Abstract: n Part I of our sequence of 2 papers, we provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with C^α initial vorticity for a large range of α. We employ an adaptive mesh method using a highly effective mesh to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling formulation are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our study shows that the 3D Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α is smaller than some critical value α^∗. By properly rescaling the initial data in the z-axis, this upper bound for potential blow-up α^∗ can asymptotically approach 1/3. Compared with Elgindi's blow-up result in a similar setting [15], our potential blow-up scenario has a different Hölder continuity property in the initial data and the scaling properties of the two initial data are also quite different.

Publication: arXiv
ID: CaltechAUTHORS:20230227-194424192

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Abstract: In Part II of this sequence to our previous paper for the 3-dimensional Euler equations [8], we investigate potential singularity of the n-diemnsional axisymmetric Euler equations with C^α initial vorticity for a large range of α. We use the adaptive mesh method to solve the n-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the n-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α < α^∗, and this upper bound α∗ can asymptotically approach 1 − 2/n. Moreover, we introduce a stretching parameter δ along the z-direction. Based on a few assumptions inspired by our numerical experiments, we obtain α^∗ = 1 − 2/n by studying the limiting case of δ→0. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the n-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the n-dimensional Euler equations. As shown in [8], the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in [6].

Publication: arXiv
ID: CaltechAUTHORS:20230227-194427740

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Abstract: We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

Publication: arXiv
ID: CaltechAUTHORS:20230227-194420642

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Abstract: The physics-informed neural operator (PINO) is a machine learning architecture that has shown promising empirical results for learning partial differential equations. PINO uses the Fourier neural operator (FNO) architecture to overcome the optimization challenges often faced by physics-informed neural networks. Since the convolution operator in PINO uses the Fourier series representation, its gradient can be computed exactly on the Fourier space. While Fourier series cannot represent nonperiodic functions, PINO and FNO still have the expressivity to learn nonperiodic problems with Fourier extension via padding. However, computing the Fourier extension in the physics-informed optimization requires solving an ill-conditioned system, resulting in inaccurate derivatives which prevent effective optimization. In this work, we present an architecture that leverages Fourier continuation (FC) to apply the exact gradient method to PINO for nonperiodic problems. This paper investigates three different ways that FC can be incorporated into PINO by testing their performance on a 1D blowup problem. Experiments show that FC-PINO outperforms padded PINO, improving equation loss by several orders of magnitude, and it can accurately capture the third order derivatives of nonsmooth solution functions.

Publication: arXiv
ID: CaltechAUTHORS:20221221-004750416

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Abstract: Inspired by the numerical evidence of a potential 3D Euler singularity [Luo-Hou-14a, Luo-Hou-14b], we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. There are several essential difficulties in proving finite time blowup of 3D Euler with smooth initial data. One of the essential difficulties is to control a number of nonlocal terms that do not seem to offer any damping effect. Another essential difficulty is that the strong advection normal to the boundary introduces a large growth factor for the perturbation if we use weighted L² estimates. We overcome this difficulty by using a combination of a weighted L∞ norm and a weighted C^(1/2) norm, and develop sharp functional inequalities using the symmetry properties of the kernels and some techniques from optimal transport. Moreover we decompose the linearized operator into a leading order operator plus a finite rank operator. The leading order operator is designed in such a way that we can obtain sharp stability estimates. The contribution from the finite rank operator can be captured by an auxiliary variable and its contribution to linear stability can be estimated by constructing approximate solution in space-time. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary.

Publication: arXiv
ID: CaltechAUTHORS:20230227-191437852

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Abstract: Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and C^(1,α) initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with C^(1,α) initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in [35,52] require some strong regularity assumption on the initial data, which is not satisfied by the C^(1,α) velocity field. In this paper, we generalize the analysis of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with C^(1,α) velocity are unstable under the notion of stability introduced in [35,52]. These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. The stability analysis of the blowup solution obtained in [8,18] is based on the stability of a dynamically rescaled blowup profile in space and time, which is nonlinear in nature. The linear stability analysis in [35,52] is performed by directly linearizing the 3D Euler equations around a blowup solution in the original variables. It does not take into account the changes in the blowup time, the dynamic changes of the rescaling rate of the perturbed blowup profile and the blowup exponent of the original 3D Euler equations using a perturbed initial condition when there is an approximate self-similar blowup profile. Such information has been used in an essential way in establishing the nonlinear stability of the blowup profile in [8,18,19].

