[ { "id": "https://authors.library.caltech.edu/records/g0k6z-gvr57", "eprint_id": 114898, "eprint_status": "archive", "datestamp": "2023-08-20 06:01:45", "lastmod": "2023-10-24 15:15:04", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hamze-Bajgiran-Hamed", "name": { "family": "Hamze Bajgiran", "given": "Hamed" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Aggregation of Models, Choices, Beliefs, and Preferences", "ispublished": "unpub", "full_text_status": "public", "note": "The authors gratefully acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies) and partial support from the Air Force Office of Scientific Research under awards number FA9550-18-1-0271 (Games for Computation and Learning) and FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). \n\nThe first version of the paper was written during the first author's Ph.D. studies with many helpful comments from Federico Echenique and Kota Saito. The first author thanks his Ph.D. advisors Jaksa Cvitanic, Federico Echenique, Kota Saito, and Robert Sherman. For helpful discussions, the first author thanks Itai Ashlagi, Kim Border, Martin Cripps, David Dillenberger, Drew Fudenberg, Simone Galperti, Michihiro Kandori, Igor Kopylov, Jay Lu, Fabio Maccheroni, Thomas Palfrey, Charles Plott, Luciano Pomatto, Antonio Rangel, Pablo Schenone, Omer Tamuz, and Leeat Yariv.\n\n
Submitted - 2111.11630.pdf
", "abstract": "A natural notion of rationality/consistency for aggregating models is that, for all (possibly aggregated) models A and B, if the output of model A is f(A) and if the output model B is f(B), then the output of the model obtained by aggregating A and B must be a weighted average of f(A) and f(B). Similarly, a natural notion of rationality for aggregating preferences of ensembles of experts is that, for all (possibly aggregated) experts A and B, and all possible choices x and y, if both A and B prefer x over y, then the expert obtained by aggregating A and B must also prefer x over y. Rational aggregation is an important element of uncertainty quantification, and it lies behind many seemingly different results in economic theory: spanning social choice, belief formation, and individual decision making. Three examples of rational aggregation rules are as follows. (1) Give each individual model (expert) a weight (a score) and use weighted averaging to aggregate individual or finite ensembles of models (experts). (2) Order/rank individual model (expert) and let the aggregation of a finite ensemble of individual models (experts) be the highest-ranked individual model (expert) in that ensemble. (3) Give each individual model (expert) a weight, introduce a weak order/ranking over the set of models/experts, aggregate A and B as the weighted average of the highest-ranked models (experts) in A or B. Note that (1) and (2) are particular cases of (3). In this paper, we show that all rational aggregation rules are of the form (3). This result unifies aggregation procedures across different economic environments. Following the main representation, we show applications and extensions of our representation in various separated economics topics such as belief formation, choice theory, and social welfare economics.", "date": "2022-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220524-180312022", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220524-180312022", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Beyond Limits" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-20-1-0358" } ] }, "local_group": { "items": [ { "id": "Center-for-Autonomous-Systems-and-Technologies-(CAST)" } ] }, "doi": "10.48550/arXiv.2111.11630", "primary_object": { "basename": "2111.11630.pdf", "url": "https://authors.library.caltech.edu/records/g0k6z-gvr57/files/2111.11630.pdf" }, "resource_type": "monograph", "pub_year": "2022", "author_list": "Hamze Bajgiran, Hamed and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/aeg7b-hbw77", "eprint_id": 114900, "eprint_status": "archive", "datestamp": "2023-08-20 06:13:28", "lastmod": "2023-10-24 15:15:09", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hamze-Bajgiran-Hamed", "name": { "family": "Hamze Bajgiran", "given": "Hamed" }, "orcid": "0000-0002-6246-2783" }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Aggregation of Pareto optimal models", "ispublished": "unpub", "full_text_status": "public", "note": "The authors gratefully acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies) and partial support from the Air Force Office of Scientific Research under awards number FA9550-18-1-0271 (Games for Computation and Learning) and FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation).\n\nSubmitted - 2112.04161.pdf
", "abstract": "In statistical decision theory, a model is said to be Pareto optimal (or admissible) if no other model carries less risk for at least one state of nature while presenting no more risk for others. How can you rationally aggregate/combine a finite set of Pareto optimal models while preserving Pareto efficiency? This question is nontrivial because weighted model averaging does not, in general, preserve Pareto efficiency. This paper presents an answer in four logical steps: (1) A rational aggregation rule should preserve Pareto efficiency (2) Due to the complete class theorem, Pareto optimal models must be Bayesian, i.e., they minimize a risk where the true state of nature is averaged with respect to some prior. Therefore each Pareto optimal model can be associated with a prior, and Pareto efficiency can be maintained by aggregating Pareto optimal models through their priors. (3) A prior can be interpreted as a preference ranking over models: prior \u03c0 prefers model A over model B if the average risk of A is lower than the average risk of B. (4) A rational/consistent aggregation rule should preserve this preference ranking: If both priors \u03c0 and \u03c0\u2032 prefer model A over model B, then the prior obtained by aggregating \u03c0 and \u03c0\u2032 must also prefer A over B. Under these four steps, we show that all rational/consistent aggregation rules are as follows: Give each individual Pareto optimal model a weight, introduce a weak order/ranking over the set of Pareto optimal models, aggregate a finite set of models S as the model associated with the prior obtained as the weighted average of the priors of the highest-ranked models in S. This result shows that all rational/consistent aggregation rules must follow a generalization of hierarchical Bayesian modeling. Following our main result, we present applications to Kernel smoothing, time-depreciating models, and voting mechanisms.", "date": "2022-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220524-180318744", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220524-180318744", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Beyond Limits" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-20-1-0358" } ] }, "local_group": { "items": [ { "id": "Center-for-Autonomous-Systems-and-Technologies-(CAST)" } ] }, "doi": "10.48550/arXiv.2112.04161", "primary_object": { "basename": "2112.04161.pdf", "url": "https://authors.library.caltech.edu/records/aeg7b-hbw77/files/2112.04161.pdf" }, "resource_type": "monograph", "pub_year": "2022", "author_list": "Hamze Bajgiran, Hamed and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/kmksv-qsg95", "eprint_id": 114896, "eprint_status": "archive", "datestamp": "2023-08-20 01:51:14", "lastmod": "2023-10-24 15:14:59", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hamzi-Boumediene", "name": { "family": "Hamzi", "given": "Boumediene" }, "orcid": "0000-0002-9446-2614" }, { "id": "Maulik-Romit", "name": { "family": "Maulik", "given": "Romit" }, "orcid": "0000-0001-9731-8936" }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods", "ispublished": "unpub", "full_text_status": "public", "keywords": "Surrogate models, kernel methods, geophysical forecasting", "note": "This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research, under Contract DE-AC02-06CH11357. This research was funded in part and used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. R. M. acknowledges support from the Margaret Butler Fellowship at the Argonne Leadership Computing Facility. B. H. thanks the European Commission for funding through the Marie Curie fellowship STALDYS-792919 (Statistical Learning for Dynamical Systems). H. O. gratefully acknowledges support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning) and MURI (FA9550-20-1-0358). This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. DOE or the United States Government. \n\nAuthors' Contributions. B.H. prepared code and analysis, wrote portions of the paper. R.M. designed the investigation, prepared code and documentation, performed analyses, generated visualizations, wrote paper. H.O. prepared code and analysis, wrote portions of the paper. \n\nData Accessibility. The data that support the findings of this study are openly available in Github at https://github.com/Romit-Maulik/POD_RKHS.\n\nSubmitted - 2103.10935.pdf
