Adaptive low-rank approximations for operator equations: Accuracy control and computational complexity
DOI10.1090/conm/754/15151zbMath1478.65105arXiv1910.07052OpenAlexW3046312611MaRDI QIDQ4998630
Wolfgang Dahmen, Markus Bachmayr
Publication date: 9 July 2021
Published in: 75 Years of Mathematics of Computation (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/1910.07052
nonlinear approximationlow-rank approximationparametric PDEshigh-dimensional diffusion equationsapproximation classestensor formatsa posteriori error boundshard and soft thresholding
Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) (42C10) Multidimensional problems (41A63) Numerical solutions to equations with linear operators (65J10) Error bounds for numerical methods for ordinary differential equations (65L70) Approximation by arbitrary nonlinear expressions; widths and entropy (41A46) Preconditioners for iterative methods (65F08)
Related Items (1)
Uses Software
Cites Work
- Unnamed Item
- Unnamed Item
- Tensor Decompositions and Applications
- Tensor-Train Decomposition
- Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations
- Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations
- Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
- Adaptive near-optimal rank tensor approximation for high-dimensional operator equations
- Approximate iterations for structured matrices
- \(\mathcal H\)-matrix approximation for the operator exponential with applications
- Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients
- Adaptive wavelet methods. II: Beyond the elliptic case
- On minimal subspaces in tensor representations
- Singular value decomposition in Sobolev spaces. I
- HT-AWGM: a hierarchical Tucker-adaptive wavelet Galerkin method for high-dimensional elliptic problems
- A new scheme for the tensor representation
- Error bounds for approximations with deep ReLU networks
- Adaptive stochastic Galerkin FEM with hierarchical tensor representations
- Iterative methods based on soft thresholding of hierarchical tensors
- Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs
- Adaptive wavelet methods for elliptic partial differential equations with random operators
- Adaptive Low-Rank Methods: Problems on Sobolev Spaces
- Adaptive low-rank methods for problems on Sobolev spaces with error control in L2
- Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
- On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format
- An adaptive stochastic Galerkin method for random elliptic operators
- A literature survey of low-rank tensor approximation techniques
- The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format
- A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
- Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
- Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs
- Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
- Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
- Hierarchical Singular Value Decomposition of Tensors
- ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S
- Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
- Tensor Spaces and Numerical Tensor Calculus
- Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
- Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
- On the efficient computation of high-dimensional integrals and the approximation by exponential sums
- A Multilinear Singular Value Decomposition
- Adaptive wavelet methods for elliptic operator equations: Convergence rates
- Numerical operator calculus in higher dimensions
- Parametric PDEs: sparse or low-rank approximations?
- Approximation of high-dimensional parametric PDEs
- Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
- A projection method to solve linear systems in tensor format
- Adaptive wavelet algorithms for elliptic PDE's on product domains
- Approximation of 1/x by exponential sums in [1, ∞)
- Preconditioned Low-rank Riemannian Optimization for Linear Systems with Tensor Product Structure
This page was built for publication: Adaptive low-rank approximations for operator equations: Accuracy control and computational complexity
