Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems

Herrmann, Lukas; Keller, Magdalena; Schwab, Christoph

doi:10.1090/mcom/3615

Cited by 14 publications

(9 citation statements)

References 40 publications

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“…We remark that in [88], MC and QMC integration has been analyzed by real-variable arguments for such Gaussian priors. In [61], corresponding results have been obtain also for so-called Besov priors, again by real-variable arguments for the parametric posterior. Since the presently developed, quantified parametric holomorphy results are independent of the particular measure placed upon the parameter domain R ∞ .…”

Section: Parametric Posterior Analyticity and Sparsity In Bipsmentioning

confidence: 78%

“…Inserting into Θ(a) in (5.1), (5.2) this results in a countably-parametric density U ∋ y → Θ(a(y)), for y ∈ U = R ∞ , and the Gaussian reference measure π 0 on E in (5.1) is pushed forward into a countable Gaussian product measure on U : using (5.1) and choosing a Gaussian prior (e.g. [43,Section 2.4] or [61,79])…”

Section: Formulation and Well-posednessmentioning

confidence: 99%

“…The parametric density U → R in (5.3) which arises in Bayesian PDE inversion under Gaussian prior and also under more general, so-called Besov prior measures on U , see e.g. [43, Section 2.3], [61]. The parametric density y → φ(U (a(y)))Θ(a(y)) , inherits sparsity from the forward map y → U (a(y)), which sparsity is expressed as before in terms of summability properties of Wiener-Hermite PC expansion coefficients.…”

Section: Formulation and Well-posednessmentioning

confidence: 99%

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Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Dũng¹,

Schwab²,

Zech³

2022

Preprint

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We establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergenceform partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphybased argument allows a unified, "differentiation-free" sparsity analysis of Wiener-Hermite polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various constructive high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of anisotropic sparse-grid Hermite-Smolyak interpolation and quadrature in both forward and inverse computational uncertainty quantification.

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Section: Parametric Posterior Analyticity and Sparsity In Bipsmentioning

confidence: 78%

Section: Formulation and Well-posednessmentioning

confidence: 99%

Section: Formulation and Well-posednessmentioning

confidence: 99%

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Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Dũng¹,

Schwab²,

Zech³

2022

Preprint

Self Cite

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“…Moreover, for sufficiently smooth integrands it is possible to construct QMC rules with error bounds not depending on the number of stochastic variables while attaining faster convergence rates compared to Monte Carlo methods. For these reasons QMC methods have been very successful in applications to PDEs with random coefficients (see, e.g., [2,9,14,16,17,21,22,23,30,31,32,36,39,40]) and especially in PDE-constrained optimization under uncertainty, see [19,20]. In [29] the authors derive regularity results for the saddle point operator, which fall within the same framework as the QMC approximation of affine parametric operator equation setting considered in [40].…”

Section: Introductionmentioning

confidence: 99%

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

A.¹,

Kaarnioja²,

Kuo³

et al. 2022

Preprint

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We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem -and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

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“…These features make QMC methods ideal for heavy-duty uncertainty quantification compared to regular Monte Carlo methods (slow convergence rate), sparse grids (non-parallelizable) or approximations based on lower order moments (poor accuracy). QMC methods have been applied successfully to many problems such as Bayesian inverse problems [11,28,45,46], spectral eigenvalue problems [19], optimization under uncertainty [25,26], the Schrödinger equation [53], the wave equation [16], problems arising in quantum physics [27], and various others.…”

mentioning

confidence: 99%

Quasi-Monte Carlo and discontinuous Galerkin

Kaarnioja¹,

Rupp²

2022

Preprint

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In this study, we design and develop Quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficient. That is, we consider the affine and uniform, and the lognormal models for the input random field, and investigate the QMC cubatures to compute the response statistics (expectation and variance) of the discretized PDE. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Numerical results underline our analytical findings. Moreover, we present a novel analysis for the parametric regularity in the lognormal setting.

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Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems

Cited by 14 publications

References 40 publications

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

Quasi-Monte Carlo and discontinuous Galerkin

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