", "abstract": "Modeling geophysical processes as low-dimensional dynamical systems and regressing their vector field from data is a promising approach for learning emulators of such systems. We show that when the kernel of these emulators is also learned from data (using kernel flows, a variant of cross-validation), then the resulting data-driven models are not only faster than equation-based models but are easier to train than neural networks such as the long short-term memory neural network. In addition, they are also more accurate and predictive than the latter. When trained on geophysical observational data, for example, the weekly averaged global sea-surface temperature, considerable gains are also observed by the proposed technique in comparison to classical partial differential equation-based models in terms of forecast computational cost and accuracy. When trained on publicly available re-analysis data for the daily temperature of the North-American continent, we see significant improvements over classical baselines such as climatology and persistence-based forecast techniques. Although our experiments concern specific examples, the proposed approach is general, and our results support the viability of kernel methods (with learned kernels) for interpretable and computationally efficient geophysical forecasting for a large diversity of processes.", "date": "2022-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220524-180305206", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220524-180305206", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-AC02-06CH11357" }, { "agency": "Argonne National Laboratory" }, { "agency": "Marie Curie Fellowship", "grant_number": "792919" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-20-1-0358" } ] }, "doi": "10.48550/arXiv.2103.10935", "primary_object": { "basename": "2103.10935.pdf", "url": "https://authors.library.caltech.edu/records/kmksv-qsg95/files/2103.10935.pdf" }, "resource_type": "monograph", "pub_year": "2022", "author_list": "Hamzi, Boumediene; Maulik, Romit; et el." }, { "id": "https://authors.library.caltech.edu/records/vhc39-wjz49", "eprint_id": 114899, "eprint_status": "archive", "datestamp": "2023-08-20 06:02:22", "lastmod": "2023-10-24 15:15:07", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Lee-Jonghyeon", "name": { "family": "Lee", "given": "Jonghyeon" } }, { "id": "De-Brouwer-Edward", "name": { "family": "De Brouwer", "given": "Edward" }, "orcid": "0000-0003-0608-0155" }, { "id": "Hamzi-Boumediene", "name": { "family": "Hamzi", "given": "Boumediene" }, "orcid": "0000-0002-9446-2614" }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series", "ispublished": "unpub", "full_text_status": "public", "note": "Attribution 4.0 International (CC BY 4.0).\n\nSubmitted - 2111.13037.pdf
", "abstract": "A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF) [34] (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.", "date": "2022-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220524-180315371", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220524-180315371", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.2111.13037", "primary_object": { "basename": "2111.13037.pdf", "url": "https://authors.library.caltech.edu/records/vhc39-wjz49/files/2111.13037.pdf" }, "resource_type": "monograph", "pub_year": "2022", "author_list": "Lee, Jonghyeon; De Brouwer, Edward; et el." }, { "id": "https://authors.library.caltech.edu/records/v0cp5-ykc35", "eprint_id": 114897, "eprint_status": "archive", "datestamp": "2023-08-20 04:47:15", "lastmod": "2023-10-24 15:15:01", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hamze-Bajgiran-Hamed", "name": { "family": "Hamze Bajgiran", "given": "Hamed" } }, { "id": "Batlle-Franch-Pau", "name": { "family": "Batlle Franch", "given": "Pau" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-Clint", "name": { "family": "Scovel", "given": "Clint" }, "orcid": "0000-0001-7757-3411" }, { "id": "Shirdel-Mahdy", "name": { "family": "Shirdel", "given": "Mahdy" } }, { "id": "Stanley-Michael", "name": { "family": "Stanley", "given": "Michael" } }, { "id": "Tavallali-Peyman", "name": { "family": "Tavallali", "given": "Peyman" }, "orcid": "0000-0001-7166-5489" } ] }, "title": "Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021. California Institute of Technology. Government sponsorship acknowledged. \n\nPart of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The authors gratefully acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies) and partial support from the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning).\n\nSubmitted - 2108.10517.pdf
", "abstract": "There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample x, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in [0,1] acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near 1, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near 0, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.", "date": "2022-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220524-180308552", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220524-180308552", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NASA/JPL/Caltech" }, { "agency": "Beyond Limits" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" } ] }, "local_group": { "items": [ { "id": "Center-for-Autonomous-Systems-and-Technologies-(CAST)" } ] }, "doi": "10.48550/arXiv.2108.10517", "primary_object": { "basename": "2108.10517.pdf", "url": "https://authors.library.caltech.edu/records/v0cp5-ykc35/files/2108.10517.pdf" }, "resource_type": "monograph", "pub_year": "2022", "author_list": "Hamze Bajgiran, Hamed; Batlle Franch, Pau; et el." }, { "id": "https://authors.library.caltech.edu/records/1b23n-q5002", "eprint_id": 108529, "eprint_status": "archive", "datestamp": "2023-08-20 02:21:47", "lastmod": "2023-10-23 17:07:13", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Tavallali-Peyman", "name": { "family": "Tavallali", "given": "Peyman" }, "orcid": "0000-0001-7166-5489" }, { "id": "Hamze-Bajgiran-Hamed", "name": { "family": "Hamze Bajgiran", "given": "Hamed" } }, { "id": "Esaid-Danial-J", "name": { "family": "Esaid", "given": "Danial J." } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Decision Theoretic Bootstrapping", "ispublished": "unpub", "full_text_status": "public", "note": "Attribution 4.0 International (CC BY 4.0).\n\n\u00a9 2021. California Institute of Technology. Government sponsorship acknowledged.\n\nThis research was carried out at the Jet Propulsion Laboratory, California Institute\nof Technology, under a contract with the National Aeronautics and Space Administration\nand support from Beyond Limits (Learning Optimal Models) and AFOSR (Grant number\nFA9550-18-1-0271, Games for Computation and Learning). The authors are thankful to\nAmy Braverman, Lukas Mandrake and Kiri Wagstaff, for their insights.\n\nSubmitted - 2103.09982.pdf
", "abstract": "The design and testing of supervised machine learning models combine two fundamental distributions: (1) the training data distribution (2) the testing data distribution. Although these two distributions are identical and identifiable when the data set is infinite; they are imperfectly known (and possibly distinct) when the data is finite (and possibly corrupted) and this uncertainty must be taken into account for robust Uncertainty Quantification (UQ). We present a general decision-theoretic bootstrapping solution to this problem: (1) partition the available data into a training subset and a UQ subset (2) take m subsampled subsets of the training set and train m models (3) partition the UQ set into n sorted subsets and take a random fraction of them to define n corresponding empirical distributions \u03bc_j (4) consider the adversarial game where Player I selects a model i\u2208{1,\u2026,m}, Player II selects the UQ distribution \u03bc_j and Player I receives a loss defined by evaluating the model i against data points sampled from \u03bc_j (5) identify optimal mixed strategies (probability distributions over models and UQ distributions) for both players. These randomized optimal mixed strategies provide optimal model mixtures and UQ estimates given the adversarial uncertainty of the training and testing distributions represented by the game. The proposed approach provides (1) some degree of robustness to distributional shift in both the distribution of training data and that of the testing data (2) conditional probability distributions on the output space forming aleatory representations of the uncertainty on the output as a function of the input variable.", "date": "2021-03-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210323-130821498", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210323-130821498", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NASA/JPL/Caltech" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" } ] }, "doi": "10.48550/arXiv.2103.09982", "primary_object": { "basename": "2103.09982.pdf", "url": "https://authors.library.caltech.edu/records/1b23n-q5002/files/2103.09982.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Tavallali, Peyman; Hamze Bajgiran, Hamed; et el." }, { "id": "https://authors.library.caltech.edu/records/5mydb-afj74", "eprint_id": 106569, "eprint_status": "archive", "datestamp": "2023-08-19 22:54:20", "lastmod": "2023-10-20 23:35:50", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Do ideas have shape? Plato's theory of forms as the continuous limit of artificial neural networks", "ispublished": "unpub", "full_text_status": "public", "note": "The author gratefully acknowledges support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning). Thanks to Clint Scovel for a careful readthrough with detailed comments and feedback.\n\nSubmitted - 2008.03920.pdf
