QTT-finite-element approximation for multiscale problems. I: Model problems in one dimension

DOI10.1007/s10444-016-9491-yzbMath1375.65152OpenAlexW2549636337MaRDI QIDQ2400487

Christoph Schwab, Vladimir A. Kazeev, Ivan V. Oseledets, M. V. Rakhuba

Publication date: 1 September 2017

Published in: Advances in Computational Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10444-016-9491-y

zbMATH Keywords

homogenizationHelmholtz equationnumerical experimentexponential convergencemultiscale problemstensor decompositionsscale resolutionelliptic multiscale-diffusionhigh frequency scattering problemsquantized tensor trains

Mathematics Subject Classification ID

Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Multilinear algebra, tensor calculus (15A69) Homogenization in context of PDEs; PDEs in media with periodic structure (35B27)

Related Items (10)

Tensor rank bounds for point singularities in \(\mathbb{R}^3\) ⋮ Learning with tree tensor networks: complexity estimates and model selection ⋮ Quantized Tensor FEM for Multiscale Problems: Diffusion Problems in Two and Three Dimensions ⋮ Low-rank tensor methods for partial differential equations ⋮ Exponential ReLU neural network approximation rates for point and edge singularities ⋮ Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions ⋮ Limitations of neural network training due to numerical instability of backpropagation ⋮ Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs ⋮ High dimensional finite elements for time-space multiscale parabolic equations ⋮ Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDE

Uses Software

• redbKIT

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Tensor-Train Decomposition
• Low-rank tensor structure of linear diffusion operators in the TT and QTT formats
• Some estimates for \(h\)-\(p\)-\(k\)-refinement in isogeometric analysis
• \(O(d \log N)\)-quantics approximation of \(N\)-\(d\) tensors in high-dimensional numerical modeling
• A fast iteration method for solving elliptic problems with quasiperiodic coefficients
• Approximation of matrices with logarithmic number of parameters
• Finite element analysis of acoustic scattering
• Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions
• Constructive representation of functions in low-rank tensor formats
• Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs
• A new scheme for the tensor representation
• Error-bounds for finite element method
• Convergence analysis of the Generalized Empirical Interpolation Method
• Adaptive low-rank methods for problems on Sobolev spaces with error control in L₂
• Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity
• A literature survey of low-rank tensor approximation techniques
• The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format
• Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
• Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems
• Tensor Approximation of Stationary Distributions of Chemical Reaction Networks
• Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs
• Hierarchical Singular Value Decomposition of Tensors
• Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
• Numerical Solution of the Hartree–Fock Equation in Multilevel Tensor-Structured Format
• 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
• Неравенство Для Алгебраических Многоченов и зависимостб Между Наилучщими СтепеннЫми приближениями $$E(f)_{L_p } E(f)_{L_q } $$ Фуhкций $$f(x)\varepsilon L_p $$
• High-Dimensional Finite Elements for Elliptic Problems with Multiple Scales
• Tensor approximations of matrices generated by asymptotically smooth functions
• Low-Rank Tensor Structure of Solutions to Elliptic Problems with Jumping Coefficients
• Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse
• Solution of Linear Systems and Matrix Inversion in the TT-Format
• Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker--Planck Equation
• A projection method to solve linear systems in tensor format
• Reduced Basis Methods for Partial Differential Equations
• Preconditioned Low-rank Riemannian Optimization for Linear Systems with Tensor Product Structure
• Exponential Convergence of the hp Version of Isogeometric Analysis in 1D

This page was built for publication: QTT-finite-element approximation for multiscale problems. I: Model problems in one dimension