Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs

Martin Eigel; Nando Farchmin; Philipp Trunschke; Sebastian Heidenreich

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Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs

Martin Eigel, Nando Farchmin, Sebastian Heidenreich, Philipp Trunschke

Numerical Analysis · Submitted 2 Dec 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.

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arXiv ID2112.01285 (version 2)

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CitationMartin Eigel, Nando Farchmin, Philipp Trunschke, Sebastian Heidenreich. "Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs." arXiv:2112.01285 (2021). https://pith.science/paper/HDITKSMG

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Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods.	background 1.00	0.05	0.04	hedged possibility 0.63
s1	This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation.	contribution 0.98	0.80	0.09	asserted 0.99
s2	It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods.	method 0.69	0.80	0.08	asserted 1.00
s3	The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format.	method 0.97	0.86	0.07	asserted 1.00
s4	We derive an adaptive refinement algorithm which is steered by a reliable error estimator.	contribution 0.55	0.95	0.14	asserted 1.00
s5	Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters.	contribution 0.68	0.70	0.20	asserted 1.00
s6	Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.	result 0.99	0.93	0.57	asserted 1.00

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Original abstract

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.

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There are various open problems in this direction; for example, how to compute a reduced dimension space $_K$ when $$ is a Banach space, or how to perform adaptive mesh refinement (see, e.g., [ref] ).

Near-optimal learning of Banach-valued, high-dimensional functions via deep neural networks · Conclusions and open problems · 0.83

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Compares with 1 paper

The rates for both FE discretizations are the same as observed in [ref] in the affine case.

Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM · Adaptive convergence results · 0.99

Cites as support 1 paper

Given a sufficient amount of training samples, which can be estimated by a heuristic criterion, the obtained approximation is equivalent to the stochastic Galerkin projection with high probability [ref] .

Convergence bounds for nonlinear least squares and applications to tensor recovery · Introduction · 0.98

Cites as background 2 papers

ASGFEM in hierachical tensor formats are presented in [ref] for affine coefficients.

An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains · Introduction · 0.84
For a more detailed overview, we refer the reader to [ref] and the references therein.

Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format · The Tensor Train format · 0.99

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