A new numerical method for backward parabolic problems in the maximum-norm setting

SIAM Journal on Numerical Analysis

Volumn 40, Issue 4, 2002, Pages 1405-1420

  Marban J M ^a   Palencia, C ^a  

^a UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

Backward parabolic problems; Chebyshev nodes; Harmonic measure; Holomorphic recovery; Ill posed problems; Least squares method; Maximum norm

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; HARMONIC ANALYSIS; PARAMETER ESTIMATION; PROBLEM SOLVING;

LEAST SQUARES METHOD;

CHEBYSHEV APPROXIMATION;

EID: 0141869065     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901386422     Document Type: Article

References (53)

1. K. A. AMES AND J. F. EPPERSON, A kernel-based method for the approximate, solution of backward parabolic problems, SIAM J. Numer. Anal., 34 (1997), pp. 1357-1390.
2. R. ANSORGE, Differenzenapproximationen partieller Anfangswertaufgaben, Teubner, Stuttgart, 1978.
3. N. Yu. BAKAEV, Resolvent estimates for a multidimensional elliptic difference operator, Funct.-Diff. Eqs., Perm', PPI (1991), pp. 118-126 (in Russian).
4. N. Yu. BAKAEV, On variable stepsize Runge-Kutta approximations of a Cauchy problem for the evolution equation, BIT, 38 (1998), pp. 462-485.
5. N. Yu. BAKAEV, V. THOMÉE, AND L. B. WAHLBIN, Maximum-Norm Estimates for Resolvents of Elliptic Finite Elements Operators, preprint, Department of Mathematics, Chalmers University of Technology, Göteborg University, Göteborg, Sweden, 2001.
6. P. BRENNER AND V. THOMÉE, On rational approximations of semigroups, SIAM J. Numer, Anal., 16 (1979), pp. 683-694.
7. K. BURRAGE AND S. PISKAREV, Stochastic methods for ill-posed problems, BIT, 40 (2000), pp. 226-240.
8. B. L. BUZBEE AND A. CARASSO, On the numerical computation of parabolic problems for preceding times, Math. Comp., 27 (1973), pp. 237-265.
9. A. CARASSO, The backward beam equation: Two A-stable schemes for parabolic problems, SIAM J. Numer. Anal., 9 (1972), pp. 406-434.
10. A. S. CARASSO, Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Numer. Anal., 31 (1994), pp. 1535-1557.
11. A. S. CARASSO, Error bounds in nonsmooth image deblurring, SIAM J. Math. Anal., 28 (1997), pp. 656-668.
12. A. S. CARASSO, J. G. SANDERSON, AND J. M. HYMAN, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal., 15 (1978), pp. 344-367.
13. D. COLTON AND J. WIMP, The construction of solutions to heat equation backward in time, Math. Meth. Appl. Sci., 1 (1979), pp. 32-39.
14. M. CROUZEIX, S. LARSSON, S. PISKAREV, AND V. THOMÉE, The stability of rational approximations of analytic semigroups, BIT, 33 (1993), pp. 74-84.
15. L. ELDEN, Time discretization in the backward solution of parabolic equations, I, Math. Comp., 39 (1982), pp. 53-68.
16. L. ELDEN, Time discretization in the backward solution of parabolic equations, II, Math. Comp., 39 (1982), pp. 69-84.
17. R. E. EWING, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal., 6 (1975), pp. 283-294.
18. H. O. FATTORINI, The Cauchy Problem, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA, 1983.
19. S. D. FISHER, Function Theory in Planar Domains, John Wiley, New York, 1983.
20. J. N. FRANKLIN, On Tikhonov's method for ill-posed problems, Math. Comp., 18 (1974), pp. 889-907.
21. D. GAIER, Lectures on Complex Approximation, Birkhäuser, Boston, 1987.
22. G. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., The John Hopkins University Press, Baltimore, MD, 1989.
23. E. HAIRER AND G. WANNER, Solving Ordinary Equations II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin, 1996.
24. D. N. HÁO, A mollification method for ill-posed problems, Numer. Math., 68 (1994), pp. 469-506.
25. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1998.
26. R. HERSH AND T. KATO, High-accuracy stable difference schemes for well-posed initial-value problems, SIAM J. Numer. Anal., 16 (1979), pp. 670-682.
27. K. ITO AND F. KAPPEL, The Trotter-Kato theorem and approximations of PDEs, Math. Comp., 67 (1998), pp. 21-44.
28. F. JOHN, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl., 40 (1955), pp. 129-142.
29. T. KATO, Theory for Linear Operators, Springer-Verlag, New York, 1966.
30. R. LATTES AND J. L. LIONS, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.
31. A. LUNARDI, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
32. P. MANSELLI AND K. MILLER, Dimensionality reduction methods for efficient numerical solution, backward in time, of parabolic equations with variable coefficients, SIAM J. Math. Anal., 11 (1980), pp. 147-159.
33. J. M. MARBÁN AND C. PALENCIA, On the numerical recovery of a holomorphic mapping from a finite set of approximate values, Numer. Math., 91 (2002), pp. 57-75.
34. K. MILLER, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal., 1 (1970), pp. 52-74.
35. K. MILLER, Efficient numerical methods for backward solution of parabolic problems with variable coefficients, in Improperly Posed Boundary Value Problems, Res. Notes Math. 1, A. Carasso and A. P. Stone, eds., Pitman, Bath, 1975, pp. 54-64.
36. K. MILLER, Logarithmic convexity results for holomorphic semigroups, Pacific J. Math., 58 (1975), pp. 549-551.
37. C. PALENCIA, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal., 30 (1993), pp. 1373-1384.
38. C. PALENCIA, On the stability of variable stepsize approximations of holomorphic semigroups, Math. Comp., 62 (1994), pp. 93-103.
39. C. PALENCIA, Maximum norm analysis of completely discrete finite element methods for parabolic problems, SIAM J. Numer. Anal., 33 (1996), pp. 1654-1668.
40. C. PALENCIA AND J. M. SANZ-SERNA, An extension of the Lax-Richtmyer theory, Numer. Math., 44 (1984), pp. 279-283.
41. J. R. PARTINGTON, Interpolation, Identification and Sampling, London Math. Soc. Monogr. (N.S.) 17, Oxford University Press, New York, 1997.
42. L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations, CBMS-NSF Reg. Conf. Ser. in Appl. Math. 22, SIAM, Philadelphia, 1975.
43. A. PAZY, Semigroups of Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
44. T. RANSFORD, Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts 28, Cambridge University Press, Cambridge, UK, 1995.
45. 0-projection into finite element spaces, Math. Comp., 38 (1982), pp. 1-21.
46. T. I. SEIDMAN, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), pp. 162-170.
47. R. E. SHOWALTER, Quasi-reversibility of first and second order parabolic evolution equations, in Improperly Posed Boundary Value Problems, Res. Notes Math. 1, A. Carasso and A. P. Stone, eds., Pitman, Bath, 1975, pp. 76-84.
48. E. SINESTRARI, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), pp. 16-65.
49. H. B. STEWART, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc., 199 (1974), pp. 141-162.
50. H. B. STEWART, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259 (1980), pp. 299-310.
51. V. THOMÉE, Galerkin Finite Elements for Parabolic Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
52. A. N. TIKHONOV, Regularization of incorrectly posed problems, Dokl. Akad. Nauk. SSSR, 153 (1963), pp. 49-52 (in Russian); Soviet. Math. Dokl., 4 (1963), pp. 1624-1627 (in English).
53. A. N. TIKHONOV, Regularization of incorrectly posed problems, Dokl. Akad. Nauk. SSSR, 153 (1963), pp. 49-52 (in Russian); Soviet. Math. Dokl., 4 (1963), pp. 1624-1627 (in English).