2018/05/30 by Mazen Ali, Ali, Mazen, Karsten Urban +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1805.12016
openalex publication_date 2018/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with the construction, analysis and realization of a\nnumerical method to approximate the solution of high dimensional elliptic\npartial differential equations. We propose a new combination of an Adaptive\nWavelet Galerkin Method (AWGM) and the well known Hierarchical Tensor (HT)\nformat. The arising HT-AWGM is adaptive both in the wavelet representation of\nthe low dimensional factors and in the tensor rank of the HT representation.\nThe point of departure is an adaptive wavelet method for the HT format using\napproximate Richardson iterations from [1] and an AWGM method as described in\n[13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncated\npreconditioned conjugate gradient (PCG) algorithm from [33] in combination with\na tensor-based preconditioner from [3]. Our analysis starts by showing\nconvergence of the truncated conjugate gradient method. The next step is to add\nroutines realizing the adaptive refinement. The resulting HT-AWGM is analyzed\nconcerning convergence and complexity. We show that the performance of the\nscheme asymptotically depends only on the desired tolerance with convergence\nrates depending on the Besov regularity of low dimensional quantities and the\nlow rank tensor structure of the solution. The complexity in the ranks is\nalgebraic with powers of four stemming from the complexity of the tensor\ntruncation. Numerical experiments show the quantitative performance.\n