On the Numerical Stability in Solving Ill-Posed Problems

Abstract

For the numerical solution of Fredholm integral equations of the first kind or ill-posed problems in general, a method is proposed which gives the minimum norm solution by solving a nonlinear matrix problem. However, as in the finite element method, the entries of some matrices have to be evaluated by quadrature formulas. These formulas are introducing an undesired discretization error. Since the problems usually turn out to be ill-conditioned, this error may have a disastrous effect on the final result. In this paper, this effect has been studied for the above mentioned algorithm. The main theorem gives an upper bound for the relative error of the minimum norm solution f₀ of the problem for the case where the perturbations of the matrices (of order at most n, say) lie below some given bound. For instance, if the perturbations are of the order of (math) for some p ≧ 5/2 then the relative error of f₀ is of the order of (Math) where λ_n is a small positive parameter determined by the algorithm and such that lim_n λ_n = 0.