", "abstract": "We show that ResNets converge, in the infinite depth limit, to a generalization of image registration algorithms. In this generalization, images are replaced by abstractions (ideas) living in high dimensional RKHS spaces, and material points are replaced by data points. Whereas computational anatomy aligns images via deformations of the material space, this generalization aligns ideas by via transformations of their RKHS. This identification of ResNets as idea registration algorithms has several remarkable consequences. The search for good architectures can be reduced to that of good kernels, and we show that the composition of idea registration blocks with reduced equivariant multi-channel kernels (introduced here) recovers and generalizes CNNs to arbitrary spaces and groups of transformations. Minimizers of L2 regularized ResNets satisfy a discrete least action principle implying the near preservation of the norm of weights and biases across layers. The parameters of trained ResNets can be identified as solutions of an autonomous Hamiltonian system defined by the activation function and the architecture of the ANN. Momenta variables provide a sparse representation of the parameters of a ResNet. The registration regularization strategy provides a provably robust alternative to Dropout for ANNs. Pointwise RKHS error estimates lead to deterministic error estimates for ANNs.", "date": "2020-11-10", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20201109-155524397", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201109-155524397", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" } ] }, "doi": "10.48550/arXiv.2008.03920", "primary_object": { "basename": "2008.03920.pdf", "url": "https://authors.library.caltech.edu/records/5mydb-afj74/files/2008.03920.pdf" }, "resource_type": "monograph", "pub_year": "2020", "author_list": "Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/sshn9-4d642", "eprint_id": 106570, "eprint_status": "archive", "datestamp": "2023-08-19 22:21:53", "lastmod": "2023-10-20 23:35:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hamzi-Boumediene", "name": { "family": "Hamzi", "given": "Boumediene" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Learning dynamical systems from data: a simple cross-validation perspective", "ispublished": "unpub", "full_text_status": "public", "note": "B. H. thanks the European Commission for funding through the Marie Curie fellowship STALDYS-792919 (Statistical Learning for Dynamical Systems). H. O. gratefully acknowledges support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning). We thank Deniz Ero\u011flu, Yoshito Hirata, Jeroen Lamb, Edmilson Roque, Gabriele Santin and Yuzuru Sato for useful comments.\n\nSubmitted - 2007.05074.pdf
", "abstract": "Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. We present variants of cross-validation (Kernel Flows [31] and its variants based on Maximum Mean Discrepancy and Lyapunov exponents) as simple approaches for learning the kernel used in these emulators.", "date": "2020-11-10", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20201109-155527819", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201109-155527819", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Marie Curie Fellowship", "grant_number": "792919" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" } ] }, "doi": "10.48550/arXiv.2007.05074", "primary_object": { "basename": "2007.05074.pdf", "url": "https://authors.library.caltech.edu/records/sshn9-4d642/files/2007.05074.pdf" }, "resource_type": "monograph", "pub_year": "2020", "author_list": "Hamzi, Boumediene and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/tpvre-65819", "eprint_id": 106491, "eprint_status": "archive", "datestamp": "2023-08-19 21:54:35", "lastmod": "2023-10-20 23:32:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Sch\u00e4fer-F", "name": { "family": "Sch\u00e4fer", "given": "Florian" } }, { "id": "Anandkumar-A", "name": { "family": "Anandkumar", "given": "Anima" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Competitive Mirror Descent", "ispublished": "unpub", "full_text_status": "public", "note": "AA is supported in part by Bren endowed chair, DARPA PAIHR00111890035, LwLL grants, Raytheon, BMW, Microsoft, Google, Adobe faculty fellowships, and DE Logi grant. FS gratefully acknowledges support by the Ronald and Maxine Linde Institute of Economic and Management Sciences at Caltech. FS and HO gratefully acknowledge support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning) and the Office of Naval Research under award number N00014-18-1-2363.\n\nSubmitted - 2006.10179.pdf
", "abstract": "Constrained competitive optimization involves multiple agents trying to minimize conflicting objectives, subject to constraints. This is a highly expressive modeling language that subsumes most of modern machine learning. In this work we propose competitive mirror descent (CMD): a general method for solving such problems based on first order information that can be obtained by automatic differentiation. First, by adding Lagrange multipliers, we obtain a simplified constraint set with an associated Bregman potential. At each iteration, we then solve for the Nash equilibrium of a regularized bilinear approximation of the full problem to obtain a direction of movement of the agents. Finally, we obtain the next iterate by following this direction according to the dual geometry induced by the Bregman potential. By using the dual geometry we obtain feasible iterates despite only solving a linear system at each iteration, eliminating the need for projection steps while still accounting for the global nonlinear structure of the constraint set. As a special case we obtain a novel competitive multiplicative weights algorithm for problems on the positive cone.", "date": "2020-11-06", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20201106-120218966", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201106-120218966", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Bren Professor of Computing and Mathematical Sciences" }, { "agency": "Defense Advanced Research Projects Agency (DARPA)", "grant_number": "HR00111890035" }, { "agency": "Learning with Less Labels (LwLL)" }, { "agency": "BMW" }, { "agency": "Microsoft Faculty Fellowship" }, { "agency": "Google Faculty Research Award" }, { "agency": "Adobe" }, { "agency": "Caltech De Logi Fund" }, { "agency": "Linde Institute of Economic and Management Science" }, { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" }, { "agency": "Office of Naval Research (ONR)", "grant_number": "N00014-18-1-2363" } ] }, "doi": "10.48550/arXiv.2006.10179", "primary_object": { "basename": "2006.10179.pdf", "url": "https://authors.library.caltech.edu/records/tpvre-65819/files/2006.10179.pdf" }, "resource_type": "monograph", "pub_year": "2020", "author_list": "Sch\u00e4fer, Florian; Anandkumar, Anima; et el." }, { "id": "https://authors.library.caltech.edu/records/rsq9f-6w197", "eprint_id": 98806, "eprint_status": "archive", "datestamp": "2023-08-19 16:49:05", "lastmod": "2023-10-18 17:39:12", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-Clint", "name": { "family": "Scovel", "given": "Clint" }, "orcid": "0000-0001-7757-3411" }, { "id": "Yoo-Gene-Ryan", "name": { "family": "Yoo", "given": "Gene Ryan" }, "orcid": "0000-0002-5319-5599" } ] }, "title": "Kernel Mode Decomposition and programmable/interpretable regression networks", "ispublished": "unpub", "full_text_status": "public", "note": "The authors gratefully acknowledge support by the Air Force\nOffice of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning).\n\nSubmitted - 1907.08592.pdf
