Well-posed problem

Physics

Mathematical optimization

Chebyshev polynomials

Mathematical analysis

Inverse problem

Krylov subspace

Tikhonov regularization

Artificial intelligence

Tensor (intrinsic definition)

Eigenvalues and eigenvectors

Iterative method

Chebyshev iteration

Pure mathematics

Geometry

Regularization (linguistics)

Applied mathematics

Mathematics

Computer science

Tensor product

Coefficient matrix

Quantum mechanics

Linear system

Discretization

System of linear equations

Applied Numerical Mathematics

We consider the solution of linear discrete ill-posed systems of equations with a certain tensor product structure. Two aspects of this kind of problems are investigated: They are transformed to large linear systems of equations and the conditioning of the matrix of the latter system is analyzed. Also, the distance of this matrix to symmetry and skew-symmetry is investigated. The aim of our analysis is to shed light on properties of linear discrete ill-posed problems and to study the feasibility of using Krylov subspace iterative methods in conjunction with Tikhonov regularization to solve Sylvester tensor equations with severely ill-conditioned coefficient matrices. The performance of several proposed algorithms is studied numerically. Applications include color image restoration and the solution of a 3D radiative transfer equation that is discretized by a Chebyshev collocation spectral method.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations