This thesis deals with linear ill-posed problems related to compact operators, and iterative Krylov subspace methods for solving discretized versions of these. Linear compact operators in infinite dimensional Hilbert spaces will be investigated and several results on the singular values and eigenvalues for such will be presented. A large subset of linear compact operators consists of integral o...

Given a totally positive matrix, can one insert line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson Smith who gave an algorithm that results in possible insertion. In this work, we revisit problem. First, show every matrix be associated with certain vertex-weighted graph such way the entries of are equal to sums ov...

We consider systems of second-order, quasilinear, ordinary differential equations with an identically degenerate matrix coefficient the principal term and well-posed initial conditions. Fundamental differences between such problems solved respect to second derivative are indicated. In terms polynomials, we formulate conditions existence uniqueness solutions in a neighborhood starting point.

Abstract. The main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. Properties of the Legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. Afterwards, an iterative algorithm is examined for solvin...

The aim of this paper is to describe an eecient adaptive strategy for discretizing ill-posed linear operator equations of the rst kind: we consider Tikhonov-Phillips regularization x = (A A + I) ?1 A y with a nite dimensional approximation A n instead of A. We propose a sparse matrix structure which still leads to optimal convergences rates but requires substantially less scalar products for co...

Initial value problems for the integrable discrete equations on quadgraphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are di...

Flexible GMRES, introduced by Saad, is a generalization of the standard GMRES method for the solution of large linear systems of equations. It is based on the flexible Arnoldi process for reducing a large square matrix to a small matrix. We describe how the flexible Arnoldi process can be applied to implement one-parameter and multi-parameter Tikhonov regularization of linear discrete ill-posed...

We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first pos...

an ecient method, based on the legendre wavelets, is proposed to solve thesecond kind fredholm and volterra integral equations of hammerstein type.the properties of legendre wavelet family are utilized to reduce a nonlinearintegral equation to a system of nonlinear algebraic equations, which is easilyhandled with the well-known newton's method. examples assuring eciencyof the method and ...

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linea...