", "abstract": "Mode decomposition is a prototypical pattern recognition problem that can be addressed from the (a priori distinct) perspectives of numerical approximation, statistical inference and deep learning. Could its analysis through these combined perspectives be used as a Rosetta stone for deciphering mechanisms at play in deep learning? Motivated by this question we introduce programmable and interpretable regression networks for pattern recognition and address mode decomposition as a prototypical problem. The programming of these networks is achieved by assembling elementary modules decomposing and recomposing kernels and data. These elementary steps are repeated across levels of abstraction and interpreted from the equivalent perspectives of optimal recovery, game theory and Gaussian process regression (GPR). The prototypical mode/kernel decomposition module produces an optimal approximation (w\u2081,w\u2082,\u22ef,w_m) of an element (v\u2081,v\u2082,\u2026,v_m) of a product of Hilbert subspaces of a common Hilbert space from the observation of the sum v:=v\u2081+\u22ef+v_m. The prototypical mode/kernel recomposition module performs partial sums of the recovered modes w_i based on the alignment between each recovered mode w_i and the data v. We illustrate the proposed framework by programming regression networks approximating the modes v_i=a_i(t)y_i(\u03b8_i(t)) of a (possibly noisy) signal \u2211_iv_i when the amplitudes a_i, instantaneous phases \u03b8_i and periodic waveforms y_i may all be unknown and show near machine precision recovery under regularity and separation assumptions on the instantaneous amplitudes a_i and frequencies \u03b8_i. The structure of some of these networks share intriguing similarities with convolutional neural networks while being interpretable, programmable and amenable to theoretical analysis.", "date": "2019-09-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190923-153747161", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190923-153747161", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-18-1-0271" } ] }, "doi": "10.48550/arXiv.1907.08592", "primary_object": { "basename": "1907.08592.pdf", "url": "https://authors.library.caltech.edu/records/rsq9f-6w197/files/1907.08592.pdf" }, "resource_type": "monograph", "pub_year": "2019", "author_list": "Owhadi, Houman; Scovel, Clint; et el." }, { "id": "https://authors.library.caltech.edu/records/91xq4-c7r09", "eprint_id": 92171, "eprint_status": "archive", "datestamp": "2023-08-19 01:41:38", "lastmod": "2023-10-20 00:02:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Campbell-T", "name": { "family": "Campbell", "given": "Tom" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Sauvageauz-J", "name": { "family": "Sauvageauz", "given": "Joe" } }, { "id": "Watkinson-D", "name": { "family": "Watkinson", "given": "David" } } ] }, "title": "On testing the simulation theory", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Lorena Buitrago for her help with Figure 4. We also thank an anonymous referee whose detailed comments and suggestions have lead to significant improvements.\n\nSubmitted - 1703.00058.pdf
", "abstract": "Can the theory that reality is a simulation be tested? We investigate this question based on the assumption that if the system performing the simulation is finite (i.e. has limited resources), then to achieve low computational complexity, such a system would, as in a video game, render content (reality) only at the moment that information becomes available for observation by a player and not at the moment of detection by a machine (that would be part of the simulation and whose detection would also be part of the internal computation performed by the Virtual Reality server before rendering content to the player). Guided by this principle we describe conceptual wave/particle duality experiments aimed at testing the simulation theory.", "date": "2019-01-09", "date_type": "published", "id_number": "CaltechAUTHORS:20190109-112815344", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190109-112815344", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "collection": "CaltechAUTHORS", "doi": "10.48550/arXiv.1703.00058", "primary_object": { "basename": "1703.00058.pdf", "url": "https://authors.library.caltech.edu/records/91xq4-c7r09/files/1703.00058.pdf" }, "resource_type": "monograph", "pub_year": "2019", "author_list": "Campbell, Tom; Owhadi, Houman; et el." }, { "id": "https://authors.library.caltech.edu/records/hkp94-1wn28", "eprint_id": 78885, "eprint_status": "archive", "datestamp": "2023-08-19 03:15:31", "lastmod": "2023-10-26 14:25:48", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-C", "name": { "family": "Scovel", "given": "Clint" }, "orcid": "0000-0001-7757-3411" } ] }, "title": "Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 31 Mar 2017 (v1), last revised 29 May 2017 (this version, v2)) \n\nThe authors gratefully acknowledges this work supported by the Air Force Office of Scientific Research and the DARPA EQUiPS Program under award number FA9550-16-1-0054 (Computational Information Games).\n\nSubmitted - 1703.10761.pdf
", "abstract": "We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space B endowed with a quadratic norm \u2225\u22c5\u2225, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of u \u2208 B, given partial measurements [\u03d5_i,u] with \u03d5_i \u2208 B^\u2217, using relative error in \u2225\u22c5\u2225-norm as a loss) is a centered Gaussian field \u03be solely determined by the norm \u2225\u22c5\u2225, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm \u2225\u22c5\u2225 and induce a multi-resolution decomposition of B that is adapted to the eigensubspaces of the operator defining the norm \u2225\u22c5\u2225. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operators (including arbitrary continuous linear bijections from H^s_0 to H^(\u2212s) or to L^2) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to O(NpolylogN) solvers and eigenspace adapted Multiresolution Analysis (resulting in near linear complexity approximation of all eigensubspaces).", "date": "2017-07-10", "date_type": "published", "id_number": "CaltechAUTHORS:20170710-085210757", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170710-085210757", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-16-1-0054" }, { "agency": "Defense Advanced Research Projects Agency (DARPA)" } ] }, "doi": "10.48550/arXiv.1703.10761", "primary_object": { "basename": "1703.10761.pdf", "url": "https://authors.library.caltech.edu/records/hkp94-1wn28/files/1703.10761.pdf" }, "resource_type": "monograph", "pub_year": "2017", "author_list": "Owhadi, Houman and Scovel, Clint" }, { "id": "https://authors.library.caltech.edu/records/0171g-e5p13", "eprint_id": 64730, "eprint_status": "archive", "datestamp": "2023-08-19 21:08:07", "lastmod": "2023-10-17 21:49:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Bou-Rabee-N-M", "name": { "family": "Bou-Rabee", "given": "Nawaf" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Ballistic Transport at Uniform Temperature", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 8 Oct 2007 (v1), last revised 17 Oct 2007 (this version, v2). 17 February 2013.\n\nSubmitted - 0710.1565.pdf
", "abstract": "A paradigm for isothermal, mechanical rectification of stochastic uctuations is introduced in this paper. The central idea is to transform energy injected by random\nperturbations into rigid-body rotational kinetic energy. The prototype considered in this paper is a mechanical system consisting of a set of rigid bodies in interaction\nthrough magnetic fields. The system is stochastically forced by white noise and dissipative through mechanical friction. The Gibbs-Boltzmann distribution at a specific\ntemperature defines the unique invariant measure under the \nflow of this stochastic process and allows us to define \"the temperature\" of the system. This measure is also\nergodic and strongly mixing. Although the system does not exhibit global directed motion, it is shown that global ballistic motion is possible (the mean-squared displacement\ngrows like t^2). More precisely, although work cannot be extracted from thermal energy by the second law of thermodynamics, it is shown that ballistic transport from thermal energy is possible. In particular, the dynamics is characterized by a meta-stable state in which the system exhibits directed motion over random time scales. This phenomenon is caused by interaction of three attributes of the system: a non at (yet bounded) potential energy landscape, a rigid body effect (coupling translational momentum and angular momentum through friction) and the degeneracy of the noise/friction tensor on the momentums (the fact that noise is not applied to all degrees of freedom).", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-101438724", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-101438724", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.0710.1565", "primary_object": { "basename": "0710.1565.pdf", "url": "https://authors.library.caltech.edu/records/0171g-e5p13/files/0710.1565.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Bou-Rabee, Nawaf and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/bgn30-19p88", "eprint_id": 64713, "eprint_status": "archive", "datestamp": "2023-08-20 08:45:50", "lastmod": "2023-10-17 21:37:08", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-C", "name": { "family": "Scovel", "given": "Clint" }, "orcid": "0000-0001-7757-3411" } ] }, "title": "Brittleness of Bayesian inference and new Selberg formulas", "ispublished": "unpub", "full_text_status": "public", "keywords": "Bayesian inference, misspecification, robustness, uncertainty quantification, optimal uncertainty quantification, reproducing kernel Hilbert spaces (RKHS), Selberg integral formulas", "note": "(Submitted on 26 Apr 2013 (v1), last revised 24 Oct 2014 (this version, v2)). October 27, 2014. \n\nWe would like to thank G\u00e9rard Letac for his helpful comments, in particular for his substantial simplification, included here, of our previous proof of Lemma 4.1. We would also like to thank one of the referees for many helpful comments which we also feel improved the manuscript. The authors gratefully acknowledge this work supported by the Air Force Office of Scientific Research under Award Number FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators).\n\nSubmitted - 1304.7046.pdf
