Existence of dynamical low-rank approximations to parabolic problems

DOI10.1090/mcom/3626zbMath1482.65136arXiv2002.12197OpenAlexW3117824799WikidataQ114094315 ScholiaQ114094315MaRDI QIDQ4992232

André Uschmajew, Emil Kieri, Markus Bachmayr, Henrik Eisenmann

Publication date: 7 June 2021

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2002.12197

zbMATH Keywords

low-rank approximations

Mathematics Subject Classification ID

Eigenvalues, singular values, and eigenvectors (15A18) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Numerical methods for initial value problems involving ordinary differential equations (65L05) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Initial value problems for second-order parabolic equations (35K15) Weak solutions to PDEs (35D30) Multilinear algebra, tensor calculus (15A69) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02) PDEs on manifolds (35R01) Numerical methods for low-rank matrix approximation; matrix compression (65F55)

Related Items (5)

A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds ⋮ Low-rank tensor methods for partial differential equations ⋮ A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations ⋮ Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations ⋮ An unconventional robust integrator for dynamical low-rank approximation

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Tensor-Train Decomposition
• Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations
• On the approximation of high-dimensional differential equations in the hierarchical Tucker format
• Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations
• Finite element approximation of the Navier-Stokes equations
• Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations
• Dynamically orthogonal field equations for continuous stochastic dynamical systems
• Adaptive near-optimal rank tensor approximation for high-dimensional operator equations
• From quantum to classical molecular dynamics: Reduced models and numerical analysis.
• Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree-Fock equations: a sufficient condition
• Numerical low-rank approximation of matrix differential equations
• Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions
• On the Dirac-Frenkel variational principle on tensor Banach spaces
• Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure
• Applied functional analysis. Main principles and their applications
• A new scheme for the tensor representation
• Approximation rates for the hierarchical tensor format in periodic Sobolev spaces
• A projector-splitting integrator for dynamical low-rank approximation
• Adaptive Low-Rank Methods: Problems on Sobolev Spaces
• Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values
• Guarantees of Riemannian Optimization for Low Rank Matrix Recovery
• Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors
• Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
• Variational-splitting time integration of the multi-configuration time-dependent Hartree-Fock equations in electron dynamics
• Dynamical Tensor Approximation
• Tensor Spaces and Numerical Tensor Calculus
• Geometric Methods on Low-Rank Matrix and Tensor Manifolds
• An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
• A Multilinear Singular Value Decomposition
• A Low-Rank Projector-Splitting Integrator for the Vlasov--Poisson Equation
• Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods
• Convergence of a Low-Rank Lie--Trotter Splitting for Stiff Matrix Differential Equations
• Time Integration of Tensor Trains
• Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs
• A Geometric Approach to Dynamical Model Order Reduction
• Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics
• Dynamical Low‐Rank Approximation
• Introduction

This page was built for publication: Existence of dynamical low-rank approximations to parabolic problems