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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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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

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