Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Bachmayr, Markus; Cohen, Albert; Dũng, Đinh; Schwab, Christoph

doi:10.1137/17m111626x

Cited by 37 publications

(76 citation statements)

References 20 publications

(45 reference statements)

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“…We point out that the additional spacial regularity of the parametric solution family {u(y) : y ∈ U } ⊂ X s will, in general, entail reduced summability of the • X s -norms of the gpc coefficients. This was observed in previous work [28,2] to be essential for improved convergence rates when measured in terms of the total number of degrees of freedom. The following theorem accounts for that by providing two summability indices p 0 and p 1 for X resp., for X s -regularity.…”

Section: Auxiliary Results On Interpolation Of Sequence Spacessupporting

confidence: 56%

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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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Section: Auxiliary Results On Interpolation Of Sequence Spacessupporting

confidence: 56%

“…As is well-known by now, see e.g. [2], multilevel approximations of parametric solution families involve space discretization of instances of the parametric solution, incurring a discretization error. To bound it, we require "spacial regularity" of parametric solutions.…”

Section: Auxiliary Results On Interpolation Of Sequence Spacesmentioning

confidence: 99%

Multilevel approximation of parametric and stochastic PDES

Zech

Dũng

Schwab

2019

Math. Models Methods Appl. Sci.

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“…Moreover, the results in Figure 3(a) also demonstrate that in the present case with m = 1, one indeed only obtains u ∈ A s (S ×F) with s ≈ 2 3 α. In other words, the statement in Proposition 6.6(iv), shown in [3], appears to be sharp also for m = 1. This is a surprising difference to the corresponding results for m = 2, 3 with s up to α m , which are necessarily sharp.…”

Section: Anisotropic Dependence On Infinitely Many Parametersmentioning

confidence: 74%

Parametric PDEs: sparse or low-rank approximations?

Bachmayr

Cohen

Dahmen

2017

IMA Journal of Numerical Analysis

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We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations separating spatial and parametric variables, and hierarchical tensor decompositions separating all variables. We describe corresponding adaptive algorithms based on a common generic template and show their near-optimality with respect to natural approximability assumptions for each type of approximation. A central ingredient in the resulting bounds for the total computational complexity are new operator compression results for the case of infinitely many parameters. We conclude with a comparison of the complexity estimates based on the actual approximability properties of classes of parametric model problems, which shows that the computational costs of optimized low-rank expansions can be significantly lower or higher than those of sparse polynomial expansions, depending on the particular type of parametric problem.

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“…We remark that a related estimate to Theorem 4.6 has been derived in Theorem 6.1 of [6] without taking into account the spatial weight function.…”

Section: Discretizationmentioning

confidence: 94%

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Herrmann

Schwab

2019

ESAIM: M2AN

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We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ R d . The multilevel algorithm Q * L which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411-449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline-or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63-102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827-2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ R d , d = 2, 3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen-Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.Mathematics Subject Classification. 65D30, 65N30, 60G60.

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Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Cited by 37 publications

References 20 publications

Multilevel approximation of parametric and stochastic PDES

Multilevel approximation of parametric and stochastic PDES

Parametric PDEs: sparse or low-rank approximations?

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

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