiaoyun'}, 'show_email': 'NO'}]}
Year: 2000
DOI: 10.7907/1f3r-va82
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Part I:
The relationship between hard vs. soft bounds and probabilistic vs. worst-case problem formulations for robustness analysis has been a source of some apparent confusion in the control community, and this thesis attempts to clarify some of these issues. Essentially, worst-case analysis involves computing the maximum of a function which measures performance over some set of uncertainty. Probabilistic analysis assumes some distribution on the uncertainty and computes the resulting probability measure on performance. Exact computation in each case is intractable in general. In the past most research focused on computing hard bounds on worst-case performance. This thesis explores the use of both hard and soft bounds in probabilistic robustness analysis, and investigates the computational complexity of the problems through extensive numerical experimentation. We focus on the simplest possible problem formulations that we believe reveal the difficulties associated with more general probabilistic analysis.
By extending the standard structured singular value [...] framework to allow for probabilistic descriptions of uncertainty, probabilistic [...] is defined, which characterizes the probability distribution of some performance function. The computation of probabilistic [...] involves approximating the level surface of the function in the parameter space, which is even more complex than the worst-case [...] computation, a well-known NP-hard problem. In particular, providing sufficiently tight bounds in the tail of the distribution is extremely difficult. This thesis proposes three different methods for computing a hard upper bound on probabilistic [...] whose tightness can be tested by comparison with the soft bound provided by Monte-Carlo simulations. At the same time, the efficiency of the soft bounds can be significantly improved with the information from the hard bound computation. Among the three algorithms proposed, the LC-BNB algorithm is proven by numerical experiments to provide the best average performance on random examples. One particular example is shown in the end to demonstrate the effectiveness of the method.
Part II:
The design of robust and reliable networks and network services has become an increasingly challenging task in today's Internet world. To achieve this goal, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic. Moreover, a direct link has been established between the self-similar nature of measured aggregate network traffic and the underlying heavy-tailed distributions of the Web traffic at the source level.
This thesis provides a natural and plausible explanation for the origin of heavy tails in Web traffic by introducing a series of simplified models for optimal Web layout design with varying levels of realism and analytic tractability. The basic approach is to view the minimization of the average file download time as a generalization of standard source coding for data compression, but with the design of the Web layout rather than the codewords. The results, however, are quite different from standard source coding, as all assumptions produce power law distributions for a wide variety of user behavior models.
In addition, a simulation model of more complex Web site layouts is proposed, with more detailed hyperlinks and user behavior. The throughput of a Web site can be maximized by taking advantage of information on user access patterns and rearranging (splitting or merging) files on the Web site accordingly, with a constraint on available resources. A heuristic optimization on random graphs is formulated, with user navigation modeled as Markov Chains. Simulations on different classes of graphs as well as more realistic models with simple geometries in individual Web pages all produce power law tails in the resulting size distributions of the files transferred from the Web sites. This again verifies our conjecture that heavy-tailed distributions result naturally from the tradeoff between the design objective and limited resources, and suggests a methodology for aiding in the design of high-throughput Web sites.https://thesis.library.caltech.edu/id/eprint/1607Multiscale numerical methods for the singularly perturbed convection-diffusion equation
https://resolver.caltech.edu/CaltechETD:etd-02272006-094856
Authors: {'items': [{'email': 'peter_park@harvard.edu', 'id': 'Park-P-J', 'name': {'family': 'Park', 'given': 'Peter J.'}, 'show_email': 'NO'}]}
Year: 2000
DOI: 10.7907/4C7T-3440
We develop efficient and robust numerical methods in the finite element framework for numerical solutions of the singularly perturbed convection-diffusion equation and of a degenerate elliptic equation. The standard methods for purely elliptic or hyperbolic problems perform poorly when there are sharp boundary and internal layers in the solution caused by the dominant convective effect. We offer a new approach in which we design the finite element basis functions that capture the local behavior correctly.
When the structure of the layers can be determined locally, we apply the multiscale finite element method in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. When the layer structure is nonlocal, we use a variational principle to gain additional information. For a random velocity field, this variational principle provides correct scaling results. This allows us to design asymptotic basis functions that can capture the global layers correctly.
The same approach is also extended to elliptic problems with high contrast coefficients. When an asymptotic result is available, it is incorporated naturally into the finite element setting developed earlier. When there is a strong singularity due to a discontinuous coefficient, we construct the basis functions using the infinite element method. Our methods can handle singularities efficiently and are not sensitive to the large contrast.
https://thesis.library.caltech.edu/id/eprint/783Magnetohydrodynamic modeling of solar magnetic arcades using exponential propagation methods
https://resolver.caltech.edu/CaltechETD:etd-02062006-154529
Authors: {'items': [{'email': 'mtokman@ucmerced.edu', 'id': 'Tokman-M', 'name': {'family': 'Tokman', 'given': 'Mayya'}, 'show_email': 'YES'}]}
Year: 2001
DOI: 10.7907/PDCS-GX15
Advanced numerical methods based on exponential propagation have been applied to magnetohydrodynamic (MHD) simulations. This recently developed numerical technique evolves the system of nonlinear equations using exponential propagation of the Jacobian matrix. The exponential of the matrix is approximated by projecting it onto the Krylov subspace using the Arnoldi algorithm. The primary advantage of the exponential propagation method is that it allows time steps exceeding the Courant-Friedrichs-Lewy (CFL) limit. Another important aspect is faster convergence of the iteration computing the Krylov subspace projection compared to solving an implicit formulation of the system with similar iterative methods. Since the time scales in the resistive MHD equations are widely separated, the exponential propagation methods are especially advantageous for computing the long term evolution of a low-beta plasma. We analyze several types of exponential propagation methods and highlight important issues in the development of such techniques. Our analysis also suggests new ways to construct schemes of this type. Implementation issues, including scalability properties of exponential propagation methods, and performance are also discussed.
In the second part of this work we present numerical MHD models which are constructed using exponential propagation methods and which describe the evolution of the magnetic arcades in the solar corona. Since these numerical methods have not been used before for large evolutionary systems like resistive MHD, we first validate our approach by demonstrating application of the exponential schemes to two existing magnetohydrodynamic models. We simulate the reconnection process resulting from shearing the footpoints of two-dimensional magnetic arcades and compute the three-dimensional linear force-free states of plasma configurations. Analysis of these calculations leads us to new insights about the topology of the solutions. The final chapter of this work is dedicated to a new three-dimensional numerical model of the dynamics of coronal plasma configurations. The model is motivated by observations and laboratory experiments simulating the evolution of solar arcades. We analyze the results of numerical simulations and demonstrate that our numerical approach provides an accurate and stable way to compute the solution to the zero-resistive MHD system. Based on comparisons of the simulation results and the observational data, we offer an explanation for the observed structure of eruptive events in the corona called coronal mass ejections (CME). We argue that the diversity of the images of CMEs obtained by the observational instruments can be explained as two-dimensional projections of a unique three-dimensional plasma configuration and suggest an eruption mechanism.https://thesis.library.caltech.edu/id/eprint/519Three-Dimensional Cohesive Modeling of Impact Damage of Composites
https://resolver.caltech.edu/CaltechTHESIS:10112010-130530819
Authors: {'items': [{'email': 'rena@uclm.es', 'id': 'Yu-Chengxiang-Rena', 'name': {'family': 'Yu', 'given': 'Chengxiang Rena'}, 'orcid': '0000-0003-4176-0324', 'show_email': 'YES'}]}
Year: 2001
DOI: 10.7907/nd8e-tc84
<p>The objective of this work is to establish the applicability of cohesive theories of fracture in situations involving material interface, material heterogeneity (e.g., layered composites), material anisotropy(e.g., fiber-reinforced composites), shear cracks, intersonic dynamic crack growth and dynamic crack branching. The widely used cohesive model is extended to orthotropic range. The so-developed computational tool, completed by a self-adaptive fracture procedure and a frictional contact algorithm, is capable of following the evolution of three-dimensional damage processes, modeling the progressive decohesion of interfaces and anisotropic materials. The material parameters required by cohesive laws are directly obtained from static experiments. The ability of the methodology to simulate diverse problems such as delamination between fibers of graphite/epoxy composites, as well as sandwich structures and branching within brittle bulk materials has been demonstrated.</p>https://thesis.library.caltech.edu/id/eprint/6126Diffusion-Mediated Regulation Endocrine Networks
https://resolver.caltech.edu/CaltechETD:etd-12052003-095049
Authors: {'items': [{'email': 'petrasek@caltech.edu', 'id': 'Petrasek-Danny', 'name': {'family': 'Petrasek', 'given': 'Danny'}, 'orcid': '0000-0003-4178-4844', 'show_email': 'NO'}]}
Year: 2002
DOI: 10.7907/776t-vs55
In endocrine glands, vigorous and coordinated responses are often elicited by modest changes in the concentration of the organist molecule. The mammalian parathyroid gland is a representative case. Small (5%) changes in serum calcium result in tenfold (1000%) changes in glandular parathyroid hormone (PTH) release. In vitro, single isolated cells are observed to secrete fewer hormones than cells residing within a connected group, suggesting that a network has emergent regulatory properties. In PTH secreting tumors however, the ability to quickly respond to changes in calcium is strongly damped. A unifying hypothesis that accounts for these phenomena is realized by extra-cellular modulation of calcium diffusivity. A theoretical model and computational experiments demonstrate qualitative agreement with published experimental results. Our results suggest that in addition to the cellular mechanisms, endocrine glandular networks may have regulatory prowess at the level of interstitial transport. The extra-cellular diffusional mechanism proposed provides a consistent argument for 1) higher secretion of single cells in a connected network compared to isolated cells, 2) the rapid nonlinear response seen in healthy glands as well as 3) the pathological responses seen in hyperplasia and adenoma. Since the proposed diffusional regulation strongly depends on the existence of a connected cell network (gland), it also suggests a rationale for the advantages of cell networks as organs versus a dispersed system of isolated cells (in the case of the parathyroid gland).https://thesis.library.caltech.edu/id/eprint/4788Dynamics of Phase Transitions in Strings, Beams and Atomic Chains
https://resolver.caltech.edu/CaltechETD:etd-11072006-100058
Authors: {'items': [{'email': 'purohit@seas.upenn.edu', 'id': 'Purohit-Prashant-Kishore', 'name': {'family': 'Purohit', 'given': 'Prashant Kishore'}, 'show_email': 'NO'}]}
Year: 2002
DOI: 10.7907/DP97-XH80
This thesis presents a theory for dynamical martensitic phase transitions in strings and beams. Shape memory alloys that rely on such phase transitions for their unique properties are often used in slender configurations like beams and rods. Yet most studies of phase transformations are in one dimension and consider only extension. The theory presented in this thesis to model these slender structures is based on the general continuum mechanical framework of thermoelasticity with a non-convex Helmholtz free energy. This non-convexity allows for the simultaneous existence of several metastable phases in a material; in particular, it leads to the formation of phase boundaries. The study of the laws governing the propagation of phase boundaries is the object of this thesis.
Phase boundaries in strings are studied first. It is demonstrated that the motion of phase boundaries is not fully described by the usual balance laws of mass, momentum and energy. Additional constitutive information must be furnished from outside, and this additional information is referred to as the kinetic relation. While this notion is well-accepted in continuum theory, there is no definitive experiment or theoretical framework to determine the kinetic relation. This study of strings proposes a simple experiment to determine the kinetic relation. It also proposes a numerical method that accurately describes the complex behaviour of strings with phase boundaries.
The kinetic relation can also be viewed from the atomic scale. Phase transformations involve a complex rearrangement of the atoms the explicit details of which are averaged in a continuum theory. The kinetic relation may be viewed as an aggregate of those aspects of the atomistic rearrangement that have a bearing on macroscopic phenomena. This view is explored using a simple one dimensional model of an atomic chain with non-convex interaction potentials. A kinetic relation is obtained from dynamic simulations of impact experiments on the chain.
The latter part of this thesis studies beams made of materials capable of phase transitions. It develops a conceptual framework that accounts for extension, shear and flexure in such beams using a non-convex stored energy function. Specific constitutive assumptions that relate to the underlying crystallography are developed. The theory is applied to design a simple experiment on single crystals of martensitic materials with the objective of measuring the kinetic relation.
