English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

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