A double optimal descent algorithm for iteratively solving ill-posed linear inverse problems

Abstract

In the iterative solution of an ill-posed linear system , how to select a fast and easily established descent direction to reduce the residual is an important issue. A mathematical procedure to find a double optimal descent direction in , without inverting , is developed in an -dimensional Krylov subspace. The novelty is that we expand in an affine Krylov subspace with undetermined coefficients, and then two optimization techniques are used to determine these coefficients in closed form, which can greatly accelerate the convergence speed in solving the ill-posed linear problems. The double optimal descent algorithm is proven to be absolutely convergent very fast, accurate and robust against noise, which is confirmed by numerical tests of several linear inverse problems, including the heat source identification problem, the backward heat conduction problem, the inverse Cauchy problem and the external force identification problem.