A comparison of regularizations for an ill-posed problem

Abstract:

We consider numerical methods for a “quasi-boundary value” regularization of the backward parabolic problem given by \[ \begin {cases} u_t+Au=0, & 0<t<T \\ u(T)=f, & \end {cases} \] where $A$ is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value $u(T)$ by adding $\alpha u(0)$, where $\alpha$ is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson.