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HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems

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Abstract

This paper is concerned with the construction, analysis, and realization of a numerical method to approximate the solution of high-dimensional elliptic partial differential equations. We propose a new combination of an adaptive wavelet Galerkin method (AWGM) and the well-known hierarchical tensor (HT) format. The arising HT-AWGM is adaptive both in the wavelet representation of the low-dimensional factors and in the tensor rank of the HT representation. The point of departure is an adaptive wavelet method for the HT format using approximate Richardson iterations and an AWGM for elliptic problems. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm in combination with a tensor-based preconditioner. Our analysis starts by showing convergence of the truncated conjugate gradient method. The next step is to add routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed concerning convergence and complexity. We show that the performance of the scheme asymptotically depends only on the desired tolerance with convergence rates depending on the Besov regularity of low-dimensional quantities and the low-rank tensor structure of the solution. The complexity in the ranks is algebraic with powers of four stemming from the complexity of the tensor truncation. Numerical experiments show the quantitative performance.

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Acknowledgments

We would like to thank Markus Bachmayr, Rob Stevenson, and Wolfgang Dahmen for their very helpful comments on this work. This paper was partly written when Mazen Ali was a visiting researcher at Centrale Nantes in collaboration with Anthony Nouy. We acknowledge Anthony Nouy for the helpful discussions and financial support.

Funding

This study was financially supported by the European Model Reduction Network (TD COST Action TD1307).

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Authors and Affiliations

  1. Ulm University, Inst. f. Numerical Mathematics, Helmholtzstr. 20, 89081, Ulm, Germany

    Mazen Ali & Karsten Urban

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  1. Mazen Ali
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  2. Karsten Urban
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Correspondence to Mazen Ali.

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Communicated by: Alexander Barnett

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Ali, M., Urban, K. HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems. Adv Comput Math 46, 59 (2020). https://doi.org/10.1007/s10444-020-09797-9

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  • DOI: https://doi.org/10.1007/s10444-020-09797-9

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