Finally, propulsion at small scales is discussed as an application of beams made of phase transforming material. The goal is to mimic the flagellum of a micro-organism by propagating phase boundaries through a shearbale rod.https://thesis.library.caltech.edu/id/eprint/4442Phase Boundary Propagation in Heterogeneous Media
https://resolver.caltech.edu/CaltechTHESIS:10082010-142653040
Authors: {'items': [{'id': 'Craciun-Bogdan', 'name': {'family': 'Craciun', 'given': 'Bogdan'}, 'show_email': 'NO'}]}
Year: 2002
DOI: 10.7907/JXG6-W865
<p>There has been much recent progress in the study of free boundary problems motivated by phase transformations in materials science. Much of this literature considers fronts propagating in homogeneous media. However, usual materials are heterogeneous due to the presence of defects, grains and precipitates. This thesis addresses the propagation of phase boundaries in heterogeneous media.</p>
<p>A particular motivation is a material undergoing martensitic phase transformation. Given a martensitic material with many non-transforming inclusions, there are well established microscopic laws that give the complex evolution of a particular twin or phase boundary as it encounters the many inclusions. The issue of interest is the overall evolution of this interface and the effect of defects and impurities on this evolution. In particular, if the defects are small, it is desirable to find the effective macroscopic law that governs the overall motion, without having to follow all the microscopic details but implicitly taking them into account. Using a theory of phase transformations based on linear elasticity, we show that the normal velocity of the martensitic phase or twin boundary may be written as a sum of several terms: first a homogeneous (but non-local) term that one would obtain for the propagation of the boundary in a homogeneous medium, second a heterogeneous term describing the effects of the inclusions but completely independent of the phase or twin boundary and third an interfacial energy term proportional to the mean curvature of the boundary.</p>
<p>As a guide to understanding this problem, we begin with two simplified settings which are also of independent interest. First, we consider the homogenization for the case when the normal velocity depends only on position (the heterogeneous term only). This is equivalent to the homogenization of a Hamilton-Jacobi equation. We establish several variational principles which give useful formulas to characterize the effective Hamiltonian. We illustrate the usefulness of these results through examples and we also provide a qualitative study of the effective normal velocity.</p>
<p>Second, we address the case when the interfacial energy is not negligible, so we keep the heterogeneous and curvature terms. This leads to a problem of homogenization of a degenerate parabolic initial value problem. We prove a homogenization theorem and obtain a characterization for the effective normal velocity, which however proves not to be too useful a tool for actual calculations. We therefore study some interesting examples and limiting cases and provide explicit formula in these situations. We also provide some numerical examples.</p>
<p>We finally address the problem in full generality in the setting of anti-plane shear. We explicitly evaluate the term induced by the presence of the inclusions and we propose a numerical method that allows us to trace the evolution of the phase boundary. We use this numerical method to evaluate the effect of the inclusions and show that their effect is quite localized. We use it to explain some experimental observations in NiTi.</p>https://thesis.library.caltech.edu/id/eprint/6122A Front Tracking Method for Modelling Thermal Growth
https://resolver.caltech.edu/CaltechETD:etd-03042003-115138
Authors: {'items': [{'email': 'beth@jetcafe.org', 'id': 'Howard-Elizabeth-Anne', 'name': {'family': 'Howard', 'given': 'Elizabeth Anne'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/HRNV-DA03
Several important thermal growth problems involve a solid growing into an undercooled liquid. The heat that is released at the interface diffuses into both the solid and the liquid phases. This is a free boundary problem where the position of the interface is an unknown which must be found as part of the solution. The problem can conveniently be represented as an integral equation for the unknown interface. However, a history integral must be evaluated at each time step which requires information about the boundary position at all previous times. The time and memory required to perform this calculation quickly becomes unreasonable. We develop an alternative way to deal with the problems that the history integral presents. By taking advantage of properties of the diffusion equation, we can use a method with a constant operation count and amount of memory required for each time step. We show that a numerical algorithm can be implemented for a two-dimensional, symmetric problem with equal physical parameters in both phases. The results agree well with the exact solution for the expanding circle case and microscopic solvability theory. We also extend the method to the nonsymmetric case. Additionally, a stability analysis is done of a simple, parabolic moving front to perturbations on the surface. As the eigenvalues of our problem increase, the interface becomes more increasingly oscillatory. https://thesis.library.caltech.edu/id/eprint/860Efficient Algorithms for Solving Static Hamilton-Jacobi Equations
https://resolver.caltech.edu/CaltechETD:etd-05202003-170423
Authors: {'items': [{'id': 'Mauch-Sean-Patrick', 'name': {'family': 'Mauch', 'given': 'Sean Patrick'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/5R5P-Y603
<p>We present an algorithm for computing the closest point transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for the points in a grid. We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm solves the eikonal equation |grad u| = 1 with the method of characteristics. For many problems, the computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold.</p>
<p>Many query problems can be aided by using orthogonal range queries (ORQ). There are several standard data structures for performing ORQ's in 3-D, including kd-trees, octrees, and cell arrays. We develop additional data structures based on cell arrays. We study the characteristics of each data structure and compare their performance.</p>
<p>We present a new algorithm for solving the single-source, non-negative weight, shortest-paths problem. Dijkstra's algorithm solves this problem with computational complexity O((E + V) log V) where E is the number of edges and V is the number of vertices. The new algorithm, called Marching with a Correctness Criterion (MCC), has computational complexity O(E + R V), where R is the ratio of the largest to smallest edge weight.</p>
<p>Sethian's Fast Marching Method (FMM) may be used to solve static Hamilton-Jacobi equations. It has computational complexity O(N log N), where N is the number of grid points. The FMM has been regarded as an optimal algorithm because it is closely related to Dijkstra's algorithm. The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations. This algorithm requires difference schemes that difference not only in coordinate directions, but in diagonal directions as well. It has computational complexity O(R N), where R is the ratio of the largest to smallest propagation speed and N is the number of grid points.</p>https://thesis.library.caltech.edu/id/eprint/1888Variational Methods for Nonsmooth Mechanics
https://resolver.caltech.edu/CaltechETD:etd-05222003-110241
Authors: {'items': [{'email': 'van@math.sfu.ca', 'id': 'Fetecau-Razvan-Constantin', 'name': {'family': 'Fetecau', 'given': 'Razvan Constantin'}, 'orcid': '0000-0001-9059-0283', 'show_email': 'YES'}]}
Year: 2003
DOI: 10.7907/VXBJ-R447
<p>In this thesis we investigate nonsmooth classical and continuum mechanics and its discretizations by means of variational numerical and geometric methods.</p>
<p>The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.</p>
<p>Next, we develop the foundations of a multisymplectic treatment of nonsmooth classical and continuum mechanics. This work may be regarded as a PDE generalization of the previous formulation of a variational approach to collision problems. The multisymplectic formulation includes a wide collection of nonsmooth dynamical models such as rigid-body collisions, material interfaces, elastic collisions, fluid-solid interactions and lays the groundwork for a treatment of shocks.</p>
<p>Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically and the longtime stable energy behavior typical of variational methods is demonstrated.</p>https://thesis.library.caltech.edu/id/eprint/1934Energy-Minimizing Microstructures in Multiphase Elastic Solids
https://resolver.caltech.edu/CaltechETD:etd-05252004-131315
Authors: {'items': [{'email': 'Isaac.Chenchiah : bristol.ac.uk', 'id': 'Chenchiah-Isaac-Vikram', 'name': {'family': 'Chenchiah', 'given': 'Isaac Vikram'}, 'orcid': '0000-0002-8618-620X', 'show_email': 'YES'}]}
Year: 2004
DOI: 10.7907/RXE5-9A33
<p>This thesis concerns problems of microstructure and its macroscopic consequences in multiphase elastic solids, both single crystals and polycrystals.</p>
<p>The elastic energy of a two-phase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated extremal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution (precipitation, coarsening, etc.), homogenization of composites and optimal design. Mathematically, the problem is to determine the relaxation under fixed volume fraction of a two-well energy.</p>
<p>We compute the relaxation under fixed volume fraction for a two-well linearized elastic energy in two dimensions with no restrictions on the elastic moduli and transformation strains; and show that there always exist rank-I or rank-II laminates that are extremal. By minimizing over the volume fraction we obtain the quasiconvex envelope of the energy. We relate these results to experimental observations on the equilibrium morphology and behavior under external loads of precipitates in Nickel superalloys. We also compute the relaxation under fixed volume fraction for a two-well linearized elastic energy in three dimensions when the elastic moduli are isotropic (with no restrictions on the transformation strains) and show that there always exist rank-I, rank-II or rank-III laminates that are extremal.</p>
<p>Shape memory effect is the ability of a solid to recover on heating apparently plastic deformation sustained below a critical temperature. Since utility of shape memory alloys critically depends on their polycrystalline behavior, understanding and predicting the recoverable strains of shape memory polycrystals is a central open problem in the study of shape memory alloys. Our contributions to the solution of this problem are twofold:</p>
<p>We prove a dual variational characterization of the recoverable strains of shape memory polycrystals and show that dual (stress) fields could be signed Radon measures with finite mass supported on sets with Lebesgue measure zero. We also show that for polycrystals made of materials undergoing cubic-tetragonal transformations the strains fields associated with macroscopic recoverable strains are related to the solutions of hyperbolic partial differential equations.</p>https://thesis.library.caltech.edu/id/eprint/2044Robust Flow Stability: Theory, Computations and Experiments in Near Wall Turbulence
https://resolver.caltech.edu/CaltechETD:etd-05282004-143324
Authors: {'items': [{'id': 'Bobba-Kumar-Manoj', 'name': {'family': 'Bobba', 'given': 'Kumar Manoj'}, 'show_email': 'NO'}]}
Year: 2004
DOI: 10.7907/0J1D-1B18
Helmholtz established the field of hydrodynamic stability with his pioneering work in 1868. From then on, hydrodynamic stability became an important tool in understanding various fundamental fluid flow phenomena in engineering (mechanical, aeronautics, chemical, materials, civil, etc.) and science (astrophysics, geophysics, biophysics, etc.), and turbulence in particular. However, there are many discrepancies between classical hydrodynamic stability theory and experiments. In this thesis, the limitations of traditional hydrodynamic stability theory are shown and a framework for robust flow stability theory is formulated. A host of new techniques like gramians, singular values, operator norms, etc. are introduced to understand the role of various kinds of uncertainty. An interesting feature of this framework is the close interplay between theory and computations. It is shown that a subset of Navier-Stokes equations are globally, non-nonlinearly stable for all Reynolds number. Yet, invoking this new theory, it is shown that these equations produce structures (vortices and streaks) as seen in the experiments. The experiments are done in zero pressure gradient transiting boundary layer on a flat plate in free surface tunnel. Digital particle image velocimetry, and MEMS based laser Doppler velocimeter and shear stress sensors have been used to make quantitative measurements of the flow. Various theoretical and computational predictions are in excellent agreement with the experimental data. A closely related topic of modeling, simulation and complexity reduction of large mechanics problems with multiple spatial and temporal scales is also studied. A nice method that rigorously quantifies the important scales and automatically gives models of the problem to various levels of accuracy is introduced. Computations done using spectral methods are presented.
