We prove Hölder-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space. Specifically, given a positive self-adjoint operator D in a Hilbert space, we consider the ill-posed evolution problem du(t) dt = A(t,D)u(t) 0 ≤ t < T

The condition number of a problem measures the sensitivity of the answer to small changes in the input, where “small” refers to some distance measure. A problem is called illconditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to th...

We consider numerical methods for a “quasi-boundary value” regularization of the backward parabolic problem given by { ut + Au = 0 , 0 < t < T u(T ) = f, where A is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value u(T ) by adding αu(0), where α is a small parameter. We show how this leads very naturally to a reformulation of the pro...

For an ill-posed problem which has its origin in several applications (e.g. electro-cardiology) a weak formulation is given over a Hilbert space without any constraints. This is achieved by means of Lagrangian multipliers. Beside theoretical questions (e.g. existence and uniqueness of a solutuion) a nite element approximation is considered. Error estimates, an investigation of the condition num...