A weighted reduced basis method for parabolic PDEs with random data

Spannring, Christopher; Ullmann, Sebastian; Lang, Jens

doi:10.48550/arxiv.1712.07393

Cited by 3 publications

(2 citation statements)

References 7 publications

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“…A variant is the quasi-random sampling using low-discrepancy sequences [55,59], such as Halton or Sobol sequence, which tends to provide more equidistributed samples in the parameter space. For different probability distributions of the parameter, weighted reduced basis/POD methods [26,62,64] were developed by sampling from the probability distribution with a weighted a-posteriori error estimator for the construction of the reduced basis space. Structured sampling methods using quadrature/collocation points such as Chebyshev points and Gauss Legendre/Hermite points have also been investigated [27] in comparison with the random sampling methods.…”

Section: A Short Survey Of Sampling Methodsmentioning

confidence: 99%

Hessian-Based Sampling for High-Dimensional Model Reduction

Chen

Ghattas

2019

Int. J. UncertaintyQuantification

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In this work we develop a Hessian-based sampling method for the construction of goal-oriented reduced order models with high-dimensional parameter inputs. Model reduction is known very challenging for high-dimensional parametric problems whose solutions also live in high-dimensional manifolds. However, the manifold of some quantity of interest (QoI) depending on the parametric solutions may be low-dimensional. We use the Hessian of the QoI with respect to the parameter to detect this low-dimensionality, and draw training samples by projecting the high-dimensional parameter to a low-dimensional subspace spanned by the eigenvectors of the Hessian corresponding to its dominating eigenvalues. Instead of forming the full Hessian, which is computationally intractable for a high-dimensional parameter, we employ a randomized algorithm to efficiently compute the dominating eigenpairs of the Hessian whose cost does not depend on the nominal dimension of the parameter but only on the intrinsic dimension of the QoI. We demonstrate that the Hessian-based sampling leads to much smaller errors of the reduced basis approximation for the QoI compared to a random sampling for a diffusion equation with random input obeying either uniform or Gaussian distributions.

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Section: A Short Survey Of Sampling Methodsmentioning

confidence: 99%

Hessian-Based Sampling for High-Dimensional Model Reduction

Chen

Ghattas

2019

Int. J. UncertaintyQuantification

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“…Many different methods are available to solve such problems numerically, such as stochastic Galerkin [22], stochastic collocation [1,17,25], as well as Monte-Carlo [22]. More recently, methods based on reduced order models have been developed [6,7,8,11,21,23]. In practice, they aim to construct low-dimensional approximation subspaces in a weighted fashion, i.e.…”

Section: Introductionmentioning

confidence: 99%

A Weighted POD Method for Elliptic PDEs with Random Inputs

2018

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In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L 2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

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Hessian-based sampling for high-dimensional model reduction

Chen

Ghattas

2018

Preprint

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A weighted reduced basis method for parabolic PDEs with random data

Cited by 3 publications

References 7 publications

Hessian-Based Sampling for High-Dimensional Model Reduction

Hessian-Based Sampling for High-Dimensional Model Reduction

A Weighted POD Method for Elliptic PDEs with Random Inputs

Hessian-based sampling for high-dimensional model reduction

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