Optimal algorithms in a Krylov subspace for solving linear inverse problems by MFS

The method of fundamental solutions (MFS) is used to solve backward heat conduction problem, inverse heat source problem, inverse Cauchy problem and inverse Robin problem. In order to overcome the ill-posedness of resulting linear equations, two optimal algorithms with optimal descent vectors that consist of m vectors in a Krylov subspace are developed, of which the m weighting parameters are determined by minimizing a properly defined merit function in terms of a quadratic quotient. The optimal algorithms OA1 and OA2 are convergent fast, accurate and robust against large noise, which are confirmed through numerical tests.