https://thesis.library.caltech.edu/id/eprint/2192Foundations of Computational Geometric Mechanics
https://resolver.caltech.edu/CaltechETD:etd-03022004-000251
Authors: {'items': [{'email': 'mleok@math.ucsd.edu', 'id': 'Leok-Melvin', 'name': {'family': 'Leok', 'given': 'Melvin'}, 'orcid': '0000-0002-8326-0830', 'show_email': 'YES'}]}
Year: 2004
DOI: 10.7907/KDV0-WR34
<p>Geometric mechanics involves the study of Lagrangian and Hamiltonian mechanics using geometric and symmetry techniques. Computational algorithms obtained from a discrete Hamilton's principle yield a discrete analogue of Lagrangian mechanics, and they exhibit excellent structure-preserving properties that can be ascribed to their variational derivation.</p>
<p>We construct discrete analogues of the geometric and symmetry methods underlying geometric mechanics to enable the systematic development of computational geometric mechanics. In particular, we develop discrete theories of reduction by symmetry, exterior calculus, connections on principal bundles, as well as generalizations of variational integrators.</p>
<p>Discrete Routh reduction is developed for abelian symmetries, and extended to systems with constraints and forcing. Variational Runge-Kutta discretizations are considered in detail, including the extent to which symmetry reduction and discretization commute. In addition, we obtain the Reduced Symplectic Runge-Kutta algorithm, which is a discrete analogue of cotangent bundle reduction.</p>
<p>Discrete exterior calculus is modeled on a primal simplicial complex, and a dual circumcentric cell complex. Discrete notions of differential forms, exterior derivatives, Hodge stars, codifferentials, sharps, flats, wedge products, contraction, Lie derivative, and the Poincar?emma are introduced, and their discrete properties are analyzed. In examples such as harmonic maps and electromagnetism, discretizations arising from discrete exterior calculus commute with taking variations in Hamilton's principle, which implies that directly discretizing these equations yield numerical schemes that have the structure-preserving properties associated with variational schemes.</p>
<p>Discrete connections on principal bundles are obtained by introducing the discrete Atiyah sequence, and considering splittings of the sequence. Equivalent representations of a discrete connection are considered, and an extension of the pair groupoid composition that takes into account the principal bundle structure is introduced. Discrete connections provide an intrinsic coordinatization of the reduced discrete space, and the necessary discrete geometry to develop more general discrete symmetry reduction techniques.</p>
<p>Generalized Galerkin variational integrators are obtained by discretizing the action integral through appropriate choices of finite-dimensional function space and numerical quadrature. Explicit expressions for Lie group, higher-order Poincaré, higher-order symplectic-energy-momentum, and pseudospectral variational integrators are presented, and extensions such as spatio-temporally adaptive and multiscale variational integrators are briefly described.</p>https://thesis.library.caltech.edu/id/eprint/831Localized Non-bBlowup Conditions for 3D Incompressible Euler Flows and Related Equations
https://resolver.caltech.edu/CaltechETD:etd-05302005-161405
Authors: {'items': [{'email': 'xinwei.yu@gmail.com', 'id': 'Yu-Xinwei', 'name': {'family': 'Yu', 'given': 'Xinwei'}, 'show_email': 'YES'}]}
Year: 2005
DOI: 10.7907/K5E1-W938
<p>In this thesis, new results excluding finite time singularities with localized assumptions/conditions are obtained for the 3D incompressible Euler equations.</p>
<p>The 3D incompressible Euler equations are some of the most important nonlinear equations in mathematics. They govern the motion of ideal fluids. After hundreds of years of study, they are still far from being well-understood. In particular, a long-outstanding open problem asks whether finite time singularities would develop for smooth initial values. Much theoretical and numerical study on this problem has been carried out, but no conclusion can be drawn so far.</p>
<p>In recent years, several numerical experiments have been carried out by various authors, with results indicating possible breakdowns of smooth solutions in finite time. In these numerical experiments, certain properties of the velocity and vorticity field are observed in near-singular flows. These properties violate the assumptions of existing theoretical theorems which exclude finite time singularities. Thus there is a gap between current theoretical and numerical results. To narrow this gap is the main purpose of the work presented in this thesis.</p>
<p>In this thesis, a new framework of investigating flows carried by divergence-free velocity fields is developed. Using this new framework, new, localized sufficient conditions for the flow to remain smooth are obtained rigorously. These new results can deal with fast shrinking large vorticity regions and are applicable to recent numerical experiments. The application of the theorems in this thesis reveals new subtleties, and yields new understandings of the 3D incompressible Euler flow.</p>
<p>This new framework is then further applied to a two-dimensional model equation, the 2D quasi-geostrophic equation, for which global existence is still unproved. Under certain assumptions, we obtain new non-blowup results for the 2D quasi-geostrophic equation.</p>
<p>Finally, future plans of applying this new framework to some other PDEs as well as other possibilities of attacking the 3D Euler and 2D quasi-geostrophic singularity problems are discussed.</p>https://thesis.library.caltech.edu/id/eprint/2302Upscaling for Two-Phase Flows in Porous Media
https://resolver.caltech.edu/CaltechETD:etd-05252005-085744
Authors: {'items': [{'email': 'an_westhead@yahoo.com', 'id': 'Westhead-Andrew-Neil', 'name': {'family': 'Westhead', 'given': 'Andrew Neil'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/T7Q0-FG76
<p>The understanding and modeling of flow through porous media is an important issue in several branches of engineering. In petroleum engineering, for instance, one wishes to model the "enhanced oil recovery" process, whereby water or steam is injected into an oil saturated porous media in an attempt to displace the oil so that it can be collected. In groundwater contaminant studies the transport of dissolved material, such as toxic metals or radioactive waste, and how it affects drinking water supplies, is of interest.</p>
<p>Numerical simulation of these flow are generally difficult. The principal reason for this is the presence of many different length scales in the physical problem, and resolving all these is computationally expensive. To circumvent these difficulties a class of methods known as upscaling methods has been developed where one attempts to solve only for large scale features of interest and model the effect of the small scale features.</p>
<p>In this thesis, we review some of the previous efforts in upscaling and introduce a new scheme that attempts to overcome some of the existing shortcomings of these methods. In our analysis, we consider the flow problem in two distinct stages: the first is the determination of the velocity field which gives rise to an elliptic partial differential equation (PDE) and the second is a transport problem which gives rise to a hyperbolic PDE.</p>
<p>For the elliptic part, we make use of existing upscaling methods for elliptic equations. In particular, we use the multi-scale finite element method of Hou et al. to solve for the velocity field on a coarse grid, and yet still be able to obtain fine scale information through a special means of interpolation.</p>
<p>The analysis of the hyperbolic part forms the main contribution of this thesis. We first analyze the problem by restricting ourselves to the case where the small scales have a periodic structure. With this assumption, we are able to derive a coupled set of equations for the large scale average and the small scale fluctuations about this average. This is done by means of a special averaging, which is done along the fine scale streamlines. This coupled set of equations provides better starting point for both the modeling of the largescale small-scale interactions and the numerical implementation of any scheme. We derive an upscaling scheme from this by tracking only a sub-set of the fluctuations, which are used to approximate the scale interactions. Once this model has been derived, we discuss and present a means to extend it to the case where the fluctuations are more general than periodic.</p>
<p>In the sections that follow we provide the details of the numerical implementation, which is a very significant part of any practical method. Finally, we present numerical results using the new scheme and compare this with both resolved computations and some existing upscaling schemes.</p>https://thesis.library.caltech.edu/id/eprint/2050Mathematical Modeling and Simulation of Aquatic and Aerial Animal Locomotion
https://resolver.caltech.edu/CaltechETD:etd-05272005-004852
Authors: {'items': [{'email': 'svgaby@gmail.com', 'id': 'Stredie-Valentin-Gabriel', 'name': {'family': 'Stredie', 'given': 'Valentin Gabriel'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/8PWE-RK28
<p>In this thesis we investigate the locomotion of fish and birds by applying both new and well known mathematical techniques.</p>
<p>The two-dimensional model is first studied using Krasny's vortex blob method, and then a new numerical method based on Wu's theory is developed. To begin with, we will implement Krasny's ideas for a couple of examples and then switch to the numerical implementation of the nonlinear analytical mathematical model presented by Wu. We will demonstrate the superiority of this latter method both by applying it to some specific cases and by comparing with the experiments. The nonlinear effects are very well observed and this will be shown by analyzing Wagner's result for a wing abruptly undergoing an increase in incidence angle, and also by analyzing the vorticity generated by a wing in heaving, pitching and bending motion. The ultimate goal of the thesis is to accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet.</p>
<p>In the second part of this work we will introduce a three-dimensional method for a flat plate advancing perpendicular to the flow. The accuracy of the method will be shown both by comparing its results with the two-dimensional ones and by validating them versus the experimental results obtained by Ringuette in the towing tank of the Aeronautics Department at Caltech.</p>https://thesis.library.caltech.edu/id/eprint/2142Upscaling Immiscible Two-Phase Flows in an Adaptive Frame
https://resolver.caltech.edu/CaltechETD:etd-02192006-165348
Authors: {'items': [{'email': 'theofilos13@yahoo.com', 'id': 'Strinopoulos-Theofilos', 'name': {'family': 'Strinopoulos', 'given': 'Theofilos'}, 'show_email': 'YES'}]}
Year: 2006
DOI: 10.7907/AP0P-8S11
<p>We derive the two-scale limit of a linear or nonlinear saturation equation with a flow-based coordinate transformation. This transformation consists of the pressure and the streamfunction. In this framework the saturation equation is decoupled to a family of one-dimensional nonconservative transport equations along streamlines. This simplifies the derivation of the two-scale limit. Moreover it allows us to obtain the convergence independent of the assumptions of periodicity and scale separation. We provide a rigorous estimate on the convergence rate. We combine the two-scale limit with Tartar's method to complete the homogenization.</p>
<p>To design an efficient numerical method, we use an averaging approach across the streamlines on the two-scale limit equations. The resulting numerical method for the saturation has all the advantages in terms of adaptivity that methods have. We couple it with a moving mesh along the streamlines to resolve the shock more efficiently. We use the multiscale finite element method to upscale the pressure equation because it gives access to the fine scale velocity, which enters in the saturation equation, through the basis functions. We propose to solve the pressure equation in the coordinate frame of the initial pressure and saturation, which is similar to the modified multiscale finite element method.</p>
<p>We test our numerical method in realistic permeability fields, such as the Tenth SPE Comparative Solution Project permeabilities, for accuracy and computational cost.</p>https://thesis.library.caltech.edu/id/eprint/680Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations
https://resolver.caltech.edu/CaltechETD:etd-05182006-173710
Authors: {'items': [{'id': 'Luo-Wuan', 'name': {'family': 'Luo', 'given': 'Wuan'}, 'show_email': 'NO'}]}
Year: 2006
DOI: 10.7907/RPKX-BN02
<p>Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. In this thesis, we study a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing. WCE represents a stochastic solution as a spectral expansion with respect to a set of random basis. By deriving a governing equation for the expansion coefficients, we can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation. All the statistical information of the solution can be recovered from the deterministic coefficients using very simple formulae.</p>
<p>We apply the WCE-based method to solve stochastic Burgers equations, Navier-Stokes equations and nonlinear reaction-diffusion equations with either additive or multiplicative random forcing. Our numerical results demonstrate convincingly that the new method is much more efficient and accurate than MC simulations for solutions in short to moderate time. For a class of model equations, we prove the convergence rate of the WCE method. The analysis also reveals precisely how the convergence constants depend on the size of the time intervals and the variability of the random forcing. Based on the error analysis, we design a sparse truncation strategy for the Wiener chaos expansion. The sparse truncation can reduce the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rates.</p>
<p>For long time solutions, we propose a new computational strategy where MC simulations are used to correct the unresolved small scales in the sparse Wiener chaos solutions. Numerical experiments demonstrate that the WCE-MC hybrid method can handle SPDEs in much longer time intervals than the direct WCE method can. The new method is shown to be much more efficient than the WCE method or the MC simulation alone in relatively long time intervals. However, the limitation of this method is also pointed out.</p>
