Data-sparse approximation to a class of operator-valued functions

DOI10.1090/S0025-5718-04-01703-XzbMath1066.65060MaRDI QIDQ4654016

Boris N. Khoromskij, Ivan P. Gavrilyuk, Wolfgang Hackbusch

Publication date: 1 March 2005

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

zbMATH Keywords

Lyapunov equationRiccati equationelliptic operatorquadrature formulaeoperator-valued functiondata-sparse approximationDunford-Cauchy integralfunction-to-operator mappingsGauss-Lobatto quadrature \(\mathcal{H}\)-matricesLyapunov-Sylvester equationsoperator-to-operator mappingsequence of operators-to-operator mappingSinc quadrature

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Numerical solutions to equations with linear operators (65J10) Other special methods applied to PDEs (35A25)

Related Items

Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations ⋮ A numerical study of the homogeneous elliptic equation with fractional boundary conditions ⋮ Unnamed Item ⋮ Matrix Structures and Matrix Functions ⋮ Numerical solution of time-dependent problems with fractional power elliptic operator ⋮ Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels ⋮ A quadrature based method for evaluating exponential-type functions for exponential methods ⋮ Weighted estimates for boundary value problems with fractional derivatives ⋮ Approximate iterations for structured matrices ⋮ Hierarchical Kronecker tensor-product approximations ⋮ Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity ⋮ Tensor-product approximation to operators and functions in high dimensions ⋮ Weighted estimates of the Cayley transform method for abstract differential equations ⋮ Exponentially Convergent Method for them-Point Nonlocal Problem for a First Order Differential Equation in Banach Space ⋮ Fast tensor method for summation of long‐range potentials on 3D lattices with defects ⋮ An exponentially convergent algorithm for nonlinear differential equations in Banach spaces ⋮ Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems ⋮ Adaptive application of the operator exponential ⋮ Numerical solution of time-dependent problems with a fractional-power elliptic operator ⋮ Wavelets for density matrix computation in electronic structure calculation ⋮ Numerically solving an equation for fractional powers of elliptic operators ⋮ Decay bounds for the numerical quasiseparable preservation in matrix functions ⋮ High order numerical schemes for solving fractional powers of elliptic operators ⋮ Quasi-optimal rank-structured approximation to multidimensional parabolic problems by Cayley transform and Chebyshev interpolation ⋮ Tucker tensor analysis of Matérn functions in spatial statistics ⋮ Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation ⋮ High order approximations of the operator Lyapunov equation have low rank ⋮ Numerical approximation of fractional powers of elliptic operators ⋮ Sinc-approximations of fractional operators: a computing approach ⋮ Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators I. Separable approximation of multi-variate functions ⋮ Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators II. HKT representation of certain operators

Uses Software

• Algorithm 432

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices
• Solution of Lyapunov equations by alternating direction implicit iteration
• Iterative solution of the Lyapunov matrix equation
• A sparse matrix arithmetic based on \({\mathfrak H}\)-matrices. I: Introduction to \({\mathfrak H}\)-matrices
• Strongly \(P\)-positive operators and explicit representations of the solutions of initial value problems for second-order differential equations in Banach space
• Construction and arithmetics of \(\mathcal H\)-matrices
• \(\mathcal H\)-matrix approximation for the operator exponential with applications
• A sparse \({\mathcal H}\)-matrix arithmetic: General complexity estimates
• A sparse \({\mathcal H}\)-matrix arithmetic. II: Application to multi-dimensional problems
• Lyapunov matrix equations in system stability and control.
• Spectral Properties of Elementary Operators. II
• A Hessenberg-Schur method for the problem AX + XB= C
• Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations
• Collocating Convolutions
• On Krylov Subspace Approximations to the Matrix Exponential Operator
• A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature
• Data-sparse approximation to the operator-valued functions of elliptic operator
• Linear Operator Equations
• Algorithm 432 [C2: Solution of the matrix equation AX + XB = C [F4]]
• Explicit Solutions of Linear Matrix Equations