", "abstract": "The incorporation of priors in the Optimal Uncertainty Quantification (OUQ) framework reveals brittleness in Bayesian inference; a model may share an arbitrarily\nlarge number of finite-dimensional marginals with, or be arbitrarily close (in Prokhorov or total variation metrics) to, the data-generating distribution and still make the largest possible prediction error after conditioning on an arbitrarily large number of samples. The initial purpose of this paper is to unwrap this brittleness mechanism by providing (i) a quantitative version of the Brittleness Theorem of and (ii) a detailed and comprehensive analysis of its application to the revealing example of estimating the mean of a random variable on the unit interval [0, 1] using priors that exactly capture the distribution of an arbitrarily large number of Hausdorff moments. However, in doing so, we discovered that the free parameter associated with Markov and Kre\u0129n's canonical representations of truncated Hausdorff moments generates reproducing\nkernel identities corresponding to reproducing kernel Hilbert spaces of polynomials. Furthermore, these reproducing identities lead to biorthogonal systems of Selberg integral formulas.\nThis process of discovery appears to be generic: whereas Karlin and Shapley used Selberg's integral formula to first compute the volume of the Hausdorff moment space\n(the polytope defined by the first n moments of a probability measure on the interval [0, 1]), we observe that the computation of that volume along with higher order moments of the uniform measure on the moment space, using different finite-dimensional representations of subsets of the infinite-dimensional set of probability measures on [0, 1] representing the first n moments, leads to families of equalities corresponding to classical and new Selberg identities.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-073833523", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-073833523", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-12-1-0389" } ] }, "doi": "10.48550/arXiv.1304.7046", "primary_object": { "basename": "1304.7046.pdf", "url": "https://authors.library.caltech.edu/records/bgn30-19p88/files/1304.7046.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Owhadi, Houman and Scovel, Clint" }, { "id": "https://authors.library.caltech.edu/records/r4cp9-5jy65", "eprint_id": 64709, "eprint_status": "archive", "datestamp": "2023-08-20 08:06:33", "lastmod": "2023-10-17 21:36:55", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-C", "name": { "family": "Scovel", "given": "Clint" }, "orcid": "0000-0001-7757-3411" } ] }, "title": "Conditioning Gaussian measure on Hilbert space", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 13 Jun 2015 (v1), last revised 1 Sep 2015 (this version, v2)). September 2, 2015. \n\nThe authors gratefully acknowledge this work supported by the Air Force Office of Scientific Research under Award Number FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators).\n\nSubmitted - 1506.04208.pdf
", "abstract": "For a Gaussian measure on a separable Hilbert space with covariance operator C, we show that the family of conditional measures associated with conditioning on a\nclosed subspace S^\u22a5 are Gaussian with covariance operator the short S(C) of the operator C to S. We provide two proofs. The first uses the theory of Gaussian Hilbert\nspaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship\nbetween the shorted operator and C-symmetric oblique projections onto S^\u22a5. To obtain the assertion when such projections do not exist, we develop an approximation result for the shorted operator by showing, for any positive operator A, how to construct a sequence of approximating operators A^n which possess A^n- symmetric oblique projections onto S^\u22a5 such that the sequence of shorted operators S(A^n) converges to S(A) in the weak operator topology. This result combined with the martingale convergence of random variables associated with the corresponding approximations C^n establishes the main assertion in general. Moreover, it in turn strengthens the approximation theorem for shorted operator when the operator is trace class; then the sequence of shorted operators S(A^n) converges to S(A) in trace norm.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-065740350", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-065740350", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Air Force Office of Scientific Research (AFOSR)", "grant_number": "FA9550-12-1-0389" } ] }, "doi": "10.48550/arXiv.1506.04208", "primary_object": { "basename": "1506.04208.pdf", "url": "https://authors.library.caltech.edu/records/r4cp9-5jy65/files/1506.04208.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Owhadi, Houman and Scovel, Clint" }, { "id": "https://authors.library.caltech.edu/records/vbh75-mwe93", "eprint_id": 64721, "eprint_status": "archive", "datestamp": "2023-08-19 03:53:38", "lastmod": "2023-10-17 21:37:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Sullivan-T-J", "name": { "family": "Sullivan", "given": "T. J." } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "H." }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Equivalence of concentration inequalities for linear and non-linear functions", "ispublished": "unpub", "full_text_status": "public", "keywords": "words and phrases. concentration of measure, large deviations, quasiconvexity, normal\ndistance.", "note": "(Submitted on 24 Sep 2010). Date: September 27, 2010. \n\nThe authors acknowledge portions of this work supported by the United States Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613 through the California Institute of Technology's ASC/PSAAP Center for the Predictive Modeling and Simulation of High Energy Density Dynamic Response of Materials.\n\nSubmitted - 1009.4913.pdf
", "abstract": "We consider a random variable X that takes values in a (possibly infinite-dimensional) topological vector space X. We show that, with respect to an appropriate \"normal distance\" on X, concentration inequalities for linear and non-linear functions of X are equivalent. This normal distance corresponds naturally to the concentration rate in classical concentration results such as Gaussian concentration and concentration on the Euclidean and Hamming cubes. Under suitable assumptions on the roundness of the sets of interest, the concentration inequalities so obtained are asymptotically optimal in the high-dimensional limit.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-082333411", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-082333411", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE) National Nuclear Security Administration", "grant_number": "DE-FC52-08NA28613" } ] }, "doi": "10.48550/arXiv.1009.4913", "primary_object": { "basename": "1009.4913.pdf", "url": "https://authors.library.caltech.edu/records/vbh75-mwe93/files/1009.4913.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Sullivan, T. J. and Owhadi, H." }, { "id": "https://authors.library.caltech.edu/records/ska9y-fwf53", "eprint_id": 64731, "eprint_status": "archive", "datestamp": "2023-08-19 19:32:46", "lastmod": "2023-10-17 21:49:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Bou-Rabee-N-M", "name": { "family": "Bou-Rabee", "given": "Nawaf" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 23 Oct 2007 (v1), last revised 10 Apr 2008 (this version, v4). February 16, 2013.\n\nSubmitted - 0710.4259.pdf