<p>Using the sparse WCE truncation, we can resolve the probability distributions of a stochastic Burgers equation numerically and provide direct evidence for the existence of a unique stationary measure. Using the WCE-MC hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows. Our numerical results confirm the conjecture by Jack Xin that the front propagation speed obeys a quadratic enhancing law.</p>
<p>Using the machinery we have developed for the Wiener chaos method, we resolve a few technical difficulties in solving stochastic elliptic equations by Karhunen-Loeve-based polynomial chaos method. We further derive an upscaling formulation for the elliptic system of the Wiener chaos coefficients. Eventually, we apply the upscaled Wiener chaos method for uncertainty quantification in subsurface modeling, combined with a two-stage Markov chain Monte Carlo sampling method we have developed recently.</p>https://thesis.library.caltech.edu/id/eprint/1861Curvelets, Wave Atoms, and Wave Equations
https://resolver.caltech.edu/CaltechETD:etd-05262006-133555
Authors: {'items': [{'email': 'demanet@gmail.com', 'id': 'Demanet-Laurent', 'name': {'family': 'Demanet', 'given': 'Laurent'}, 'orcid': '0000-0001-7052-5097', 'show_email': 'NO'}]}
Year: 2006
DOI: 10.7907/1TEF-RQ51
<p>We argue that two specific wave packet families---curvelets and wave atoms---provide powerful tools for representing linear systems of hyperbolic differential equations with smooth, time-independent coefficients. In both cases, we prove that the matrix representation of the Green's function is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals, whose location is predicted by geometrical optics.</p>
<p>This result holds only when the basis elements obey a precise parabolic balance between oscillations and support size, shared by curvelets and wave atoms but not wavelets, Gabor atoms, or any other such transform.</p>
<p>A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.</p>
<p>We also provide fast digital implementations of tight frames of curvelets and wave atoms in two dimensions. In both cases the complexity is O(N² log N) flops for N-by-N Cartesian arrays, for forward as well as inverse transforms.</p>
<p>Finally, we present a geometric strategy based on wave atoms for the numerical solution of wave equations in smoothly varying, 2D time-independent periodic media. Our algorithm is based on sparsity of the matrix representation of Green's function, as above, and also exploits its low-rank block structure after separation of the spatial indices. As a result, it becomes realistic to accurately build the full matrix exponential using repeated squaring, up to some time which is much larger than the CFL timestep. Once available, the wave atom representation of the Green's function can be used to perform 'upscaled' timestepping.</p>
<p>We show numerical examples and prove complexity results based on a priori estimates of sparsity and separation ranks. They beat the O(N^3) bottleneck on an N-by-N grid, for a wide range of physically relevant situations. In practice, the current wave atom solver can become competitive over a pseudospectral method in the regime when the wave equation should be solved several times with different initial conditions, as in reflection seismology.</p>https://thesis.library.caltech.edu/id/eprint/2112Structure and Evolution of Martensitic Phase Boundaries
https://resolver.caltech.edu/CaltechETD:etd-05292007-211950
Authors: {'items': [{'email': 'patrick.dondl@mathematik.uni-freiburg.de', 'id': 'Dondl-Patrick-Werner', 'name': {'family': 'Dondl', 'given': 'Patrick Werner'}, 'orcid': '0000-0003-3035-7230', 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/89AW-3S87
<p>This work examines two major aspects of martensitic phase boundaries. The first part studies numerically the deformation of thin films of shape memory alloys by using subdivision surfaces for discretization. These films have gained interest for their possible use as actuators in microscale electro-mechanical systems, specifically in a pyramid-shaped configuration. The study of such configurations requires adequate resolution of the regions of high strain gradient that emerge from the interplay of the multi-well strain energy and the penalization of the strain gradient through a surface energy term. This surface energy term also requires the spatial numerical discretization to be of higher regularity, i.e., it needs to be continuously differentiable. This excludes the use of a piecewise linear approximation. It is shown in this thesis that subdivision surfaces provide an attractive tool for the numerical examination of thin phase transforming structures. We also provide insight in the properties of such tent-like structures.</p>
<p>The second part of this thesis examines the question of how the rate-independent hysteresis that is observed in martensitic phase transformations can be reconciled with the linear kinetic relation linking the evolution of domains with the thermodynamic driving force on a microscopic scale. A sharp interface model for the evolution of martensitic phase boundaries, including full elasticity, is proposed. The existence of a solution for this coupled problem of a free discontinuity evolution to an elliptic equation is proved. Numerical studies using this model show the pinning of a phase boundary by precipitates of non-transforming material. This pinning is the first step in a stick-slip behavior and therefore a rate-independent hysteresis.</p>
<p>In an approximate model, the existence of a critical pinning force as well as the existence of solutions traveling with an average velocity are proved rigorously. For this shallow phase boundary approximation, the depinning behavior is studied numerically. We find a universal power-law linking the driving force to the average velocity of the interface. For a smooth local force due to an inhomogeneous but periodic environment we find a critical exponent of 1/2.</p>
https://thesis.library.caltech.edu/id/eprint/2251Hamilton-Pontryagin Integrators on Lie Groups
https://resolver.caltech.edu/CaltechETD:etd-06052007-153115
Authors: {'items': [{'email': 'nawaf.bourabee@rutgers.edu', 'id': 'Bou-Rabee-Nawaf-Mohammed', 'name': {'family': 'Bou-Rabee', 'given': 'Nawaf Mohammed'}, 'orcid': '0000-0001-9280-9808', 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/0EC4-2042
<p>In this thesis structure-preserving time integrators for mechanical systems whose configuration space is a Lie group are derived from a Hamilton-Pontryagin (HP) variational principle. In addition to its attractive properties for degenerate mechanical systems, the HP viewpoint also affords a practical way to design discrete Lagrangians, which are the cornerstone of variational integration theory. The HP principle states that a mechanical system traverses a path that extremizes an HP action integral. The integrand of the HP action integral consists of two terms: the Lagrangian and a kinematic constraint paired with a Lagrange multiplier (the momentum). The kinematic constraint relates the velocity of the mechanical system to a curve on the tangent bundle. This form of the action integral makes it amenable to discretization.</p>
<p>In particular, our strategy is to implement an s-stage Runge-Kutta-Munthe-Kaas (RKMK) discretization of the kinematic constraint. We are motivated by the fact that the theory, order conditions, and implementation of such methods are mature. In analogy with the continuous system, the discrete HP action sum consists of two parts: a weighted sum of the Lagrangian using the weights from the Butcher tableau of the RKMK scheme, and a pairing between a discrete Lagrange multiplier (the discrete momentum) and the discretized kinematic constraint. In the vector space context, it is shown that this strategy yields a well-known class of symplectic partitioned Runge-Kutta methods including the Lobatto IIIA-IIIB pair which generalize to higher-order accuracy.</p>
<p>In the Lie group context, the strategy yields an interesting and novel family of variational partitioned Runge-Kutta methods. Specifically, for mechanical systems on Lie groups we analyze the ideal context of EP systems. For such systems the HP principle can be transformed from the Pontryagin bundle to a reduced space. To set up the discrete theory, a continuous reduced HP principle is also analyzed. It is this reduced HP principle that we apply our discretization strategy to. The resulting integrator describes an update scheme on the reduced space. As in RKMK we parametrize the Lie group using coordinate charts whose model space is the Lie algebra and that approximate the exponential map. Since the Lie group is non abelian, the structure of these integrators is not the same as in the vector space context.</p>
<p>We carry out an in-depth study of the simplest integrators within this family that we call variational Euler integrators; specifically we analyze the integrator's efficiency, global error, and geometric properties. Because of their variational character, the variational Euler integrators preserve a discrete momentum map and symplectic form. Moreover, since the update on the configuration space is explicit, the configuration updates exhibit no drift from the Lie group. We also prove that the global error of these methods is second order. Numerical experiments on the free rigid body and the chaotic dynamics of an underwater vehicle reveal that these reduced variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.</p>
<p>In addition we discuss how the HP integrators extend to a wider class of mechanical systems with, e.g., configuration dependent potentials and non trivial shape-space dynamics.</p>https://thesis.library.caltech.edu/id/eprint/2465Simulations and Analysis of Two- and Three-Dimensional Single-Mode Richtmyer-Meshkov Instability using Weighted Essentially Non-Oscillatory and Vortex Methods
https://resolver.caltech.edu/CaltechETD:etd-12082006-124547
Authors: {'items': [{'email': 'mlatini@acm.caltech.edu', 'id': 'Latini-Marco', 'name': {'family': 'Latini', 'given': 'Marco'}, 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/1397-GZ04
<p>An incompressible vorticity-streamfunction (VS) method is developed to investigate the single-mode Richtmyer-Meshkov instability in two and three dimensions. The initial vortex sheet (representing the initial shocked interface) is thickened to regularize the limit of classical Lagrangian vortex methods. In the limit of smaller thickness, the initial velocity converges to the velocity of a vortex sheet. The vorticity on the Cartesian grid follows the vorticity evolution equation augmented by the baroclinic vorticity production term (to capture the effects of the instability on the layer) and a viscous dissipation term. The equations are discretized using a fourth-order in space and third-order in time semi-implicit Adams-Bashforth backward differentiation scheme. The convergence properties of the method with respect to varying the diffuse interface thickness and viscosity are investigated. It is shown that the small-scale structures within the roll-up are more sensitive to the diffuse interface thickness than to the viscosity. By contrast, the large-scale quantities, including the perturbation, bubble, and spike amplitudes are less sensitive. Fourth-order point-wise convergence is achieved, provided that a sufficiently fine grid is used.</p>
<p>In two dimensions, the VS method is applied to investigate late-time nonlinear effects of the single-mode Mach 1.3 air(acetone)/SF_6 shock tube experiment of Jacobs and Krivets. The results are also compared to those from compressible ninth-order weighted essentially non-oscillatory (WENO) simulations. The density fields from the WENO and VS methods agree with the experimental PLIF images in the large-scale structures but differ in the small-scale structures. The WENO method exhibits small-scale disordered structure similar to that in the experiment, while the VS method does not capture such structure, but shows a strong rotating core. The perturbation amplitudes from the two methods are in good agreement and match the experimental data points well. The WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. Comparing amplitudes from simulations with varying Mach number shows that as the Mach number increases, the differences in the bubble and spike amplitudes increase due to intensifying pressure perturbations not present in the incompressible VS method. The perturbation amplitude from the WENO and VS methods is also compared to the predictions of nonlinear amplitude growth models in which the growth rate was reduced to account for the diffuse initial interface. In general, the model predictions agree with the simulation amplitudes at early-to-intermediate times and underpredict at later times, corresponding to the late nonlinear regime.</p>
<p>The WENO simulation is used to investigate reshock, which occurs when the transmitted shock reflects from the end wall of the test section and interacts with the evolving layer. The post-reshock mixing layer width agrees well with the predictions of reshock models for short times until the interaction of the reflected rarefaction with the layer.</p>
<p>The VS simulation was also compared to classical Lagrangian and vortex-in-cell simulations as the Atwood number was varied. For low Atwood numbers, all three simulations agree. As the Atwood number increases, the VS simulation shows differences in the bubble and spike amplitudes compared to the Lagrangian and VIC simulations, as the baroclinic vorticity production for a diffuse layer is different from that of a thin layer. The simulation amplitudes agree with the predictions of nonlinear amplitude growth models at early times. The growth models underpredict the amplitudes at later times.</p>
<p>The investigation is extended to three dimensions, where the initial perturbation is a product of sinusoids and the initial vorticity deposition is given by linear instability analysis. The instability evolution and dynamics of vorticity are visualized using the mass fraction and enstrophy isosurface, respectively. For the WENO and VS methods, two roll-ups corresponding to the bubble and spike regions form, and the vorticity shows the formation of a ring-like structure. The perturbation amplitudes from the WENO and VS methods are in excellent agreement. The bubble and spike amplitude are in good agreement at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike. The nonlinear three-dimensional Zhang-Sohn model agrees with the simulation amplitudes at early times, and underpredicts later. In three dimensions, the enstrophy iso-surface after reshock shows significant fragmentation and the formation of small, short, tubular structures. Simulations with different initial amplitudes show that the mixing layer width after reshock does not depend on the pre-shock amplitude. Finally, the effects of Atwood number are investigated using the VS method and the amplitudes are compared to the predictions of the Zhang-Sohn model. The simulation and the models are in agreement at early times, while the models underpredict later.</p>
