Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems

Hansen, Markus; Schwab, Christoph

doi:10.1007/s10013-013-0011-9

Cited by 37 publications

(58 citation statements)

References 19 publications

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“…Since s > K/2, the Sobolev Embedding Theorem furthermore implies that G and Φ are continuous. Examples of model problems satisfying Assumption 2.2 include linear elliptic and parabolic partial differential equations [10,38] and non-linear ordinary differential equations [42,17]. A specific example is given in section 5.…”

Section: Bayesian Inverse Problemsmentioning

confidence: 99%

Posterior consistency for Gaussian process approximations of Bayesian posterior distributions

2017

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We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings.

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Section: Bayesian Inverse Problemsmentioning

confidence: 99%

Posterior consistency for Gaussian process approximations of Bayesian posterior distributions

2017

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“…Importantly, it has been shown in [11] that the sets Λ n achieving the convergence rate (1.10) for the problem (1.7) can be chosen from the restricted class of monotone subsets of F. While we develop, as in [11], the algorithms and theory for (1.7), we hasten to add that all results and algorithms presented in the present paper apply, without any modifications, to the adaptive numerical solution of more general parametric equations: all that is required is bounded invertibility of the parametric equation for all instances of the parameter sequene and a characterization of the parametric solution families' dependence on the parameters in the sequence. Such characterizations seem to hold for broad classes of parametric problems (we refer to [22,23,20,21] for details).…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, various strategies have been proposed for the numerical treatment of parametric partial differential equations [1,3,5,4,7,11,12,13,17,18,21,22,23,24,28,27,30,31]. Such equations have the general form D(u, y) = 0, (1.1) 1 where u → D(u, y) is a partial differential operator that depends on d parameters represented by the vector y = (y 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…We point out that the results in this paper apply to a wide class of parametric problems. Problem (1.7) is chosen for the sake of illustration only; regularity estimates for the parametric solution analogous to those used in the present paper have been obtained for nonlinear, elliptic PDE in [20], for nonlinear, initial value ODEs in [21], and for parametric, parabolic evolution problems in [22], and for parametric wave equations in [23]. We also note that regularity for elliptic systems with operators depending on the parameter sequence y have been considered in [29].…”

Section: Introductionmentioning

confidence: 99%

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High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

2013

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International audienceWe consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalized polynomial chaos discretizations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on standard principle of tensorization of a one dimensional interpolation scheme and sparsifi-cation. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in [11] that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's

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“…The results in the mentioned references were mostly (with the exception of [34,15]) for model linear parametric elliptic diffusion problems. An extension of the convergence analysis to generic parametric solution families taking values in (separable) Hilbert spaces was developed in [43]: the focus of this work is on the collocation error analysis in the parameter space.…”

Section: Previous Approximation Results For Parametric Solution Familiesmentioning

confidence: 99%

Multilevel approximation of parametric and stochastic PDES

Zech

Dũng

Schwab

2019

Math. Models Methods Appl. Sci.

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We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Under the provision of a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric solutions are obtained as solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, Journ. Math. Pures et Appliquees 103(2) 400-428 (2015)] by means of stable, finite dimensional approximations, for example nonlinear Petrov-Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The results considerably generalize several earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov-Galerkin and discrete Petrov-Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel, general computational strategy to localize sequences of nested index sets is given for the anisotropic Smolyak scheme realizing best n-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in gpc expansions due to symmetries of the probability measure [J. Zech and Ch. Schwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich].Several examples illustrating the abstract theory include domain uncertainty quantification ("UQ" for short) for general, linear, second order, elliptic advection-reaction-diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. For these equations, we also consider a combined sparse-grid scheme in physical and parameter space, affording complexity similar to the recent multiindex stochastic collocation approach. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems. ). 1 2 Computational implementation of interpolatory or collocation approximations in the parameter domain requires, as a rule, also discretization of the corresponding operator equation. Here, sparsity of sequences of discretization schemes has been identified as a crucial ingredient in viable numerical approximation schemes; cases in point are multilevel Monte Carlo Methods (see, e.g., [27] and references therein), and generalized spar...

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Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems

Cited by 37 publications

References 19 publications

Posterior consistency for Gaussian process approximations of Bayesian posterior distributions

Posterior consistency for Gaussian process approximations of Bayesian posterior distributions

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

Multilevel approximation of parametric and stochastic PDES

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