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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A comparison of regularizations for an ill-posed problem
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by Karen A. Ames, Gordon W. Clark, James F. Epperson and Seth F. Oppenheimer PDF
Math. Comp. 67 (1998), 1451-1471 Request permission

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
  • 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.)
  • © Copyright 1998 American Mathematical Society
  • 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