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Low-rank tensor methods for partial differential equations
Low-rank tensor representations can provide highly compressed approximations of
functions. These concepts, which essentially amount to generalizations of classical …
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
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 …
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?
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 …
of many processes and systems in physical, biological and other sciences. To simulate such …
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 …
scientific computing to address large‐scale linear and multilinear algebra problems, which …
Computational methods for linear matrix equations
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 …
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 …
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
In various engineering and applied science applications, repetitive numerical simulations of
partial differential equations (PDEs) for varying input parameters are often required (eg …
partial differential equations (PDEs) for varying input parameters are often required (eg …
Alternating minimal energy methods for linear systems in higher dimensions
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 …
(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
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear
Hamilton--Jacobi--Bellman equations arising in optimal feedback control of nonlinear …
Hamilton--Jacobi--Bellman equations arising in optimal feedback control of nonlinear …
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
Partial differential equations (PDEs) with random input data, such as random loadings and
coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of …
coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of …