Sparse quadrature for high-dimensional integration with Gaussian measure

Chen, Peng

doi:10.1051/m2an/2018012

Cited by 26 publications

(49 citation statements)

References 53 publications

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“…where C < ∞ may depend on f as well as the univariate nodal sets. The line of proof we present here follows and builds upon the works [9,26,4]. We complement this convergence rate with a bound on the number of collocation points associated with a given multi-index set.…”

Section: Convergence Analysismentioning

confidence: 97%

“…These were recently extended by Bachmayr et al [4] using a different analytical approach employing a weighted 2 -summability of the coefficients of the Hermite expansion of the solution and their relation to partial derivatives. The theoretical tools provided in [4] enabled a convergence analysis for adaptive sparse quadrature [9] employing, e.g., Gauss-Hermite nodes for Banach space-valued functions of countably many Gaussian random variables.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we address the convergence of sparse polynomial collocation for functions of infinitely many Gaussian random variables, such as the solution to the lognormal diffusion problem (1). Specifically, we follow the approach of [4] and [9] to prove an algebraic convergence rate with respect to the number of grid points for sparse collocation based on Gauss-Hermite interpolation nodes in the case of countably many variables. In particular, the result applies to solutions u of (1) where a is a lognormal random field.…”

Section: Introductionmentioning

confidence: 99%

“…First, we outline in Subsection 3.1 the general approaches to prove algebraic convergence rates as they can be found in the works mentioned above. Later, we follow in Subsection 3.2 the approach of [4,9] and derive sufficient conditions for the underlying univariate interpolation nodes in order to obtain such rates when approximating "countably-variate" functions of certain smoothness. Finally, in Subsection 3.3 we verify these conditions for Gauss-Hermite nodes, provide bounds for the number of nodes in the resulting sparse grids, and state a convergence result with respect to this number.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Ernst¹,

Sprungk²,

Tamellini³

2018

SIAM J. Numer. Anal.

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We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general L 2 -convergence theory based on previous work by Bachmayr et al. [4] and Chen [9] and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

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Section: Convergence Analysismentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Ernst¹,

Sprungk²,

Tamellini³

2018

SIAM J. Numer. Anal.

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“…Third, further computational savings can be achieved in combination with model reduction for the state PDE and the linearized PDEs [77,29]. Fourth, one can replace the Monte Carlo estimator in the variance reduction by a sparse quadrature [31] to possibly achieve faster convergence of the approximation error. Fifth, application of the Taylor approximation and variance reduction in a multifidelity framework [63] may lead to additional computational saving.…”

Section: 2mentioning

confidence: 99%

Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty

Chen

Villa

Ghattas

2019

Journal of Computational Physics

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In this work we develop a scalable computational framework for the solution of PDE-constrained optimal control under high-dimensional uncertainty. Specifically, we consider a mean-variance formulation of the control objective and employ a Taylor expansion with respect to the uncertain parameter either to directly approximate the control objective or as a control variate for variance reduction. The expressions for the mean and variance of the Taylor approximation are known analytically, although their evaluation requires efficient computation of the trace of the (preconditioned) Hessian of the control objective. We propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the Hessian on a small number of random directions. Then, the computational work does not depend on the nominal dimension of the uncertain parameter, but depends only on the effective dimension (i.e., the rank of the preconditioned Hessian), thus ensuring scalability to high-dimensional problems. Moreover, to increase the estimation accuracy of the mean and variance of the control objective by the Taylor approximation, we use it as a control variate for variance reduction, which results in considerable computational savings (several orders of magnitude) compared to a plain Monte Carlo method. In summary, our approach amounts to solving an optimal control constrained by the original PDE and a set of linearized PDEs, which arise from the computation of the gradient and Hessian of the control objective with respect to the uncertain parameter. We use the Lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient-based optimization method to solve the problem. We demonstrate the accuracy, efficiency, and scalability of the proposed computational method for two examples with high-dimensional uncertain parameters: subsurface flow in a porous medium modeled as an elliptic PDE, and turbulent jet flow modeled by the Reynolds-averaged Navier-Stokes equations coupled with a nonlinear advection-diffusion equation characterizing model uncertainty. In particular, for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field.

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