A semi-Lagrangian Vlasov solver in tensor train format

In this article, we derive a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Grid-based methods for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the tensor train (TT) format provides efficient basic linear algebra routines for low-rank representations of tensors. In this paper, we develop interpolation formulas for a semi-Lagrangian solver in TT format. In order to efficiently implement the method, we propose a compression of the matrix representing the interpolation step and an efficient implementation of the Hadamard product. We show numerical simulations for standard test cases in two, four and six dimensional phase space. Depending on the test case, the memory requirements reduce by a factor $10^2-10^3$ in four and a factor $10^5-10^6$ in six dimensions compared to the full-grid method

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

On Algorithms for and Computing with the Tensor Ring Decomposition

Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage-cost representations in the tensor ring format, which is an extension of the tensor train format with variable end-ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage-cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph-based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.Comment: 24 pages, 3 figures, 6 tables, implementation of algorithms available at https://github.com/oscarmickelin/tensor-ring-decompositio