This paper describes a new numerical method for the solution of large linear discrete ill-posed problems, whose matrix is a Kronecker product. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel. The available data (right-hand side) of many linear discrete ill-posed problems that aris...

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate s...

The solution, x, of the linear system of equations Ax ≈ b arising from the discretization of an ill-posed integral equation g(s) = ∫ H(s, t)f(t) dt with a square integrable kernel H(s, t) is considered. The Tikhonov regularized solution x(λ) approximating the Galerkin coefficients of f(t) is found as the minimizer of J(x) = {‖Ax− b‖2 +λ‖Lx‖2}, where b is given by the Galerkin coefficients of g(...

Singular value decomposition (SVD) is one of the most useful matrix decompositions in linear algebra. Here, a novel application SVD recovering ripped photos was exploited. Recovery done by applying truncated iteratively. Performance evaluated using Frobenius norm. Results from few experimental were decent.

In this report, we investigate the truncated multilinear singular value decomposition (MLSVD), proposed in De Lathauwer et al. (2000). Truncating the MLSVD results in an approximation, with a prescribed multilinear rank, to a tensor. We present a new error expression for an approximate Tucker decomposition with orthogonal factor matrices. From this expression, new insights are obtained which le...

The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay...

in this paper, we use modified laplace decomposition method to solving initial value problems (ivp) of the second order ordinary differential equations. theproposed method can be applied to linear and nonlinearproblems

We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train ran...

The low-field nuclear magnetic resonance (NMR) technique has been used to probe the pore size distribution and fluid composition in geophysical prospecting related fields. However, speed accuracy of existing numerical inversion methods are still challenging due ill-posed nature first kind Fredholm integral equation contamination noises. This paper proposes a novel algorithm accelerate convergen...

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularizat...