arXiv subjects

Houman Owhadi

Publications and source records attributed to Houman Owhadi.

At least 91 records · Page 5Linked to original sources

arXiv2017

Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games

We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(Ω)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE (2) enable sparse compression of the solution space in $H^1_0(Ω)$ (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multigrid method is that of an inverted pyramid in which gamblets are computed locally (by virtue of their exponential decay), hierarchically (from fine to coarse scales) and the PDE is decomposed into a hierarchy of independent linear systems with uniformly bounded condition numbers. The resulting algorithm is parallelizable both in space (via localization) and in bandwith/subscale (subscales can be computed independently from each other). Although the method is deterministic it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.

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arXiv2016

Separability of reproducing kernel spaces

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

math.FA↗

arXiv2016

Qualitative Robustness in Bayesian Inference

The practical implementation of Bayesian inference requires numerical approximation when closed-form expressions are not available. What types of accuracy (convergence) of the numerical approximations guarantee robustness and what types do not? In particular, is the recursive application of Bayes' rule robust when subsequent data or posteriors are approximated? When the prior is the push forward of a distribution by the map induced by the solution of a PDE, in which norm should that solution be approximated? Motivated by such questions, we investigate the sensitivity of the distribution of posterior distributions (i.e. posterior distribution-valued random variables, randomized through the data) with respect to perturbations of the prior and data generating distributions in the limit when the number of data points grows towards infinity.

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arXiv2016

Extreme points of a ball about a measure with finite support

We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.

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arXiv2015

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.

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arXiv2015

Temporal homogenization of linear ODEs, with applications to parametric super-resonance and energy harvest

We consider the temporal homogenization of linear ODEs of the form $\dot{x}=Ax+εP(t)x+f(t)$, where $P(t)$ is periodic and $ε$ is small. Using a 2-scale expansion approach, we obtain the long-time approximation $x(t)\approx \exp(At) \left( Ω(t)+\int_0^t \exp(-A τ) f(τ) \, dτ\right)$, where $Ω$ solves the cell problem $\dotΩ=εB Ω+ εF(t)$ with an effective matrix $B$ and an explicitly-known $F(t)$. We provide necessary and sufficient condition for the accuracy of the approximation (over a $\mathcal{O}(ε^{-1})$ time-scale), and show how $B$ can be computed (at a cost independent of $ε$). As a direct application, we investigate the possibility of using RLC circuits to harvest the energy contained in small scale oscillations of ambient electromagnetic fields (such as Schumann resonances). Although a RLC circuit parametrically coupled to the field may achieve such energy extraction via parametric resonance, its resistance $R$ needs to be smaller than a threshold $κ$ proportional to the fluctuations of the field, thereby limiting practical applications. We show that if $n$ RLC circuits are appropriately coupled via mutual capacitances or inductances, then energy extraction can be achieved when the resistance of each circuit is smaller than $nκ$. Hence, if the resistance of each circuit has a non-zero fixed value, energy extraction can be made possible through the coupling of a sufficiently large number $n$ of circuits ($n\approx 1000$ for the first mode of Schumann resonances and contemporary values of capacitances, inductances and resistances). The theory is also applied to the control of the oscillation amplitude of a (damped) oscillator.

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arXiv2015

Conditioning Gaussian measure on Hilbert space

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 closed subspace $S^{\perp}$ are Gaussian with covariance operator the short $\mathcal{S}(C)$ of the operator $C$ to $S$. We provide two proofs. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and $C$-symmetric oblique projections onto $S^{\perp}$. 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 $S^{\perp}$ such that the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ 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 $\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in trace norm.

math.PR↗

arXiv2015

Bayesian Numerical Homogenization

Numerical homogenization, i.e. the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian Inference problem in which a given PDE with rough coefficients (or multi-scale operator) is excited with noise (random right hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how Rough Polyharmonic Splines can be re-discovered as the optimal solution of a Gaussian filtering problem.

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arXiv2015

Convex Optimal Uncertainty Quantification

Optimal uncertainty quantification (OUQ) is a framework for numerical extreme-case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution. This paper presents sufficient conditions under which an OUQ problem can be reformulated as a finite-dimensional convex optimization problem, for which efficient numerical solutions can be obtained. The sufficient conditions include that the objective function is piecewise concave and the constraints are piecewise convex. In particular, we show that piecewise concave objective functions may appear in applications where the objective is defined by the optimal value of a parameterized linear program.

math.OC↗

arXiv2015

On the Brittleness of Bayesian Inference

With the advent of high-performance computing, Bayesian methods are increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is a pressing question for which there currently exist positive and negative results. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they could be generically brittle when applied to continuous systems (and their discretizations) with finite information on the data-generating distribution. If closeness is defined in terms of the total variation metric or the matching of a finite system of generalized moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusions. The mechanism causing brittlenss/robustness suggests that learning and robustness are antagonistic requirements and raises the question of a missing stability condition for using Bayesian Inference in a continuous world under finite information.

math.ST↗

arXiv2015

Brittleness of Bayesian Inference Under Finite Information in a Continuous World

We derive, in the classical framework of Bayesian sensitivity analysis, optimal lower and upper bounds on posterior values obtained from Bayesian models that exactly capture an arbitrarily large number of finite-dimensional marginals of the data-generating distribution and/or that are as close as desired to the data-generating distribution in the Prokhorov or total variation metrics; these bounds show that such models may still make the largest possible prediction error after conditioning on an arbitrarily large number of sample data measured at finite precision. These results are obtained through the development of a reduction calculus for optimization problems over measures on spaces of measures. We use this calculus to investigate the mechanisms that generate brittleness/robustness and, in particular, we observe that learning and robustness are antagonistic properties. It is now well understood that the numerical resolution of PDEs requires the satisfaction of specific stability conditions. Is there a missing stability condition for using Bayesian inference in a continuous world under finite information?

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arXiv2014

Variational and linearly-implicit integrators, with applications

We show that symplectic and linearly-implicit integrators proposed by [Zhang and Skeel, 1997] are variational linearizations of Newmark methods. When used in conjunction with penalty methods (i.e., methods that replace constraints by stiff potentials), these integrators permit coarse time-stepping of holonomically constrained mechanical systems and bypass the resolution of nonlinear systems. Although penalty methods are widely employed, an explicit link to Lagrange multiplier approaches appears to be lacking; such a link is now provided (in the context of two-scale flow convergence [Tao, Owhadi and Marsden, 2010]). The variational formulation also allows efficient simulations of mechanical systems on Lie groups.

math.NA↗

arXiv2014

Brittleness of Bayesian inference and new Selberg formulas

The incorporation of priors in the Optimal Uncertainty Quantification (OUQ) framework \cite{OSSMO:2011} reveals brittleness in Bayesian inference; a model may share an arbitrarily large 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 \cite{BayesOUQ} 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\uın's canonical representations of truncated Hausdorff moments generates reproducing kernel identities corresponding to reproducing kernel Hilbert spaces of polynomials. Furthermore, these reproducing identities lead to biorthogonal systems of Selberg integral formulas. This 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 (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.

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arXiv2013

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator $-\diiv(a\nabla \cdot)$ and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

math.NA↗

arXiv2012

Control of a Model of DNA Division via Parametric Resonance

We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (i) first reducing its dynamic to two modes exchanging energy between each other and (ii) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system.

math.DS↗

arXiv2012

Optimal Uncertainty Quantification

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.

math.PR↗

arXiv2011

Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^α\ln \frac{1}{h})$ (with $α\in [1/2,1)$) result in an $\mathcal{O}(h^{2-2α})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^α\ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

math.NA↗

arXiv2011

From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials

We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way.

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