", "abstract": "This paper introduces a geometric method for proving ergodicity of degenerate noise driven stochastic processes. The driving noise is assumed to be an arbitrary Levy process with non-degenerate diffusion component (but that may be applied to a single degree of freedom of the system). The geometric conditions are the approximate controllability of the process the fact that there exists a point in the phase space where the interior of the image of a point via a secondarily randomized version of the driving noise is non void. The paper applies the method to prove ergodicity of a sliding disk governed by Langevin-type equations (a simple stochastic rigid body system). The paper shows that a key feature of this Langevin process is that even though the diffusion and drift matrices associated to the momentums are degenerate, the system is still at uniform temperature.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-103320707", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-103320707", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.0710.4259", "primary_object": { "basename": "0710.4259.pdf", "url": "https://authors.library.caltech.edu/records/ska9y-fwf53/files/0710.4259.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Bou-Rabee, Nawaf and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/v5dr0-dv971", "eprint_id": 64728, "eprint_status": "archive", "datestamp": "2023-08-19 16:39:36", "lastmod": "2023-10-17 21:37:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Zhang-Lei", "name": { "family": "Zhang", "given": "Lei" }, "orcid": "0000-0001-9031-4318" } ] }, "title": "Metric based up-scaling", "ispublished": "unpub", "full_text_status": "public", "keywords": "Multi scale problem, compensation, homogenization, multi-fractal, numerical homogenization, compression", "note": "(Submitted on 11 May 2005 (v1), last revised 16 Nov 2005 (this version, v5)). November 15, 2005. \n\nPart of the work of the first author has been supported by CNRS. The authors would like to thank Jean-Michel Roquejoffre for indicating us the correct references on nonlinear PDEs, Mathieu Desbrun for enlightening discussions on discrete exterior calculus [58] (a powerful tool that has put into evidence the intrinsic way to define discrete differential operators on irregular triangulations), Tom Hou and Jerry Marsden for stimulating discussions on multi-scale computation, Clothilde Melot and St\u00e9phane Jaffard for stimulating discussions on multi-fractal analysis. Thanks are also due to Lexing Ying and Laurent Demanet for useful comments on the manuscript and G. Ben Arous for indicating us reference [72]. Many thanks are also due to Stefan M\u00fcller (MPI, Leipzig) for valuable suggestions and for indicating us the Hierarchical Matrices methods. We would like also to thank G. Allaire, F. Murat and S.R.S. Varadhan for stimulating discussions at the CIRM workshop on random homogenization and P. Schr\u00f6der for stimulating discussions on splines based methods. We also thank an anonymous referee for detailed comments and suggestions.\n\nSubmitted - 0505223.pdf
", "abstract": "We consider divergence form elliptic operators in dimension n \u2265 2 with L\u221e coefficients. Although solutions of these operators are only H\u00f6lder continuous, we show that they are differentiable (C1,\u03b1) with respect to harmonic\ncoordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition\nto traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-094627612", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-094627612", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Centre National de la Recherche Scientifique (CNRS)" } ] }, "doi": "10.48550/arXiv.0505223", "primary_object": { "basename": "0505223.pdf", "url": "https://authors.library.caltech.edu/records/v5dr0-dv971/files/0505223.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Owhadi, Houman and Zhang, Lei" }, { "id": "https://authors.library.caltech.edu/records/je0wb-97q10", "eprint_id": 64729, "eprint_status": "archive", "datestamp": "2023-08-19 20:59:54", "lastmod": "2023-10-17 21:37:47", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Bou-Rabee-N-M", "name": { "family": "Bou-Rabee", "given": "Nawaf" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 14 Sep 2007 (v1), last revised 23 Sep 2007 (this version, v2). December 18, 2013. \n\nWe wish to acknowledge Jerry Marsden, Sigrid Leyendecker, and Katie Whitehead for useful suggestions on this paper.\n\nSubmitted - 0709.2222.pdf
", "abstract": "Stochastic variational integrators for constrained, stochastic mechanical systems are developed in this paper. The main results of the paper are twofold: an equivalence is established between a stochastic Hamilton-Pontryagin (HP) principle in generalized coordinates and constrained coordinates via Lagrange multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are extended to this class of systems. Among these integrators are first and second-order strongly convergent RATTLE-type integrators. We prove strong order of accuracy of the methods provided. The paper also reviews the deterministic treatment of VPRK integrators from the HP viewpoint.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-100431574", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-100431574", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.0709.2222", "primary_object": { "basename": "0709.2222.pdf", "url": "https://authors.library.caltech.edu/records/je0wb-97q10/files/0709.2222.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Bou-Rabee, Nawaf and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/0qces-vpr96", "eprint_id": 64725, "eprint_status": "archive", "datestamp": "2023-08-19 02:53:20", "lastmod": "2023-10-17 21:37:35", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Tao-Molei", "name": { "family": "Tao", "given": "Molei" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Marsden-J-E", "name": { "family": "Marsden", "given": "Jerrold E." } } ] }, "title": "Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 23 Jun 2010). June 25, 2010.\n\nThis work is supported by NSF grant CMMI-092600. We thank J. M. Sanz-Serna for useful comments.\n\nSubmitted - 1006.4657.pdf
", "abstract": "Impulse methods are generalized to a family of integrators for Langevin systems with quadratic stiff potentials and arbitrary soft potentials. Uniform error bounds (independent\nfrom stiff parameters) are obtained on integrated positions allowing for coarse integration steps. The resulting integrators are explicit and structure preserving (quasi-symplectic for Langevin systems).", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-085934570", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-085934570", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CMMI-092600" } ] }, "doi": "10.48550/arXiv.1006.4657", "primary_object": { "basename": "1006.4657.pdf", "url": "https://authors.library.caltech.edu/records/0qces-vpr96/files/1006.4657.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Tao, Molei; Owhadi, Houman; et el." }, { "id": "https://authors.library.caltech.edu/records/js4rg-7dp08", "eprint_id": 64726, "eprint_status": "archive", "datestamp": "2023-08-19 03:08:00", "lastmod": "2023-10-17 21:37:37", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Tao-Molei", "name": { "family": "Tao", "given": "Molei" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Marsden-J-E", "name": { "family": "Marsden", "given": "Jerrold E." } } ] }, "title": "Temperature and Friction Accelerated Sampling of Boltzmann-Gibbs Distribution", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 6 Jul 2010). July 8, 2010. \n\nThis work is supported by NSF grant CMMI-092600. We are grateful to James L. Beck and Konstantin Zuev for insightful discussions.\n\nSubmitted - 1007.0995.pdf
", "abstract": "This paper is concerned with tuning friction and temperature in Langevin dynamics for fast sampling from the canonical ensemble. We show that near-optimal acceleration is achieved by choosing friction so that the local quadratic approximation of the Hamiltonian is a critical damped oscillator. The system is also over-heated and cooled down to its final temperature. The performances of different cooling schedules are analyzed as functions of total simulation time.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-090906859", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-090906859", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CMMI-092600" } ] }, "doi": "10.48550/arXiv.1007.0995", "primary_object": { "basename": "1007.0995.pdf", "url": "https://authors.library.caltech.edu/records/js4rg-7dp08/files/1007.0995.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "Tao, Molei; Owhadi, Houman; et el." }, { "id": "https://authors.library.caltech.edu/records/qffgv-kme54", "eprint_id": 64716, "eprint_status": "archive", "datestamp": "2023-08-19 09:52:55", "lastmod": "2023-10-17 21:37:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "McKerns-M", "name": { "family": "McKerns", "given": "M." } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "H." }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-C", "name": { "family": "Scovel", "given": "C." }, "orcid": "0000-0001-7757-3411" }, { "id": "Sullivan-T-J", "name": { "family": "Sullivan", "given": "T. J." } }, { "id": "Ortiz-M", "name": { "family": "Ortiz", "given": "M." }, "orcid": "0000-0001-5877-4824" } ] }, "title": "The optimal uncertainty algorithm in the mystic framework", "ispublished": "unpub", "full_text_status": "public", "note": "August 21, 2010. (Submitted on 6 Feb 2012). \n\nThe authors gratefully acknowledge portions of this work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613 and by the National Science Foundation under Award Number DMR-0520547.\n\nSubmitted - 1202.1055.pdf
