Enriched Krylov subspace methods for ill-posed problems

Solving Ill-Posed Cauchy Problems by a Krylov Subspace Method

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation uzz − Lu = 0 in three space dimensions (x, y, z) , where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary is sought. The problem is severely illposed. The formal solution is written as a hyperbolic cosine function ...

Krylov subspace iterative methods for nonsymmetric discrete ill-posed problems in image restoration

The BiCG and QMR methods are well-known Krylov subspace iterative methods for the solution of linear systems of equations with a large nonsymmetric, nonsingular matrix. However, little is known of the performance of these methods when they are applied to the computation of approximate solutions of linear systems of equations with a matrix of ill-determined rank. Such linear systems are known as...

Preconditioned Krylov Subspace Methods for Eigenvalue Problems

Cascadic multilevel methods for ill-posed problems

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand ...

Stochastic methods for ill-posed problems

This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as = p n ; where n is the stepsize and provides the convergence on smooth initial data. The case of t...