HT-AWGM: A Hierarchical Tucker-Adaptive Wavelet Galerkin Method for High Dimensional Elliptic Problems

TL;DR

This paper introduces HT-AWGM, a novel adaptive numerical method combining wavelet Galerkin techniques with hierarchical tensor formats to efficiently solve high-dimensional elliptic PDEs, with proven convergence and complexity analysis.

Contribution

It presents a new adaptive method integrating wavelet Galerkin and hierarchical tensor formats for high-dimensional elliptic problems, with rigorous convergence and complexity results.

Findings

Convergence of the truncated conjugate gradient method is established.

The scheme's performance depends only on the desired tolerance and solution regularity.

Numerical experiments demonstrate the method's quantitative efficiency.

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 from [1] and an AWGM method as described in [13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm from [33] in combination with a tensor-based preconditioner from [3]. 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.