<p>The VS method constitutes a useful numerical approach to investigate the Richtmyer-Meshkov instability in two and three dimensions. The VS method and, more generally, vortex methods are valid tools for predicting the large-scale instability features, including the perturbation amplitudes, into the late nonlinear regime.</p>https://thesis.library.caltech.edu/id/eprint/4868Metric Based Upscaling for Partial Differential Equations with a Continuum of Scales
https://resolver.caltech.edu/CaltechETD:etd-05162007-172755
Authors: {'items': [{'email': 'mail4lei@gmail.com', 'id': 'Zhang-Lei', 'name': {'family': 'Zhang', 'given': 'Lei'}, 'orcid': '0000-0002-2917-9652', 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/AZ06-4B54
<p>Numerical upscaling of problems with multiple scale structures have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation. Most existing methods are either based on the assumption of scale separation or heuristic arguments.</p>
<p>In this thesis, we present rigorous results on homogenization of partial differential equations with L<sup>∞</sup> coefficients which allow for a continuum of spatial and temporal scales. We propose a new type of compensation phenomena for elliptic, parabolic, and hyperbolic equations. The main idea is the use of the so-called "harmonic coordinates" ("caloric coordinates" in the parabolic case). Under these coordinates, the solutions of these differential equations have one more degree of differentiability. It has been deduced from this compensation phenomenon that numerical homogenization methods formulated as oscillating finite elements can converge in the presence of a continuum of scales, if one uses global caloric coordinates to obtain the test functions instead of using solutions of a local cell problem.</p>https://thesis.library.caltech.edu/id/eprint/1841Nonreflecting Boundary Conditions Obtained from Equivalent Sources for Time-Dependent Scattering Problems
https://resolver.caltech.edu/CaltechETD:etd-05202008-111349
Authors: {'items': [{'email': 'dhoch@acm.caltech.edu', 'id': 'Hoch-David', 'name': {'family': 'Hoch', 'given': 'David'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/5M0P-NR33
<p>In many engineering applications, scattering of acoustic or electromagnetic waves from a body of arbitrary shape is considered in an infinite medium. Solving the underlying partial differential equations with a standard numerical method such as finite elements or finite differences requires truncating the unbounded domain of definition into a finite computational region. As a consequence, an appropriate boundary condition must be prescribed at the artificial boundary. Many approaches have been proposed for this fundamental problem in the field of wave scattering. All of them fall into one of three main categories.</p>
<p>The first class of methods is based on mathematical approximations or physical heuristics. These boundary conditions are easy to implement and run in short computing times. However, these approaches give rise to spurious reflections at the artificial boundary, which travel back into the computational domain and corrupt the solution.</p>
<p>A second group consists of accurate and convergent methods. However, these formulations are usually harder to implement and often more expensive than the computation of the interior scheme itself.</p>
<p>Finally, there are methods which are accurate and fast. The drawback of these approaches lies in the fact that the outer boundary must be taken to be either a sphere, a plane, or a cylinder. For many applications of interest, this may require use of a computational domain much larger than actually needed, which leads to an expensive overall numerical scheme.</p>
<p>This work introduces a new methodology in order to compute the fields at the artificial boundary. Like the second class of methods described above, the proposed algorithm is accurate and numerically convergent, yet its computational cost is less than the underlying portion of the volumetric calculation. And, unlike the third category, this new approach allows us to choose the artificial boundary to be arbitrarily close to the scatterer. This method is based on a novel concept of "equivalent source' representations which allows a highly accurate and fast evaluation of the boundary condition.</p>https://thesis.library.caltech.edu/id/eprint/1899Effective Behavior of Dielectric Elastomer Composites
https://resolver.caltech.edu/CaltechETD:etd-08272007-145455
Authors: {'items': [{'email': 'lixiutian@gmail.com', 'id': 'Tian-Lixiu', 'name': {'family': 'Tian', 'given': 'Lixiu'}, 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/CZNF-JB47
<p>The class of electroactive polymers has been developed to a point where real life applications as ``artificial muscles" are conceivable. These actuator materials provide attractive advantages: they are soft, lightweight, can undergo large deformation, possess fast response time and are resilient. However, widespread application has been hindered by their limitations: the need for a large electric field, relatively small forces and energy density. However, recent experimental work shows great promise that this limitation can be overcome by making composites of two materials with high contrast in their dielectric modulus. In this thesis, a theoretical framework is derived to describe the electrostatic effect of the dielectric elastomers. Numerical experiments are conducted to explain the reason for the promising experimental results and to explore better microstructures of the composites to enhance the favorable properties.</p>
<p>The starting point of this thesis is a general variational principle, which characterizes the behavior of solids under combined mechanical and electrical loads. Based on this variational principle, we assume the electric field is small as of order ε½, assume further the deformation is caused by the electrostatic effects; the deformation field is then of order ε. Using the tool of Γ-convergence, we derive a small-strain model in which the electric field and the deformation field are decoupled which results in a huge simplification of the problem.</p>
<p>Based on this small-strain model, employing the powerful tool of two-scale convergence, we derive the effective properties for dielectric composites conducting small strains. A formula of the effective electromechanical coupling coefficients is given in terms of the unit cell solutions.</p>
<p>Armed with these theoretical results, we carry out numerical experiments about the effective properties of different kind of composites. A very careful analysis of the numerical results provides a deep understanding of the mechanism of the enhancement in strain by making composites of different microstructures.</p>https://thesis.library.caltech.edu/id/eprint/3248Discrete Geometric Homogenisation and Inverse Homogenisation of an Elliptic Operator
https://resolver.caltech.edu/CaltechETD:etd-05212008-164705
Authors: {'items': [{'email': 'rdonald@acm.caltech.edu', 'id': 'Donaldson-Roger-David', 'name': {'family': 'Donaldson', 'given': 'Roger David'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/S4S7-8T31
We show how to parameterise a homogenised conductivity in R² by a scalar function s(x), despite the fact that the conductivity parameter in the related up-scaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of s(x), and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated domains. We apply the parameterisation in three contexts: (i) sampling s(x) produces a family of stiffness matrices representing the elliptic operator over a hierarchy of scales; (ii) the curvature of s(x) directs the construction of meshes well-adapted to the anisotropy of the operator, improving the conditioning of the stiffness matrix and interpolation properties of the mesh; and (iii) using electric impedance tomography to reconstruct s(x) recovers the up-scaled conductivity, which while anisotropic, is unique. Extensions of the parameterisation to R³ are introduced.https://thesis.library.caltech.edu/id/eprint/1928Geometric Discretization of Lagrangian Mechanics and Field Theories
https://resolver.caltech.edu/CaltechETD:etd-12312008-173851
Authors: {'items': [{'email': 'astern@acm.caltech.edu', 'id': 'Stern-Ari-Joshua', 'name': {'family': 'Stern', 'given': 'Ari Joshua'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/K943-VJ44
This thesis presents a unified framework for geometric discretization of highly oscillatory mechanics and classical field theories, based on Lagrangian variational principles and discrete differential forms. For highly oscillatory problems in mechanics, we present a variational approach to two families of geometric numerical integrators: implicit-explicit (IMEX) and trigonometric methods. Next, we show how discrete differential forms in spacetime can be used to derive a structure-preserving discretization of Maxwell's equations, with applications to computational electromagnetics. Finally, we sketch out some future directions in discrete gauge theory, providing foundations based on fiber bundles and Lie groupoids, as well as discussing applications to discrete Riemannian geometry and numerical general relativity.
https://thesis.library.caltech.edu/id/eprint/5173High-Order Solution of Elliptic Partial Differential Equations in Domains Containing Conical Singularities
https://resolver.caltech.edu/CaltechETD:etd-08042008-005339
Authors: {'items': [{'email': 'zhiyi@acm.caltech.edu', 'id': 'Li-Zhiyi', 'name': {'family': 'Li', 'given': 'Zhiyi'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/VEEB-AV75
In this thesis we introduce an algorithm, based on the boundary integral equation method, for the numerical evaluation of singular solutions of the Laplace equation in three dimensional space, with singularities induced by a conical point on an otherwise smooth boundary surface. This is a model version of a fundamental problem in science and engineering: accurate evaluation of solutions of Partial Differential Equations in domains whose boundaries contain geometric singularities. For simplicity we assume a small region near the conical point coincides with a straight cone of given cross section; otherwise the boundary surface is not restricted in any way. Our numerical results demonstrate excellent convergence as discretizations are refined, even at the singular point where the solutions tend to infinity.https://thesis.library.caltech.edu/id/eprint/3013Multiscale Methods for Elliptic Partial Differential Equations and Related Applications
https://resolver.caltech.edu/CaltechETD:etd-07312009-095021
Authors: {'items': [{'email': 'ccjaychu@gmail.com', 'id': 'Chu-Chia-Chieh', 'name': {'family': 'Chu', 'given': 'Chia-Chieh'}, 'show_email': 'YES'}]}
Year: 2010
DOI: 10.7907/PFGA-YY17
<p>Multiscale problems arise in many scientific and engineering disciplines. A typical example is the modelling of flow in a porous medium containing a number of low and high permeability embedded in a matrix. Due to the high degrees of variability and the multiscale nature of formation properties, not only is a complete analysis of these problems extremely difficult, but also numerical solvers require an excessive amount of CPU time and storage. In this thesis, we study multiscale numerical methods for the elliptic equations arising in interface and two-phase flow problems. The model problems we consider are motivated by the multiscale computations of flow and transport of two-phase flow in strongly heterogeneous porous media. Although the analysis is carried out for simplified model problems, it does provide valuable insight in designing accurate multiscale methods for more realistic applications.</p>
<p>In the first part, we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. The method is H¹-conforming, and has an optimal convergence rate of O(h) in the energy norm and O(h²) in the L₂ norm, where h is the mesh diameter and the hidden constants in these estimates are independent of the "contrast" (i.e. ratio of largest to smallest value) of the PDE's coefficients. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. We conduct some numerical experiments to confirm the optimal rate of convergence of the proposed method and its independence from the aspect ratio of the coefficients.</p>
<p>In the second part, we propose a flow-based oversampling method where the actual two-phase flow boundary conditions are used to construct oversampling auxiliary functions. Our numerical results show that the flow-based oversampling approach is several times more accurate than the standard oversampling method. A partial theoretical explanation is provided for these numerical observations.</p>
<p>In the third part, we discuss "metric-based upscaling" for the pressure equation in two-phase flow problem. We show a compensation phenomenon and design a multiscale method for the pressure equation with highly oscillatory permeability.</p>
https://thesis.library.caltech.edu/id/eprint/5275Lagrangian and Vortex-Surface Fields in Turbulence
https://resolver.caltech.edu/CaltechTHESIS:02212011-233246689
Authors: {'items': [{'email': 'yangyue00@gmail.com', 'id': 'Yang-Yue', 'name': {'family': 'Yang', 'given': 'Yue'}, 'show_email': 'NO'}]}
Year: 2011
DOI: 10.7907/DF3E-G629
<p>In this thesis, we focus on Lagrangian investigations of isotropic turbulence, wall-bounded turbulence and vortex dynamics. In particular, the evolutionary multi-scale geometry of Lagrangian structures is quantified and analyzed. Additionally, we also study the dynamics of vortex-surface fields for some simple viscous flows with both Taylor--Green and Kida--Pelz initial conditions.</p>
