Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

Eigel, Martin; Neumann, Johannes; Schneider, Reinhold; Wolf, Sebastian

doi:10.1515/cmam-2018-0028

Cited by 18 publications

(19 citation statements)

References 76 publications

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“…Thus, the central notion of this scheme is to learn the solution operator y → a( • , y) → u( • , y) from generated data u( • , y i ). The computation of the data u( • , y i ) is completely non-intrusive, meaning that standard FE solvers can be employed to compute u( • , y i ) and perform the optimization of the mean squared errors in a standard tensor recovery algorithm as described in [50].…”

Section: Parametric Pdes As a Common Benchmark Example We Introducementioning

confidence: 99%

“…In fact, this allows for the computation of the H 1 0 (D) norm of functions in the FE space more efficiently and provides numerical stability to the minimization problem (5). Note also that one can use a standard tensor reconstruction algorithm as in [50] since u H 1 0 (D) = u 2 . However, the resulting tensor represents the solution w.r.t.…”

Section: Parametric Pdes As a Common Benchmark Example We Introducementioning

confidence: 99%

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Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

et al. 2019

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A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.

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Section: Parametric Pdes As a Common Benchmark Example We Introducementioning

confidence: 99%

Section: Parametric Pdes As a Common Benchmark Example We Introducementioning

confidence: 99%

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

et al. 2019

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“…In spite of this success, we have to point out that optimizing over S d g,ρ took about 10 times longer than optimizing over S d,aug g,ρ . Finally, we observe that the recovery in T 14 (V 10 6 ) produces unexpectedly large relative errors when compared to previous results in [13]. This suggests that the rank-adaptive algorithm from [13] has a strong regularizing effect that improves the sample efficiency.…”

Section: Quantities Of Interestmentioning

confidence: 54%

“…In the context of optimal control tensor train networks have been utilized for solving the Hamilton-Jacobi-Bellman equation in [8,9], for solving backward stochastic differential equations in [10] and for the calculation of stock options prices in [11,12]. In the context of uncertainty quantification they are used in [13][14][15] and in the context of image classification they are used in [16,17].…”

Section: Introductionmentioning

confidence: 99%

“…Although, not directly suitable for recovery tasks, it became apparent that DMRG and ALS can be adapted to work in this context. Two of these extensions to the ALS algorithm are the stablilized ALS approximation (SALSA) [21] and the block alternating steepest descent for Recovery (bASD) algorithm [13]. Both adapt the tensor network ranks and are better suited to the problem of data identification.…”

Section: Introductionmentioning

confidence: 99%

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A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression

Götte

Schneider

Trunschke

2021

Front. Appl. Math. Stat.

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Low-rank tensors are an established framework for the parametrization of multivariate polynomials. We propose to extend this framework by including the concept of block-sparsity to efficiently parametrize homogeneous, multivariate polynomials with low-rank tensors. This provides a representation of general multivariate polynomials as a sum of homogeneous, multivariate polynomials, represented by block-sparse, low-rank tensors. We show that this sum can be concisely represented by a single block-sparse, low-rank tensor.We further prove cases, where low-rank tensors are particularly well suited by showing that for banded symmetric tensors of homogeneous polynomials the block sizes in the block-sparse multivariate polynomial space can be bounded independent of the number of variables.We showcase this format by applying it to high-dimensional least squares regression problems where it demonstrates improved computational resource utilization and sample efficiency.

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Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

2022

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This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.

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Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

Cited by 18 publications

References 76 publications

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression

Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

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