", "abstract": "We have recently proposed a rigorous framework for Uncertainty Quantification (UQ) in which UQ objectives and assumption/information set are brought into\nthe forefront, providing a framework for the communication and comparison of UQ\nresults. In particular, this framework does not implicitly impose inappropriate assumptions nor does it repudiate relevant information.\nThis framework, which we call Optimal Uncertainty Quantification (OUQ), is\nbased on the observation that given a set of assumptions and information, there\nexist bounds on uncertainties obtained as values of optimization problems and that\nthese bounds are optimal. It provides a uniform environment for the optimal solution of the problems of validation, certification, experimental design, reduced order\nmodeling, prediction, extrapolation, all under aleatoric and epistemic uncertainties.\nOUQ optimization problems are extremely large, and even though under general\nconditions they have finite-dimensional reductions, they must often be solved numerically. This general algorithmic framework for OUQ has been implemented in the\nmystic optimization framework. We describe this implementation, and demonstrate\nits use in the context of the Caltech surrogate model for hypervelocity impact.", "date": "2016-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20160224-080348129", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160224-080348129", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE) National Nuclear Security Administration", "grant_number": "DE-FC52-08NA28613" }, { "agency": "NSF", "grant_number": "DMR-0520547" } ] }, "doi": "10.48550/arXiv.1202.1055", "primary_object": { "basename": "1202.1055.pdf", "url": "https://authors.library.caltech.edu/records/qffgv-kme54/files/1202.1055.pdf" }, "resource_type": "monograph", "pub_year": "2016", "author_list": "McKerns, M.; Owhadi, H.; et el." }, { "id": "https://authors.library.caltech.edu/records/36dys-2ya70", "eprint_id": 27172, "eprint_status": "archive", "datestamp": "2023-08-20 02:26:25", "lastmod": "2024-01-13 05:43:07", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Desbrun-M", "name": { "family": "Desbrun", "given": "Mathieu" }, "orcid": "0000-0003-3424-6079" }, { "id": "Donaldson-R-D", "name": { "family": "Donaldson", "given": "Roger D." } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Discrete Geometric Structures in Homogenization and Inverse Homogenization with Application to EIT", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted - Caltech_ACM_TR_2009_02.pdf
", "abstract": "We introduce a new geometric approach for the homogenization and\ninverse homogenization of the divergence form elliptic operator with rough\nconductivity coefficients \u03c3(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free\nmatrices and convex functions s(x) over the domain \u03a9. Although homogenization is a non-linear and non-injective operator when applied directly\nto conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of \n\u03a9 when re-expressed using convex\nfunctions, and is a volume averaging operator when re-expressed with\ndivergence-free matrices. We explicitly give the transformations which\nmap conductivity coefficients into divergence-free matrices and convex\nfunctions, as well as their respective inverses. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we apply\nthis geometric framework to obtain an optimally robust homogenization\nalgorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy\nto weighted Delaunay triangulations. Next, we consider inverse homogenization, that is, the recovery of the microstructure from macroscopic\ninformation, a problem which is known to be both non-linear and severly\nill-posed. We show how to decompose this reconstruction into a linear ill-posed problem and a well-posed non-linear problem. We apply this new\ngeometric approach to Electrical Impedance Tomography (EIT) in dimension two. It is known that the EIT problem admits at most one isotropic\nsolution. If an isotropic solution exists, we show how to compute it from\nany conductivity having the same boundary \nDirichlet-to-Neumann map.\nThis is of practical importance since the EIT problem always admits a\nunique solution in the space of divergence-free matrices and is stable with\nrespect to G-convergence in that space (this property fails for isotropic\nmatrices). As such, we suggest that the space of convex functions is the\nnatural space to use to parameterize solutions of the EIT problem.", "date": "2011-10-19", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20111011-163848887", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111011-163848887", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Applied-&-Computational-Mathematics" } ] }, "doi": "10.7907/XR8W-EA85", "primary_object": { "basename": "Caltech_ACM_TR_2009_02.pdf", "url": "https://authors.library.caltech.edu/records/36dys-2ya70/files/Caltech_ACM_TR_2009_02.pdf" }, "resource_type": "monograph", "pub_year": "2011", "author_list": "Desbrun, Mathieu; Donaldson, Roger D.; et el." }, { "id": "https://authors.library.caltech.edu/records/ewe01-61q87", "eprint_id": 27185, "eprint_status": "archive", "datestamp": "2023-08-20 02:26:30", "lastmod": "2024-01-13 05:43:11", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Berlyand-L", "name": { "family": "Berlyand", "given": "Leonid" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" } ] }, "title": "Flux Norm Approach to Homogenization Problems with non-separated Scales", "ispublished": "unpub", "full_text_status": "public", "note": "Part of the research of H. Owhadi is supported by the National\nNuclear Security Administration through the Predictive Science Academic Alliance Program. The work of L. Berlyand is supported in part by NSF grant DMS-0708324 and\nDOE grant DE-FG02-08ER25862. We would like to thank L. Zhang for the computations associated with Figure 1. We also thank B. Haines, L. Zhang, and O. Misiats for\ncarefully reading the manuscript and providing useful suggestions. We would like to\nthank Bj\u00f6rn Engquist, Ivo Babu\u0161ka and John Osborn for useful comments and showing us related and missing references. We are also greatly in debt to Ivo Babu\u0161ka and\nJohn Osborn for carefully reading the manuscript and providing us with very detailed\ncomments and references which have lead to substantial changes.", "abstract": "We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L^\u221e(\u03a9\u00ad), \u00ad\u03a9 \u2282 \u211d^d ) coefficients a(x) that, in particular,\nmodel media with non-separated scales and high contrast in material properties.\nWhile the homogenization of PDEs with periodic or ergodic coefficients and well\nseparated scales is now well understood, we consider here the most general case\nof arbitrary bounded coefficients. For such problems we introduce explicit finite\ndimensional approximations of solutions with controlled error estimates, which we\nrefer to as homogenization approximations. In particular, this approach allows one\nto analyze a given medium directly without introducing the mathematical concept\nof an \u2208 family of media as in classical periodic homogenization. We define the flux\nnorm as the L^2 norm of the potential part of the fluxes of solutions, which is equivalent to the usual H^1-norm. We show that in the flux norm, the error associated with\napproximating, in a properly defined finite-dimensional space, the set of solutions\nof the aforementioned PDEs with rough coefficients is equal to the error associated\nwith approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property\nas the transfer property. A simple application of this property is the construction\nof finite dimensional approximation spaces with errors independent of the regularity\nand contrast of the coefficients and with optimal and explicit convergence rates.\nThis transfer property also provides an alternative to the global harmonic change\nof coordinates for the homogenization of elliptic operators that can be extended to\nelasticity equations. The proofs of these homogenization results are based on a new\nclass of elliptic inequalities which play the same role in our approach as the div-curl\nlemma in classical homogenization.", "date": "2011-10-19", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20111012-105135181", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111012-105135181", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "National Nuclear Security Administration" }, { "agency": "NSF", "grant_number": "DMS-0708324" }, { "agency": "DOE", "grant_number": "DE-FG02-08ER25862" } ] }, "local_group": { "items": [ { "id": "Applied-&-Computational-Mathematics" } ] }, "doi": "10.7907/T5DC-SN48", "primary_object": { "basename": "Caltech_ACM_TR_2009_03.pdf", "url": "https://authors.library.caltech.edu/records/ewe01-61q87/files/Caltech_ACM_TR_2009_03.pdf" }, "resource_type": "monograph", "pub_year": "2011", "author_list": "Berlyand, Leonid and Owhadi, Houman" }, { "id": "https://authors.library.caltech.edu/records/2nmzy-qwc92", "eprint_id": 27193, "eprint_status": "archive", "datestamp": "2023-08-19 04:17:02", "lastmod": "2024-01-13 05:43:21", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Zhang-Lei", "name": { "family": "Zhang", "given": "Lei" }, "orcid": "0000-0001-9031-4318" } ] }, "title": "Localized bases for finite dimensional homogenization\n approximations with non-separated scales and high-contrast", "ispublished": "unpub", "full_text_status": "public", "note": "We thank L. Berlyand for stimulating discussions. We also\nthank Ivo Babu\u0161ka, John Osborn, George Papanicolaou and Bj\u00f6rn Engquist for pointing\nus in the direction of the localization problem. The work of H. Owhadi is partially supported\nby the National Science Foundation under Award Number CMMI-092600 and the\nDepartment of Energy National Nuclear Security Administration under Award Number\nDE-FC52-08NA28613. We thank Sydney Garstang for proofreading the manuscript.\n\nPublished - Caltech_ACM_TR_2010_04.pdf
