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A comparison of regularizations for an ill-posed problem

Authors:	Karen A Ames Gordon W Clark James F Epperson Seth F Oppenheimer

Institution:	Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, Alabama 35899 ; Department of Mathematics and Statistics, Mississippi State University, Drawer MA MSU, MS 39762 ; Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, Alabama 35899 ; Department of Mathematics and Statistics, Mississippi State University, Drawer MA MSU, MS 39762

Abstract:	We consider numerical methods for a ``quasi-boundary value' regularization of the backward parabolic problem given by $\begin{displaymath}\left\{ \begin{array}{ll} u_t+Au=0\,, & 0<t<T u(T)=f, & \end{array}\right. \end{displaymath}$ where is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value 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. We consider numerical methods for a ``quasi-boundary value' regularization of the backward parabolic problem given by $\begin{displaymath}\left\{ \begin{array}{ll} u_t+Au=0\,, & 0<t<T u(T)=f, & \end{array}\right. \end{displaymath}$ where is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value 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.

Keywords:	Quasi-reversibility final value problems ill-posed problems Freholm equations numerical methods

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