LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations

Abstract

This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one-and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub-Kahan and reducing the linear operator L to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

The Solution of Absolute Value Equations Using Two New Generalized Gauss-Seidel Iteration Methods

“…( 2) reduces to the special form (1). The AVEs are significant nondifferentiable and nonlinear problems that appear in optimization, e.g., linear programming, journal bearings, convex quadratic programming, linear complementarity problems (LCPs), and network prices [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

Solving Integral Equations by Means of Fixed Point Theory

“…This is the case, for instance, of continuity: it is possible for T 2 to be continuous without T being continuous. In this sense, some results (like Istrăţescu's fixed point theorem; see [18,19]) employing T 2 are more general than their corresponding ones with T One of the powerful applications of fixed point theory can be found in the context of integral equations, whose recent numerical treatments have made great scientific advances in this field (see, for instance, collocation methods [20], operational matrix methods [21][22][23], Galerkin methods [24,25], and Krylov subspace methods [26]).…”

A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

“…In [15], an accurate scheme is developed for solving 2D Fredholm integral equation by using the cosine-trigonometric functions. In [16], an iterative algorithm based on sparse least-squares is used to solve the problem. For some related works, we refer the readers to [17][18][19][20].…”