Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDE

DOI10.1137/19M1280156zbMath1465.65122OpenAlexW3135061116MaRDI QIDQ5857728

M. V. Rakhuba

Publication date: 7 April 2021

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/19m1280156

zbMATH Keywords

alternating direction implicit methodelliptic PDEstensor networkstensor decompositionsrobust solverQTT decomposition

Mathematics Subject Classification ID

Iterative numerical methods for linear systems (65F10) Numerical computation of matrix norms, conditioning, scaling (65F35) Second-order elliptic equations (35J15) Finite difference methods for boundary value problems involving PDEs (65N06) Multilinear algebra, tensor calculus (15A69) Numerical solution of discretized equations for boundary value problems involving PDEs (65N22) Toeplitz, Cauchy, and related matrices (15B05)

Related Items (3)

Tensor rank bounds for point singularities in \(\mathbb{R}^3\) ⋮ Low-rank tensor methods for partial differential equations ⋮ Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension

Uses Software

• cross2D
• TT Toolbox
• GitHub

Cites Work

• Unnamed Item
• Tensor-Train Decomposition
• TT-cross approximation for multidimensional arrays
• Tensor numerical methods in scientific computing
• \(O(d \log N)\)-quantics approximation of \(N\)-\(d\) tensors in high-dimensional numerical modeling
• Grid-based electronic structure calculations: the tensor decomposition approach
• Alternating direction methods for three space variables
• Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators I. Separable approximation of multi-variate functions
• Approximation of matrices with logarithmic number of parameters
• Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions
• Superfast Fourier transform using QTT approximation
• Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs
• Approximation by exponential sums revisited
• QTT-finite-element approximation for multiscale problems. I: Model problems in one dimension
• Tensor Numerical Methods in Quantum Chemistry
• Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity
• Two-Level QTT-Tucker Format for Optimized Tensor Calculus
• A literature survey of low-rank tensor approximation techniques
• Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
• Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions
• Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
• Cross approximation in tensor electron density computations
• The ADI Model Problem
• The Numerical Solution of Parabolic and Elliptic Differential Equations
• Recent computational developments in Krylov subspace methods for linear systems
• A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices
• Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse
• Fast Multidimensional Convolution in Low-Rank Tensor Formats via Cross Approximation
• Tensor numerical methods for multidimensional PDES: theoretical analysis and initial applications
• On tensor approximation of Green iterations for Kohn-Sham equations

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