Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

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
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{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?)

  • 1. Ames, K.A., and Epperson, J.F., A Kernel-based Method for the Approximate Solution of Backward Parabolic Problems, SIAM J. Num. Anal., Vol 34, no. 4, 1997, pp. 1357-1390. CMP 97:16
  • 2. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
  • 3. J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), no. 2, 218–241. MR 0448926
  • 4. Gordon W. Clark and Seth F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Differential Equations (1994), No. 08, approx. 9 pp. (electronic). MR 1302574
  • 5. Gene H. Golub and Charles F. Van Loan, Matrix computations, Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1983. MR 733103
  • 6. C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 742928
  • 7. C. T. Kelley, Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics, vol. 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. With separately available software. MR 1344684
  • 8. R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications, Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967 (French). MR 0232549
  • 9. W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc. 12 (1961), 243–247. MR 0120462, 10.1090/S0002-9939-1961-0120462-2
  • 10. K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems, Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972) Springer, Berlin, 1973, pp. 161–176. Lecture Notes in Math., Vol. 316. MR 0393903
  • 11. L. E. Payne, Some general remarks on improperly posed problems for partial differential equations, Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972) Springer, Berlin, 1973, pp. 1–30. Lecture Notes in Math., Vol. 316. MR 0410142
  • 12. Shewchuk, J.R., An introduction to the conjugate gradient method without the agonizing pain, electronically published manuscript.
  • 13. R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563–572. MR 0352644
  • 14. R. E. Showalter, Cauchy problem for hyperparabolic partial differential equations, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984) North-Holland Math. Stud., vol. 110, North-Holland, Amsterdam, 1985, pp. 421–425. MR 817519, 10.1016/S0304-0208(08)72739-7
  • 15. Ragnar Winther, Some superlinear convergence results for the conjugate gradient method, SIAM J. Numer. Anal. 17 (1980), no. 1, 14–17. MR 559456, 10.1137/0717002

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 35A35, 35R25, 65M30, 65M15

Retrieve articles in all journals with MSC (1991): 35A35, 35R25, 65M30, 65M15


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: http://dx.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