<p>First, we study the non-local geometry of finite-sized Lagrangian structures in both stationary, evolving homogenous isotropic turbulence and also with a frozen turbulent velocity field. The multi-scale geometric analysis is applied on the evolution of Lagrangian fields, obtained by a particle-backward-tracking method, to extract Lagrangian structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. Next, we report a geometric study of both evolving Lagrangian, and also instantaneous Eulerian structures in turbulent channel flow at low and moderate Reynolds numbers. A multi-scale and multi-directional analysis, based on the mirror-extended curvelet transform, is developed to quantify flow structure geometry including the averaged inclination and sweep angles of both classes of turbulent structures at multiple scales ranging from the half-height of the channel to several viscous length scales. Results for turbulent channel flow include the geometry of candidate quasi-streamwise vortices in the near-wall region, the structural evolution of near-wall vortices, and evidence for the existence and geometry of structure packets based on statistical inter-scale correlations.</p>
<p>In order to explore the connection and corresponding representations between Lagrangian kinematics and vortex dynamics, we develop a theoretical formulation and numerical methods for computation of the evolution of a vortex-surface field. Iso-surfaces of the vortex-surface field define vortex surfaces. A systematic methodology is developed for constructing smooth vortex-surface fields for initial Taylor--Green and Kida--Pelz velocity fields by using an optimization approach. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Numerical results on the evolution of vortex-surface fields clarify the continuous vortex dynamics in viscous Taylor--Green and Kida--Pelz flows including the vortex reconnection, rolling-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. This suggests a possible scenario for explaining the transition from a smooth laminar flow to turbulent flow in terms of topology and geometry of vortex surfaces.</p>
https://thesis.library.caltech.edu/id/eprint/6251Multiscale Geometric Integration of Deterministic and Stochastic Systems
https://resolver.caltech.edu/CaltechTHESIS:05262011-171044915
Authors: {'items': [{'email': 't.t.snail@gmail.com', 'id': 'Tao-Molei', 'name': {'family': 'Tao', 'given': 'Molei'}, 'show_email': 'NO'}]}
Year: 2011
DOI: 10.7907/6J83-7C18
<p>In order to accelerate computations and improve long time accuracy of numerical simulations, this thesis develops multiscale geometric integrators.</p>
<p>For general multiscale stiff ODEs, SDEs, and PDEs, FLow AVeraging integratORs (FLAVORs) have been proposed for the coarse time-stepping without any identification of the slow or the fast variables. In the special case of deterministic and stochastic mechanical systems, symplectic, multisymplectic, and quasi-symplectic multiscale integrators are easily obtained using this strategy.</p>
<p>For highly oscillatory mechanical systems (with quasi-quadratic stiff potentials and possibly high-dimensional), a specialized symplectic method has been devised to provide improved efficiency and accuracy. This method is based on the introduction of two highly nontrivial matrix exponentiation algorithms, which are generic, efficient, and symplectic (if the exact exponential is symplectic).</p>
<p>For multiscale systems with Dirac-distributed fast processes, a family of symplectic, linearly-implicit and stable integrators has been designed for coarse step simulations. An application is the fast and accurate integration of constrained dynamics.</p>
<p>In addition, if one cares about statistical properties of an ensemble of trajectories, but not the numerical accuracy of a single trajectory, we suggest tuning friction and annealing temperature in a Langevin process to accelerate its convergence.</p>
<p>Other works include variational integration of circuits, efficient simulation of a nonlinear wave, and finding optimal transition pathways in stochastic dynamical systems (with a demonstration of mass effects in molecular dynamics).</p>https://thesis.library.caltech.edu/id/eprint/6457Multiscale Modeling and Computation of 3D Incompressible Turbulent Flows
https://resolver.caltech.edu/CaltechTHESIS:05302012-081356007
Authors: {'items': [{'email': 'lanxin0106@gmail.com', 'id': 'Hu-Xin', 'name': {'family': 'Hu', 'given': 'Xin'}, 'show_email': 'YES'}]}
Year: 2012
DOI: 10.7907/K1RZ-1H07
<p>In the first part, we present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible turbulent flows. This derivation is based on a systematic multiscale analysis that expresses the Reynolds stress in terms of the solutions of local periodic cell problems. We reveal an asymptotic structure of the Reynolds stress by invoking the frame invariant property of the cell problems and an iterative dynamic homogenization of large- and small-scale solutions. The Smagorinsky model for homogeneous turbulence is recovered as an example to illustrate our mathematical derivation. Another example is turbulent channel flow, where we derive a simplified turbulence model based on the asymptotic flow structure near the wall. Additionally, we obtain a nonlinear model by using a second order approximation of the inverse flow map function. This nonlinear model captures the effects of the backscatter of kinetic energy and dispersion and is consistent with other models, such as a mixed model that combines the Smagorinsky and gradient models, and the generic nonlinear model of Lund and Novikov.</p>
<p>Numerical simulation results at two Reynolds numbers using our simplified turbulence model are in good agreement with both experiments and direct numerical simulations in turbulent channel flow. However, due to experimental and modeling errors, we do observe some noticeable differences, e.g. , root mean square velocity fluctuations at Re<sub>τ</sub> = 180.</p>
<p>In the second part, we present a new perspective on calculating fully developed turbulent flows using a data-driven stochastic method. General polynomial chaos (gPC) bases are obtained based on the mean velocity profile of turbulent channel flow in the offline part. The velocity fields are projected onto the subspace spanned by these gPC bases and a coupled system of equations is solved to compute the velocity components in the Karhunen-Loeve expansion in the online part. Our numerical results have shown that the data-driven stochastic method for fully developed turbulence offers decent approximations of statistical quantities with a coarse grid and a relatively small number of gPC base elements.</p>https://thesis.library.caltech.edu/id/eprint/7093A Systems Approach to Cardiovascular Health and Disease with a Focus on Aortic Wave Dynamics
https://resolver.caltech.edu/CaltechTHESIS:05082013-152249157
Authors: {'items': [{'email': 'niema.pahlevan@gmail.com', 'id': 'Pahlevan-Niema-Mohammed', 'name': {'family': 'Pahlevan', 'given': 'Niema Mohammed'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/Z9DR2SFM
<p>Cardiovascular diseases (CVDs) have reached an epidemic proportion in the US and worldwide with serious consequences in terms of human suffering and economic impact. More than one third of American adults are suffering from CVDs. The total direct and indirect costs of CVDs are more than $500 billion per year. Therefore, there is an urgent need to develop noninvasive diagnostics methods, to design minimally invasive assist devices, and to develop economical and easy-to-use monitoring systems for cardiovascular diseases. In order to achieve these goals, it is necessary to gain a better understanding of the subsystems that constitute the cardiovascular system. The aorta is one of these subsystems whose role in cardiovascular functioning has been underestimated. Traditionally, the aorta and its branches have been viewed as resistive conduits connected to an active pump (left ventricle of the heart). However, this perception fails to explain many observed physiological results. My goal in this thesis is to demonstrate the subtle but important role of the aorta as a system, with focus on the wave dynamics in the aorta.</p>
<p>The operation of a healthy heart is based on an optimized balance between its pumping characteristics and the hemodynamics of the aorta and vascular branches. The delicate balance between the aorta and heart can be impaired due to aging, smoking, or disease. The heart generates pulsatile flow that produces pressure and flow waves as it enters into the compliant aorta. These aortic waves propagate and reflect from reflection sites (bifurcations and tapering). They can act constructively and assist the blood circulation. However, they may act destructively, promoting diseases or initiating sudden cardiac death. These waves also carry information about the diseases of the heart, vascular disease, and coupling of heart and aorta. In order to elucidate the role of the aorta as a dynamic system, the interplay between the dominant wave dynamic parameters is investigated in this study. These parameters are heart rate, aortic compliance (wave speed), and locations of reflection sites. Both computational and experimental approaches have been used in this research. In some cases, the results are further explained using theoretical models.</p>
<p>The main findings of this study are as follows: (i) developing a physiologically realistic outflow boundary condition for blood flow modeling in a compliant vasculature; (ii) demonstrating that pulse pressure as a single index cannot predict the true level of pulsatile workload on the left ventricle; (iii) proving that there is an optimum heart rate in which the pulsatile workload of the heart is minimized and that the optimum heart rate shifts to a higher value as aortic rigidity increases; (iv) introducing a simple bio-inspired device for correction and optimization of aortic wave reflection that reduces the workload on the heart; (v) deriving a non-dimensional number that can predict the optimum wave dynamic state in a mammalian cardiovascular system; (vi) demonstrating that waves can create a pumping effect in the aorta; (vii) introducing a system parameter and a new medical index, Intrinsic Frequency, that can be used for noninvasive diagnosis of heart and vascular diseases; and (viii) proposing a new medical hypothesis for sudden cardiac death in young athletes.</p>
https://thesis.library.caltech.edu/id/eprint/7682Adaptive Methods Exploring Intrinsic Sparse Structures of Stochastic Partial Differential Equations
https://resolver.caltech.edu/CaltechTHESIS:09182012-175436855
Authors: {'items': [{'email': 'mulin.cheng@gmail.com', 'id': 'Cheng-Mulin', 'name': {'family': 'Cheng', 'given': 'Mulin'}, 'show_email': 'YES'}]}
Year: 2013
DOI: 10.7907/V638-V403
Many physical and engineering problems involving uncertainty enjoy certain low-dimensional structures, e.g., in the sense of Karhunen-Loeve expansions (KLEs), which in turn indicate the existence of reduced-order models and better formulations for efficient numerical simulations. In this thesis, we target a class of time-dependent stochastic partial differential equations whose solutions enjoy such structures at any time and propose a new methodology (DyBO) to derive equivalent systems whose solutions closely follow KL expansions of the original stochastic solutions. KL expansions are known to be the most compact representations of stochastic processes in an L<sup>2</sup> sense. Our methods explore such sparsity and offer great computational benefits compared to other popular generic methods, such as traditional Monte Carlo (MC), generalized Polynomial Chaos (gPC) method, and generalized Stochastic Collocation (gSC) method. Such benefits are demonstrated through various numerical examples ranging from spatially one-dimensional examples, such as stochastic Burgers' equations and stochastic transport equations to spatially two-dimensional examples, such as stochastic flows in 2D unit square. Parallelization is also discussed, aiming toward future industrial-scale applications. In addition to numerical examples, theoretical aspects of DyBO are also carefully analyzed, such as preservation of bi-orthogonality, error propagation, and computational complexity. Based on theoretical analysis, strategies are proposed to overcome difficulties in numerical implementations, such as eigenvalue crossing and adaptively adding or removing mode pairs. The effectiveness of the proposed strategies is numerically verified. Generalization to a system of SPDEs is considered as well in the thesis, and its success is demonstrated by applications to stochastic Boussinesq convection problems. Other generalizations, such as generalized stochastic collocation formulation of DyBO method, are also discussed. https://thesis.library.caltech.edu/id/eprint/7207Geometric Integration Applied to Moving Mesh Methods and Degenerate Lagrangians
https://resolver.caltech.edu/CaltechTHESIS:12042013-185815472
Authors: {'items': [{'email': 'tomasz.tyranowski@gmail.com', 'id': 'Tyranowski-Tomasz-Michal', 'name': {'family': 'Tyranowski', 'given': 'Tomasz Michal'}, 'show_email': 'NO'}]}
Year: 2014
DOI: 10.7907/PH3X-YH23
<p>Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.</p>
<p>In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.</p>https://thesis.library.caltech.edu/id/eprint/8038Sparse Time-Frequency Data Analysis: A Multi-Scale Approach
https://resolver.caltech.edu/CaltechTHESIS:05152014-141711934
Authors: {'items': [{'email': 'tavallali@gmail.com', 'id': 'Tavallali-Peyman', 'name': {'family': 'Tavallali', 'given': 'Peyman'}, 'show_email': 'NO'}]}
Year: 2014
DOI: 10.7907/Z9TT4NXD
In this work, we further extend the recently developed adaptive data analysis method, the Sparse Time-Frequency Representation (STFR) method. This method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. Moreover, we propose a new STFR algorithm to study intrawave signals with strong frequency modulation and analyze the convergence of this new algorithm for periodic signals. Such signals have previously remained a bottleneck for all signal processing methods. Furthermore, we propose a modified version of STFR that facilitates the extraction of intrawaves that have overlaping frequency content. We show that the STFR methods can be applied to the realm of dynamical systems and cardiovascular signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis of some cardiovascular diseases. We further explain some preliminary work on the nature of Intrinsic Mode Functions (IMFs) and how they can have different representations in different phase coordinates. This analysis shows that the uncertainty principle is fundamental to all oscillating signals.https://thesis.library.caltech.edu/id/eprint/8236Multiscale Model Reduction Methods for Deterministic and Stochastic Partial Differential Equations
https://resolver.caltech.edu/CaltechTHESIS:03312014-014047677
Authors: {'items': [{'email': 'cimaolin@gmail.com', 'id': 'Ci-Maolin', 'name': {'family': 'Ci', 'given': 'Maolin'}, 'show_email': 'NO'}]}
