An Optimal Multi-Vector Iterative Algorithm in a Krylov Subspace for Solving the Ill-Posed Linear Inverse Problems

Abstract

Abstract: An optimal m-vector descent iterative algorithm in a Krylov subspaceis developed, of which the m weighting parameters are optimized from a properlydefined objective function to accelerate the convergence rate in solving an ill-posedlinear problem. The optimal multi-vector iterative algorithm (OMVIA) is conver-gent fast and accurate, which is verified by numerical tests of several linear inverseproblems, including the backward heat conduction problem, the heat source iden-tification problem, the inverse Cauchy problem, and the external force recoveryproblem. Because the OMVIA has a good filtering effect, the numerical resultsrecovered are quite smooth with small error, even under a large noise up to 10%.Keywords: Linear inverse problems, Ill-posed linear equations system, Opti-mal multi-vector iterative algorithm (OMVIA), Future cone, Invariant-manifold,Krylov subspace method1 IntroductionThe iterative algorithm for solving algebraic equations can be derived from the dis-cretization of a certain ordinary differential equations (ODEs) system [Bhaya andKaszkurewicz (2006); Chehab and Laminie (2005); Liu and Atluri (2008)]. Partic-ularly, some descent methods can be interpreted as the discretizations of gradientflows [Helmke and Moore (1994)]. For a large scale system the major choice isusing an iterative algorithm, where an early stopping criterion is used to preventthe reconstruction of noisy component in the approximate solution. The author andhis coworkers have developed several methods to solve the ill-posed system of lin-ear algebraic equations, like using the fictitious time integration method as a filter[Liu and Atluri (2009a)], a modified polynomial expansion method [Liu and Atluri(2009b)], the Laplacian preconditioners and postconditioners [Liu, Yeih and Atluri