Publication: arXiv
ID: CaltechAUTHORS:20230227-194417020

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Abstract: We show that on the manifold of fixed-rank and symmetric positive semi-definite matrices, the Riemannian gradient descent algorithm almost surely escapes some spurious critical points on the boundary of the manifold. Our result is the first to partially overcome the incompleteness of the low-rank matrix manifold without changing the vanilla Riemannian gradient descent algorithm. The spurious critical points are some rank-deficient matrices that capture only part of the eigen components of the ground truth. Unlike classical strict saddle points, they exhibit very singular behavior. We show that using the dynamical low-rank approximation and a rescaled gradient flow, some of the spurious critical points can be converted to classical strict saddle points in the parameterized domain, which leads to the desired result. Numerical experiments are provided to support our theoretical findings.

Publication: arXiv
ID: CaltechAUTHORS:20221221-220346410

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Abstract: Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in the companion paper (arXiv:2107.05870). We present numerical evidence that the 3D Navier--Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10⁷. We have applied several blow-up criteria to study the potentially singular behavior of the Navier--Stokes equations. The Beale-Kato-Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier--Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya-Prodi-Serrin regularity criteria. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q) = (∞,2) provide strong evidence for the potentially singular behavior of the Navier--Stokes equations. Our numerical study shows that while the global L^3 norm of the velocity grows very slowly, the localized version of the L³ norm of the velocity experiences rapid dynamic growth relative to the localized L³ norm of the initial velocity. This provides further evidence for the potentially singular behavior of the NavieStokes equations.

Publication: arXiv
ID: CaltechAUTHORS:20230227-193545252

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Abstract: Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo-Hou in [31, 32], which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in [20, 21]. One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in \cite{Hou-Huang-2021,Hou-Huang-2022}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.

Publication: arXiv
ID: CaltechAUTHORS:20230227-192413744

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Abstract: In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the Riemannian gradient descent algorithm to minimize a least squares loss function, and study the asymptotic behavior as well as the exact convergence rate. We reveal a previously unknown geometric property of the low-rank matrix manifold, which is the existence of spurious critical points for the simple least squares function on the manifold. We show that under some assumptions, the Riemannian gradient descent starting from a random initialization with high probability avoids these spurious critical points and only converges to the ground truth in nearly linear convergence rate, i.e. O(log(1/ϵ) + log(n)) iterations to reach an ϵ-accurate solution. We use two applications as examples for our global analysis. The first one is a rank-1 matrix recovery problem. The second one is a generalization of the Gaussian phase retrieval problem. It only satisfies the weak isometry property, but has behavior similar to that of the first one except for an extra saddle set. Our convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations. The global analysis can be potentially extended to other data problems with random measurement structures and empirical least squares loss functions.

Publication: arXiv
ID: CaltechAUTHORS:20221221-220354531

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Abstract: In this paper, we provide some analysis on the asymptotic escape of strict saddles in manifold optimization using the projected gradient descent (PGD) algorithm. One of our main contributions is that we extend the current analysis to include non-isolated and possibly continuous saddle sets with complicated geometry. We prove that the PGD is able to escape strict critical submanifolds under certain conditions on the geometry and the distribution of the saddle point sets. We also show that the PGD may fail to escape strict saddles under weaker assumptions even if the saddle point set has zero measure and there is a uniform escape direction. We provide a counterexample to illustrate this important point. We apply this saddle analysis to the phase retrieval problem on the low-rank matrix manifold, prove that there are only a finite number of saddles, and they are strict saddles with high probability. We also show the potential application of our analysis for a broader range of manifold optimization problems.

Publication: arXiv
ID: CaltechAUTHORS:20200122-133158689

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Abstract: In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters.

Publication: arXiv
ID: CaltechAUTHORS:20200122-143531561

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Abstract: In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.

ID: CaltechAUTHORS:20160315-123826115

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Abstract: In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simultaneously sparsity approximation and fast Fourier transform, some efficient algorithms is developed. Since the information of multiple signals is used, this method is very robust to the perturbation of noise. And it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to test this method. The performances of this method are very promising.

ID: CaltechAUTHORS:20160315-152129723

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Abstract: We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which approximates the dynamics of the Euler equations on the solid boundary of a cylindrical domain. We prove the existence of a discrete family of self-similar profiles for this model and analyze their far-field properties. The self-similar profiles we find are consistent with direct simulation of the model and enjoy some stability property.

ID: CaltechAUTHORS:20160315-133921211

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Abstract: We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical study (Luo and Hou, 2013).

ID: CaltechAUTHORS:20160315-134409579

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Abstract: In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition.

ID: CaltechAUTHORS:20160315-133702384

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Abstract: In this article we apply the technique proposed in Deng-Hou-Yu [7] to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of θ, we obtain global regularity results with improved growth estimate on │∇^⊥θ│. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum │∇^⊥θ│. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of θ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of │∇^⊥θ│ observed in this and past numerical simulations.

ID: CaltechAUTHORS:20160322-071936317

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