Year: 2014
DOI: 10.7907/06ND-CY07
<p>Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.</p>
<p>For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.</p>
<p>For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.</p>
<p>For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.</p>https://thesis.library.caltech.edu/id/eprint/8174Full and Model-Reduced Structure-Preserving Simulation of Incompressible Fluids
https://resolver.caltech.edu/CaltechTHESIS:05312015-134909133
Authors: {'items': [{'email': 'gemmaellen@gmail.com', 'id': 'Mason-Gemma-Ellen', 'name': {'family': 'Mason', 'given': 'Gemma Ellen'}, 'show_email': 'NO'}]}
Year: 2015
DOI: 10.7907/Z9KK98QG
<p>This thesis outlines the construction of several types of structured integrators for incompressible fluids. We first present a vorticity integrator, which is the Hamiltonian counterpart of the existing Lagrangian-based fluid integrator. We next present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness to coarse spatial and temporal resolutions of geometric integrators, and the simplicity of homogenized boundary conditions on regular grids to deal with arbitrarily-shaped domains with sub-grid accuracy.</p>
<p>Both these numerical methods involve approximating the Lie group of volume-preserving diffeomorphisms by a finite-dimensional Lie-group and then restricting the resulting variational principle by means of a non-holonomic constraint. Advantages and limitations of this discretization method will be outlined. It will be seen that these derivation techniques are unable to yield symplectic integrators, but that energy conservation is easily obtained, as is a discretized version of Kelvin's circulation theorem.</p>
<p>Finally, we outline the basis of a spectral discrete exterior calculus, which may be a useful element in producing structured numerical methods for fluids in the future.</p>https://thesis.library.caltech.edu/id/eprint/8948Spatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models
https://resolver.caltech.edu/CaltechTHESIS:09092016-000915850
Authors: {'items': [{'email': 'pengfeiliuc@gmail.com', 'id': 'Liu-Pengfei', 'name': {'family': 'Liu', 'given': 'Pengfei'}, 'orcid': '0000-0002-6714-7387', 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9V9862G
<p>The partial differential equations (PDE) governing the motions of incompressible ideal fluid in three dimensional (3D) space are among the most fundamental nonlinear PDEs in nature and have found a lot of important applications. Due to the presence of super-critical non-linearity, the fundamental question of global well-posedness still remains open and is generally viewed as one of the most outstanding open questions in mathematics. In this thesis, we investigate the potential finite-time singularity formation of the 3D Euler equations and simplified models by studying the self-similar spatial profiles in the potentially singular solutions.</p>
<p>In the first part, we study the self-similar singularity of two 1D models, the CKY model and the HL model, which approximate the dynamics of the 3D axisymmtric Euler equations on the solid boundary of a cylindrical domain. The two models are both numerically observed to develop self-similar singularity. We prove the existence of a discrete family of self-similar profiles for the CKY model, using a combination of analysis and computer-aided verification. Then we employ a dynamic rescaling formulation to numerically study the evolution of the spatial profiles for the two 1D models, and demonstrate the stability of the self-similar singularity. We also study a singularity scenario for the HL model with multi-scale feature.</p>
<p>In the second part, we study the self-similar singularity for the 3D axisymmetric Euler equations. We first prove the local existence of a family of analytic self-similar profiles using a modified Cauchy-Kowalevski majorization argument. Then we use the dynamic rescaling formulation to investigate two types of initial data with different leading order properties. The first initial data correspond to the singularity scenario reported by Luo and Hou. We demonstrate that the self-similar profiles enjoy certain stability, which confirms the finite-time singularity reported by Luo and Hou. For the second initial data, we show that the solutions develop singularity in a different manner from the first case, which is unknown previously. The spatial profiles in the solutions become singular themselves, which means that the solutions to the Euler equations develop singularity at multiple spatial scales.</p>
<p>In the third part, we propose a family of 3D models for the 3D axisymmetric Euler and Navier-Stokes equations by modifying the amplitude of the convection terms. The family of models share several regularity results with the original Euler and Navier-Stokes equations, and we study the potential finite-time singularity of the models numerically. We show that for small convection, the solutions of the inviscid model develop self-similar singularity and the profiles behave like travelling waves. As we increase the amplitude of the velocity field, we find a critical value, after which the travelling wave self-similar singularity scenario disappears. Our numerical results reveal the potential stabilizing effect the convection terms.</p>https://thesis.library.caltech.edu/id/eprint/9920Compressing Positive Semidefinite Operators with Sparse/Localized Bases
https://resolver.caltech.edu/CaltechTHESIS:05312017-000636495
Authors: {'items': [{'email': 'zhangjiahuah@gmail.com', 'id': 'Zhang-Pengchuan', 'name': {'family': 'Zhang', 'given': 'Pengchuan'}, 'orcid': '0000-0003-1155-9507', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z91N7Z5J
<p>Given a positive semidefinite (PSD) operator, such as a PSD matrix, an elliptic operator with rough coefficients, a covariance operator of a random field, or the Hamiltonian of a quantum system, we would like to find its best finite rank approximation with a given rank. One way to achieve this objective is to project the operator to its eigenspace that corresponds to the smallest or largest eigenvalues, depending on the setting. The eigenfunctions are typically global, i.e. nonzero almost everywhere, but our interest is to find the sparsest or most localized bases for these subspaces. The sparse/localized basis functions lead to better physical interpretation and preserve some sparsity structure in the original operator. Moreover, sparse/localized basis functions also enable us to develop more efficient numerical algorithms to solve these problems.</p>
<p>In this thesis, we present two methods for this purpose, namely the sparse operator compression (Sparse OC) and the intrinsic sparse mode decomposition (ISMD). The Sparse OC is a general strategy to construct finite rank approximations to PSD operators, for which the range space of the finite rank approximation is spanned by a set of sparse/localized basis functions. The basis functions are energy minimizing functions on local patches. When applied to approximate the solution operator of elliptic operators with rough coefficients and various homogeneous boundary conditions, the Sparse OC achieves the optimal convergence rate with nearly optimally localized basis functions. Our localized basis functions can be used as multiscale basis functions to solve elliptic equations with multiscale coefficients and provide the optimal convergence rate <i>O</i>(<i>h</i><sup>k</sup>) for 2<i>k</i>'th order elliptic problems in the energy norm. From the perspective of operator compression, these localized basis functions provide an efficient and optimal way to approximate the principal eigen-space of the elliptic operators. From the perspective of the Sparse PCA, we can approximate a large set of covariance functions by a rank-<i>n</i> operator with a localized basis and with the optimal accuracy. While the Sparse OC works well on the solution operator of elliptic operators, we also propose the ISMD that works well on low rank or nearly low rank PSD operators. Given a rank-<i>n</i> PSD operator, say a <i>N</i>-by-<i>N</i> PSD matrix <i>A</i> (<i>n</i> ≤ <i>N</i>), the ISMD <i>decomposes</i> it into <i>n</i> rank-one matrices Σ<sup><i>n</i></sup><sub><i>i=1</i></sub><i>g</i><sub><i>i</i></sub><i>g</i><sup><i>T</i></sup><sub><i>i</i></sub> where the mode {<i>g</i><sub><i>i</i></sub>}<sup><i>n</i></sup><sub><i>i=1</i></sub> are required to be as sparse as possible. Under the regular-sparse assumption (see Definition 1.3.2), we have proved that the ISMD gives the optimal patchwise sparse decomposition, and is stable to small perturbations in the matrix to be decomposed. We provide several applications in both the physical and data sciences to demonstrate the effectiveness of the proposed strategies.</p>https://thesis.library.caltech.edu/id/eprint/10228Concentration Inequalities of Random Matrices and Solving Ptychography with a Convex Relaxation
https://resolver.caltech.edu/CaltechTHESIS:09022016-135721172
Authors: {'items': [{'email': 'richardchen100@gmail.com', 'id': 'Chen-Yuhua-Richard', 'name': {'family': 'Chen', 'given': 'Yuhua Richard'}, 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z9M906MF
<p>Random matrix theory has seen rapid development in recent years. In particular, researchers have developed many non-asymptotic matrix concentration inequalities that parallel powerful scalar concentration inequalities. In this thesis, we focus on three topics: 1) estimating sparse covariance matrix using matrix concentration inequalities, 2) constructing the matrix phi-entropy to derive matrix concentration inequalities, 3) developing scalable algorithms to solve the phase recovery problem of ptychography based on low-rank matrix factorization.</p>
<p>Estimation of covariance matrix is an important subject. In the setting of high dimensional statistics, the number of samples can be small in comparison to the dimension of the problem, thus estimating the complete covariance matrix is unfeasible. By assuming that the covariance matrix satisfies some sparsity assumptions, prior work has proved that it is feasible to estimate the sparse covariance matrix of Gaussian distribution using the masked sample covariance estimator. In this thesis, we use a new approach and apply non-asymptotic matrix concentration inequalities to obtain tight sample bounds for estimating the sparse covariance matrix of subgaussian distributions.</p>
<p>The entropy method is a powerful approach in developing scalar concentration inequalities. The key ingredient is the subadditivity property that scalar entropy function exhibits. In this thesis, we construct a new concept of matrix phi-entropy and prove that matrix phi-entropy also satisfies a subadditivity property similar to the scalar form. We apply this new concept of matrix phi-entropy to derive non-asymptotic matrix concentration inequalities.</p>
<p>Ptychography is a computational imaging technique which transforms low-resolution intensity-only images into a high-resolution complex recovery of the signal. Conventional algorithms are based on alternating projection, which lacks theoretical guarantees for their performance. In this thesis, we construct two new algorithms. The first algorithm relies on a convex formulation of the ptychography problem and on low-rank matrix recovery. This algorithm improves traditional approaches' performance but has high computational cost. The second algorithm achieves near-linear runtime and memory complexity by factorizing the objective matrix into its low-rank components and approximates the first algorithm's imaging quality.</p>https://thesis.library.caltech.edu/id/eprint/9911Positive Definite Matrices: Compression, Decomposition, Eigensolver, and Concentration
https://resolver.caltech.edu/CaltechTHESIS:05222020-162227420
Authors: {'items': [{'email': 'huangde0123@gmail.com', 'id': 'Huang-De', 'name': {'family': 'Huang', 'given': 'De'}, 'orcid': '0000-0003-4023-9895', 'show_email': 'YES'}]}
Year: 2020
DOI: 10.7907/g2nt-yy27
<p>For many decades, the study of positive-definite (PD) matrices has been one of the most popular subjects among a wide range of scientific researches. A huge mass of successful models on PD matrices has been proposed and developed in the fields of mathematics, physics, biology, etc., leading to a celebrated richness of theories and algorithms. In this thesis, we draw our attention to a general class of PD matrices that can be decomposed as the sum of a sequence of positive-semidefinite matrices. For this class of PD matrices, we will develop theories and algorithms on operator compression, multilevel decomposition, eigenpair computation, and spectrum concentration. We divide these contents into three main parts.</p>
<p>In the first part, we propose an adaptive fast solver for the preceding class of PD matrices which includes the well-known graph Laplacians. We achieve this by establishing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which have nearly optimal performance on both complexity and well-posedness. To develop our methods, we introduce a novel notion of energy decomposition for PD matrices and two important local measurement quantities, which provide theoretical guarantee and computational guidance for the construction of an appropriate partition and a nested adaptive basis.</p>
<p>In the second part, we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse PD matrix under the multiresolution matrix compression framework. We exploit the well-conditioned property of every decomposition components by integrating the multiresolution framework into the Implicitly Restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned.</p>
<p>In the third part, we derive concentration inequalities on partial sums of eigenvalues of random PD matrices by introducing the notion of <i>k</i>-trace. For this purpose, we establish a generalized Lieb's concavity theorem, which extends the original Lieb's concavity theorem from the normal trace to <i>k</i>-traces. Our argument employs a variety of matrix techniques and concepts, including exterior algebra, mixed discriminant, and operator interpolation.</p>https://thesis.library.caltech.edu/id/eprint/13715Nonlinear Dynamics and Stability of Viscous Free-Surface Microcapillary Flows in V-Shaped Channels and on Curved Surfaces
https://resolver.caltech.edu/CaltechTHESIS:05292022-001428228
Authors: {'items': [{'email': 'nwhite@posteo.net', 'id': 'White-Nicholas-Conlan', 'name': {'family': 'White', 'given': 'Nicholas Conlan'}, 'orcid': '0000-0002-7603-9329', 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/yd3w-ck87
