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A comparison of regularizations
for an ill-posed problem


Authors: Karen A. Ames, Gordon W. Clark, James F. Epperson and Seth F. Oppenheimer
Journal: Math. Comp. 67 (1998), 1451-1471
MSC (1991): Primary 35A35, 35R25, 65M30, 65M15
DOI: https://doi.org/10.1090/S0025-5718-98-01014-X
MathSciNet review: 1609682
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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 $A$ 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.

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 $A$ 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.


References [Enhancements On Off] (What's this?)

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Additional Information

Karen A. Ames
Affiliation: Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, Alabama 35899
Address at time of publication: Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23284
Email: ames@math.uah.edu

Gordon W. Clark
Affiliation: Department of Mathematics and Statistics, Mississippi State University, Drawer MA MSU, MS 39762
Address at time of publication: Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23284
Email: gwclark@saturn.vcu.edu

James F. Epperson
Affiliation: Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, Alabama 35899
Email: epperson@math.uah.edu, seth@math.msstate.edu

Seth F. Oppenheimer
Affiliation: Department of Mathematics and Statistics, Mississippi State University, Drawer MA MSU, MS 39762
Email: seth@math.msstate.edu

DOI: https://doi.org/10.1090/S0025-5718-98-01014-X
Keywords: Quasi-reversibility, final value problems, ill-posed problems, Freholm equations, numerical methods
Received by editor(s): April 17, 1996
Additional Notes: Partially supported by Army contract DACA 39-94-K-0018 (S.F.O.) and by NSF contract DMS-9308121 (K.A.A.)
Article copyright: © Copyright 1998 American Mathematical Society

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