", "abstract": "We construct finite-dimensional approximations of solution spaces of divergence\nform operators with L^\u221e-coefficients. Our method does not rely on concepts of\nergodicity or scale-separation, but on the property that the solution of space of\nthese operators is compactly embedded in H^1 if source terms are in the unit ball\nof L^2 instead of the unit ball of H^\u22121. Approximation spaces are generated by\nsolving elliptic PDEs on localized sub-domains with source terms corresponding to\napproximation bases for H^2. The H^1-error estimates show that O(h^\u2212d)-dimensional\nspaces with basis elements localized to sub-domains of diameter O(h^\u221e ln 1/h) (with \u03b1 \u2208 [1/2 , 1)) result in an O(h^(2\u22122\u03b1) accuracy for elliptic, parabolic and hyperbolic problems.\nFor high-contrast media, the accuracy of the method is preserved provided that\nlocalized sub-domains contain buffer zones of width O(h^\u03b1 ln 1/h ) where the contrast\nof the medium remains bounded. The proposed method can naturally be generalized\nto vectorial equations (such as elasto-dynamics).", "date": "2011-10-19", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20111012-113719601", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111012-113719601", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CMMI-092600" }, { "agency": "DOE", "grant_number": "DE-FC52-08NA28613" } ] }, "local_group": { "items": [ { "id": "Applied-&-Computational-Mathematics" } ] }, "primary_object": { "basename": "Caltech_ACM_TR_2010_04.pdf", "url": "https://authors.library.caltech.edu/records/2nmzy-qwc92/files/Caltech_ACM_TR_2010_04.pdf" }, "resource_type": "monograph", "pub_year": "2011", "author_list": "Owhadi, Houman and Zhang, Lei" }, { "id": "https://authors.library.caltech.edu/records/vgmnz-ees44", "eprint_id": 27186, "eprint_status": "archive", "datestamp": "2023-08-20 02:26:35", "lastmod": "2024-01-13 05:43:13", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Tao-Molei", "name": { "family": "Tao", "given": "Molei" } }, { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "Houman" }, "orcid": "0000-0002-5677-1600" }, { "id": "Marsden-J-E", "name": { "family": "Marsden", "given": "Jerrold E." } } ] }, "title": "Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted - Caltech_ACM_TR_2009_04.pdf
", "abstract": "We introduce a new class of integrators for stiff ODEs as well as SDEs. An\nexample of subclass of systems that we treat are ODEs and SDEs that are sums of\ntwo terms one of which has large coefficients. These integrators are (i) Multiscale:\nthey are based on \now averaging and so do not resolve the fast variables but rather\nemploy step-sizes determined by slow variables (ii) Basis: the method is based on\naveraging the \now of the given dynamical system (which may have hidden slow and\nfast processes) instead of averaging the instantaneous drift of assumed separated\nslow and fast processes. This bypasses the need for identifying explicitly (or numerically)\nthe slow or fast variables. (iii) Non intrusive: A pre-existing numerical\nscheme resolving the microscopic time scale can be used as a black box and turned\ninto one of the integrators in this paper by simply turning the large coefficients on\nover a microscopic timescale and off during a mesoscopic timescale. (iv) Convergent\nover two scales: strongly over slow processes and in the sense of measures over fast\nones. We introduce the related notion of two scale \now convergence and analyze\nthe convergence of these integrators under the induced topology. (v) Structure preserving: For stiff Hamiltonian systems (possibly on manifolds), they are symplectic,\ntime-reversible, and symmetric (under the group action leaving the Hamiltonian invariant)\nin all variables. They are explicit and apply to arbitrary stiff potentials\n(that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems\nshows accuracy and stability over 4 orders of magnitude of time scales. For\nstiff Langevin equations, they are symmetric (under a group action), time-reversible\nand Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally\nsymplectic with isotropic friction.", "date": "2011-10-19", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20111012-110532817", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111012-110532817", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Applied-&-Computational-Mathematics" } ] }, "doi": "10.7907/QZNP-SR14", "primary_object": { "basename": "Caltech_ACM_TR_2009_04.pdf", "url": "https://authors.library.caltech.edu/records/vgmnz-ees44/files/Caltech_ACM_TR_2009_04.pdf" }, "resource_type": "monograph", "pub_year": "2011", "author_list": "Tao, Molei; Owhadi, Houman; et el." }, { "id": "https://authors.library.caltech.edu/records/5j8b4-b5n05", "eprint_id": 27192, "eprint_status": "archive", "datestamp": "2023-08-19 03:41:28", "lastmod": "2024-01-13 05:43:19", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Owhadi-H", "name": { "family": "Owhadi", "given": "H." }, "orcid": "0000-0002-5677-1600" }, { "id": "Scovel-C", "name": { "family": "Scovel", "given": "C." }, "orcid": "0000-0001-7757-3411" }, { "id": "Sullivan-T-J", "name": { "family": "Sullivan", "given": "T. J." } }, { "id": "McKerns-M", "name": { "family": "McKerns", "given": "M." } }, { "id": "Ortiz-M", "name": { "family": "Ortiz", "given": "M." }, "orcid": "0000-0001-5877-4824" } ] }, "title": "Optimal Uncertainty Quantification", "ispublished": "unpub", "full_text_status": "public", "note": "The authors gratefully acknowledge portions of this work supported by the Department\nof Energy National Nuclear Security Administration under Award Number DE-FC52-\n08NA28613 through Caltech's ASC/PSAAP Center for the Predictive Modeling and\nSimulation of High Energy Density Dynamic Response of Materials. Calculations for this\npaper were performed using the mystic optimization framework [33]. We thank the Caltech\nPSAAP Experimental Science Group \u2014 Marc Adams, Leslie Lamberson, Jonathan\nMihaly, Laurence Bodelot, Justin Brown, Addis Kidane, Anna Pandolfi, Guruswami\nRavichandran and Ares Rosakis \u2014 for Formula (5.1). We thank Sydney Garstang for\nproofreading the manuscript.\n\nSubmitted - Caltech_ACM_TR_2010_03.pdf
", "abstract": "We propose a rigorous framework for Uncertainty Quantification (UQ) in which\nthe UQ objectives and the assumptions/information set are brought to the forefront.\nThis framework, which we call Optimal Uncertainty Quantification (OUQ), is based\non the observation that, given a set of assumptions and information about the problem,\nthere exist optimal bounds on uncertainties: these are obtained as extreme\nvalues of well-defined optimization problems corresponding to extremizing probabilities\nof failure, or of deviations, subject to the constraints imposed by the scenarios\ncompatible with the assumptions and information. In particular, this framework\ndoes not implicitly impose inappropriate assumptions, nor does it repudiate relevant\ninformation.\nAlthough OUQ optimization problems are extremely large, we show that under\ngeneral conditions, they have finite-dimensional reductions. As an application,\nwe develop Optimal Concentration Inequalities (OCI) of Hoeffding and McDiarmid\ntype. Surprisingly, contrary to the classical sensitivity analysis paradigm, these results\nshow that uncertainties in input parameters do not necessarily propagate to\noutput uncertainties.\nIn addition, a general algorithmic framework is developed for OUQ and is tested\non the Caltech surrogate model for hypervelocity impact, suggesting the feasibility\nof the framework for important complex systems.", "date": "2011-10-19", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20111012-113158874", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111012-113158874", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE) National Nuclear Security Administration", "grant_number": "DE-FC52-08NA28613" } ] }, "local_group": { "items": [ { "id": "Applied-&-Computational-Mathematics" } ] }, "doi": "10.7907/TTW6-QD19", "primary_object": { "basename": "Caltech_ACM_TR_2010_03.pdf", "url": "https://authors.library.caltech.edu/records/5j8b4-b5n05/files/Caltech_ACM_TR_2010_03.pdf" }, "resource_type": "monograph", "pub_year": "2011", "author_list": "Owhadi, H.; Scovel, C.; et el." } ]