<p>The last two decades have brought a revolution in miniaturization of space technology. Thanks to improved microelectronic sensors and MEMS devices, nanosatellites can perform communication and scientific studies previously limited to large satellites, significantly reducing the financial barriers to space access. But development of a reliable, long-running, small-scale propulsion system for orbital maneuvers remains a key challenge. One solution is the microfluidic electrospray propulsion (MEP) thruster under development at NASA's Jet Propulsion Laboratory (JPL).</p>
<p>This thesis analytically addresses aspects of the MEP system's propellant management, specifically, capillary flow in the groove network delivering fluid propellant from the reservoir to the emitters. Building upon the reduced-order model of viscous capillary flow in straight V-shaped channels ("V-grooves") of Weislogel (1996) and Romero and Yost (1996), we prove stability of steady-state and self-similar flows. Because the MEP design requires an electric field above the grooves, and further calls for grooves which curve and bend in three dimensions, we extend earlier V-groove models to include these effects, and also perform stability analyses of the new models. The results not only validate the use of V-grooves as a robust propellant delivery system, but also provide a theoretical basis for the design of future microfluidic devices with compact, three-dimensional designs and electric fields.</p>
<p>In order to lay the groundwork for future studies of early-time behavior of propellant on emitter tips before the Taylor cone necessary for ion emission is formed, we develop the technique of generalized linear stability analysis (Farrell and Ioannou, 1996) of capillary flow of thin viscous films coating curved surfaces (governed by the equation first developed by Roy and Schwartz, 1997). This methodology was first applied to films coating cylinders and spheres by Balestra et al. (2016, 2018); we instead apply the technique and analyze for the first time a viscous-capillary instability arising on a torus coated with a uniform thin film.</p>
<p>Besides the capillary fluid dynamics results, two additional pieces of work are included in the thesis. First, in an unorthodox application of Noether's Theorem to non-Lagrangian gradient flow equations, we show that each variational symmetry of the governing functional induces a constraint on the evolution of the system. Second, to support JPL's efforts to directly detect a "fifth force," we introduce and implement numerical methods for computation of the scalar Cubic Galileon Gravity (CGG) field at solar system scales.</p>https://thesis.library.caltech.edu/id/eprint/14648Singularity Formation in Incompressible Fluids and Related Models
https://resolver.caltech.edu/CaltechTHESIS:05172022-223804694
Authors: {'items': [{'email': 'cjiajie26@126.com', 'id': 'Chen-Jiajie', 'name': {'family': 'Chen', 'given': 'Jiajie'}, 'orcid': '0000-0002-0194-1975', 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/nqff-dh92
<p>Whether the three-dimensional (3D) incompressible Euler equations can develop a finite-time singularity from smooth initial data with finite energy is a major open problem in partial differential equations. A few years ago, Tom Hou and Guo Luo obtained strong numerical evidence of a potential finite time singularity of the 3D axisymmetric Euler equations with boundary from smooth initial data. So far, there is no rigorous justification. In this thesis, we develop a framework to study the Hou-Luo blowup scenario and singularity formation in related equations and models. In addition, we analyze the obstacle to singularity formation.</p>
<p>In the first part, we propose a novel framework of analysis based on the dynamic rescaling formulation to study singularity formation. Our strategy is to reformulate the problem of proving finite time blowup into the problem of establishing the nonlinear stability of an approximate self-similar blowup profile using the dynamic rescaling equations. Then we prove finite time blowup of the 2D Boussinesq and the 3D Euler equations with C<sup>1,α</sup> velocity and boundary. This result provides the first rigorous justification of the Hou-Luo scenario using C<sup>1,α</sup> velocity.</p>
<p>In the second part, we further develop the framework for smooth data. The method in the first part relies crucially on the low regularity of the data, and there are several essential difficulties to generalize it to study the Hou-Luo scenario with smooth data. We demonstrate that some of the challenges can be overcome by proving the asymptotically self-similar blowup of the Hou-Luo model. Applying this framework, we establish the finite time blowup of the De Gregorio (DG) model on the real line (ℝ) with smooth data. Our result resolves the open problem on the regularity of this model on ℝ that has been open for quite a long time.</p>
<p>In the third part, we investigate the competition between advection and vortex stretching, an essential difficulty in studying the regularity of the 3D Euler equations. This competition can be modeled by the DG model on S<sup>1</sup>. We consider odd initial data with a specific sign property and show that the regularity of the initial data in this class determines the competition between advection and vortex stretching. For any 0 < α < 1, we construct a finite time blowup solution from some C<sup>α</sup> initial data. On the other hand, we prove that the solution exists globally for C<sup>1</sup> initial data. Our results resolve some conjecture on the finite time blowup of this model and imply that singularities developed in the DG model and the generalized Constantin-Lax-Majda model on S<sup>1</sup> can be prevented by stronger advection.</p>https://thesis.library.caltech.edu/id/eprint/14584On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems
https://resolver.caltech.edu/CaltechTHESIS:05292023-175108484
Authors: {'items': [{'email': 'yifanc96@gmail.com', 'id': 'Chen-Yifan', 'name': {'family': 'Chen', 'given': 'Yifan'}, 'orcid': '0000-0001-5494-4435', 'show_email': 'YES'}]}
Year: 2023
DOI: 10.7907/83p4-c644
<p> This thesis focuses on numerical methods for scientific computing and scientific machine learning, specifically on solving partial differential equations and inverse problems. The design of numerical algorithms usually encompasses a spectrum that ranges from specialization to generality. Classical approaches, such as finite element methods, and contemporary scientific machine learning approaches, like neural nets, can be viewed as lying at relatively opposite ends of this spectrum. Throughout this thesis, we tackle mathematical challenges associated with both ends by advancing rigorous multiscale and statistical numerical methods. </p>
<p>Regarding the multiscale numerical methods, we present an exponentially convergent multiscale finite element method for solving high-frequency Helmholtz's equation with rough coefficients. To achieve this, we first identify the local low-complexity structure of Helmholtz's equations when the resolution is smaller than the wavelength. Then, we construct local basis functions by solving local spectral problems and couple them globally through non-overlapped domain decomposition and Galerkin's method. This results in a numerical method that achieves nearly exponentially convergent accuracy regarding the number of local basis functions, even when the solution is highly non-smooth. We also analyze the role of a subsampled lengthscale in variational multiscale methods, characterizing the tradeoff between accuracy and efficiency in the numerical upscaling of heterogeneous PDEs and scattered data approximation.</p>
<p>As for the statistical numerical methods, we discuss using Gaussian processes and kernel methods to solve nonlinear PDEs and inverse problems. This framework incorporates the flavor of scientific machine learning automation and extends classical meshless solvers. It transforms general PDE problems into quadratic optimization with nonlinear constraints. We present the theoretical underpinning of the methodology. For the scalability of the method, we develop state-of-the-art algorithms to handle dense kernel matrices in both low and high-dimensional scientific problems. For adaptivity, we analyze the convergence and consistency of hierarchical learning algorithms that adaptively select kernel functions. Additionally, we note that statistical numerical methods offer natural uncertainty quantification within the Bayesian framework. In this regard, our further work contributes to some new understanding of efficient statistical sampling techniques based on gradient flows.</p>https://thesis.library.caltech.edu/id/eprint/15224Low-Rank Matrix Recovery: Manifold Geometry and Global Convergence
https://resolver.caltech.edu/CaltechTHESIS:05302023-222447373
Authors: {'items': [{'email': 'zyzhang0907@gmail.com', 'id': 'Zhang-Ziyun', 'name': {'family': 'Zhang', 'given': 'Ziyun'}, 'orcid': '0000-0002-5794-2387', 'show_email': 'YES'}]}
Year: 2023
DOI: 10.7907/hd6q-g460
<p>Low-rank matrix recovery problems are prevalent in modern data science, machine learning, and artificial intelligence, and the low-rank property of matrices is widely exploited to extract the hidden low-complexity structure in massive datasets. Compared with Burer-Monteiro factorization in the Euclidean space, using the low-rank matrix manifold has its unique advantages, as it eliminates duplicated spurious points and reduces the polynomial order of the objective function. Yet a few fundamental questions have remained unanswered until recently. We highlight two problems here in particular, which are the global geometry of the manifold and the global convergence guarantee.</p>
<p>As for the global geometry, we point out that there exist some spurious critical points on the boundary of the low-rank matrix manifold Mᵣ, which have rank smaller than r but can serve as limit points of iterative sequences in the manifold Mᵣ. For the least squares loss function, the spurious critical points are rank-deficient matrices that capture part of the eigen spaces of the ground truth. Unlike classical strict saddle points, their Riemannian gradient is singular and their Riemannian Hessian is unbounded.</p>
<p>We show that randomly initialized Riemannian gradient descent almost surely escapes some of the spurious critical points. To prove this result, we first establish the asymptotic escape of classical strict saddle sets consisting of non-isolated strict critical submanifolds on Riemannian manifolds. We then use a dynamical low-rank approximation to parameterize the manifold Mᵣ and map the spurious critical points to strict critical submanifolds in the classical sense in the parameterized domain, which leads to the desired result. Our result is the first to partially overcome the nonclosedness of the low-rank matrix manifold without altering the vanilla gradient descent algorithm. Numerical experiments are provided to support our theoretical findings.</p>
<p>As for the global convergence guarantee, we point out that earlier approaches to many of the low-rank recovery problems only imply a geometric convergence rate toward a second-order stationary point. This is in contrast to the numerical evidence, which suggests a nearly linear convergence rate starting from a global random initialization. To establish the nearly linear convergence guarantee, we propose a unified framework for a class of low-rank matrix recovery problems including matrix sensing, matrix completion, and phase retrieval. All of them can be considered as random sensing problems of low-rank matrices with a linear measurement operator from some random ensembles. These problems share similar population loss functions that are either least squares or its variant.</p>
<p>We show that under some assumptions, for the population loss function, the Riemannian gradient descent starting from a random initialization with high probability converges to the ground truth in a nearly linear convergence rate, i.e., it takes O(log 1/ϵ + log n) iterations to reach an ϵ-accurate solution. The key to establishing a nearly optimal convergence guarantee is closely intertwined with the analysis of the spurious critical points S_# on Mᵣ. Outside the local neighborhoods of spurious critical points, we use the fundamental convergence tool by the Łojasiewicz inequality to derive a linear convergence rate. In the spurious regions in the neighborhood of spurious critical points, the Riemannian gradient becomes degenerate and the Łojasiewicz inequality could fail. By tracking the dynamics of the trajectory in three stages, we are able to show that with high probability, Riemannian gradient descent escapes the spurious regions in a small number of steps.</p>
<p>After addressing the two problems of global geometry and global convergence guarantee, we use two applications to demonstrate the broad applicability of our analytical tools. The first is the robust principal component analysis problem on the manifold Mᵣ with the Riemannian subgradient method. The second application is the convergence rate analysis of the Sobolev gradient descent method for the nonlinear Gross-Pitaevskii eigenvalue problem on the infinite dimensional sphere manifold. These two examples demonstrate that the analysis of manifold first-order algorithms can be extended beyond the previous framework, to nonsmooth functions and subgradient methods, and to infinite dimensional Hilbert manifolds. This exemplifies that the insights gained and tools developed for the low-rank matrix manifold Mᵣ can be extended to broader scientific and technological fields.</p>https://thesis.library.caltech.edu/id/eprint/15236Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks
https://resolver.caltech.edu/CaltechTHESIS:09202022-034157716
Authors: {'items': [{'email': 'zhangsm1995@gmail.com', 'id': 'Zhang-Shumao', 'name': {'family': 'Zhang', 'given': 'Shumao'}, 'orcid': '0000-0003-3071-3362', 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/8had-3a90
<p>High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ℝⁿ, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.</p>
<p>In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is C<sup>α</sup> Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor ε as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength ε < ε*, and this upper bound ε* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if ε < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.</p>
<p>In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.</p>https://thesis.library.caltech.edu/id/eprint/15033