Low-rank tensor methods for partial differential equations

M Bachmayr - Acta Numerica, 2023‏ - cambridge.org
Low-rank tensor representations can provide highly compressed approximations of
functions. These concepts, which essentially amount to generalizations of classical …

Tensor networks for dimensionality reduction and large-scale optimization: Part 2 applications and future perspectives

A Cichocki, AH Phan, Q Zhao, N Lee… - … and Trends® in …, 2017‏ - nowpublishers.com
Part 2 of this monograph builds on the introduction to tensor networks and their operations
presented in Part 1. It focuses on tensor network models for super-compressed higher-order …

Can physics-informed neural networks beat the finite element method?

TG Grossmann, UJ Komorowska, J Latz… - IMA Journal of …, 2024‏ - academic.oup.com
Partial differential equations (PDEs) play a fundamental role in the mathematical modelling
of many processes and systems in physical, biological and other sciences. To simulate such …

A literature survey of low‐rank tensor approximation techniques

L Grasedyck, D Kressner, C Tobler - GAMM‐Mitteilungen, 2013‏ - Wiley Online Library
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 …

Computational methods for linear matrix equations

V Simoncini - siam REVIEW, 2016‏ - SIAM
Given the square matrices A,B,D,E and the matrix C of conforming dimensions, we consider
the linear matrix equation A\mathbfXE+D\mathbfXB=C in the unknown matrix \mathbfX. Our …

[کتاب][B] Iterative solution of large sparse systems of equations

W Hackbusch - 1994‏ - Springer
The numerical treatment of partial differential equations splits into two different parts. The
first part are the discretisation methods and their analysis. This led to the author's …

Hypernetwork-based meta-learning for low-rank physics-informed neural networks

W Cho, K Lee, D Rim, N Park - Advances in Neural …, 2023‏ - proceedings.neurips.cc
In various engineering and applied science applications, repetitive numerical simulations of
partial differential equations (PDEs) for varying input parameters are often required (eg …

Alternating minimal energy methods for linear systems in higher dimensions

SV Dolgov, DV Savostyanov - SIAM Journal on Scientific Computing, 2014‏ - SIAM
We propose algorithms for the solution of high-dimensional symmetrical positive definite
(SPD) linear systems with the matrix and the right-hand side given and the solution sought in …

Tensor decomposition methods for high-dimensional Hamilton--Jacobi--Bellman equations

S Dolgov, D Kalise, KK Kunisch - SIAM Journal on Scientific Computing, 2021‏ - SIAM
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear
Hamilton--Jacobi--Bellman equations arising in optimal feedback control of nonlinear …

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs

C Schwab, CJ Gittelson - Acta Numerica, 2011‏ - cambridge.org
Partial differential equations (PDEs) with random input data, such as random loadings and
coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of …