Linear operator equations: Approximation and regularization

Linear Operator Equations: Approximation and Regularization

Volumn , Issue , 2009, Pages 1-249

  Nair, M Thamban ^a  

^a INDIAN INSTITUTE OF TECHNOLOGY MADRAS   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84967468297     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/7055     Document Type: Book

References (77)

1. M. AHUES, A class of strongly stable approximations, J. Austral. Math. Soc., Series B, 28 (1987) 435-442.
2. M. AHUES, A. LARGILLIER and B.V. LIMAYE, Spectral Computations for Bounded Operators, Applied Mathematics, Vol. 18, Chapman &Hall/CRC, Boca Raton, 399, 2001
3. P.M. ANESELONE, Collectively compact Operator Approximation Theory and Applications to integral Equations, Prentice-Hall, Englewood Cliffs, 1971.
4. R. ARCANGELI, Psedo-solution de l’equation Ax = y, Comptes Rendus del’ Academie des Sciences, Paris, 263 (1966) 282-285.
5. K.E. ATKINSON, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997
6. G.B. BAKER and T.M. MILLS, Lagrange Interpolation - divergence, Mathematical Chronicle, 17 (1988) 1-18.
7. R. BOULDIN, Operator approximations with stable eigenvalues, J. Austral. Math. Soc., Series A, 49 (1990) 250-257.
8. J. BAUMEISTER, Stable Solutions of Inverse Problems, Fried. Vieweg & Sohn, Braunschweig/Wiesbaden 1987.
9. H. BRACKAGE, Uber die numerische Behandlung von Integralgeichungen nach der Quadraturformelmethode, Numerische Math. 2 (1961) 183-193.
10. F. CHATELIN, Spectral Approximation of Linear Operators, Academic Press, New York, 1983.
11. F. DE HOOG, Review of Fredholm integral equations of the first kind, In: Applications and Numerical Solution of Integral equations, Eds.: R.S. An-derssen, F. de Hoog and M.A. Lucas, Sijthoff & Noordhoff, 1980, Pages: 119-134.
12. D.R. DELLWO, Accelerated refinement with applications to integral equations, SIAM J. Numer. Anal., 25 (1988) 1327-1339.
13. S. DEMKO, Spline approximation in Banach function space, In: Theory of Approximation with Applications, Eds.: A.G. Law and B.N. Shahney, Academic Press (1976) 146-154.
14. R.A. DE VORE, Degree of approximation, In: Approximation Theory II, Eds.: G.G. Lorentz, C.K. Chui and L.L. Schumaker, Academic Press (1976) 117-161.
15. H.W. ENGL, Regularization methods for the stable solution of inverse problems, Surveys Math. Indust., 3 (1993) 71-143.
16. H.W. ENGL and A. NEUBAUER, An improved version of Marti’s method for solving ill-posed linear integral equations, Math. Comp., 45 (1985) 405-416.
17. H.W. ENGL, M. HANKE and A. NEUBAUER, Regularization of inverseproblems, Kluwer Acad. Publ., Dordrecht, Boston, London, 1996.
18. M. FRIEDMAN and D.R. DELLWO, Accelerated projection methods, J. Comp. Phys., 45 (1982) 108-120.
19. M. FRIEDMAN and D.R. DELLWO, Accelerated quadrature methods for linear integral equations, J. Integral Equations, 8 (1985) 113-136.
20. S. GEORGE and M.T. NAIR, Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates, J. Optim. Th. andAppl., 83 (1994) 217-222.
21. S. GEORGE and M.T. NAIR, On a generalized Arcangeli’s method for Tikhonov regularization with inexact data, Numer. Funct. Anal. and Optim., 19 (7&8) (1998) 773-787.
22. I.G. GRAHAM and I. SLOAN, On compactness of certain integral operators, J. Math. Anal. and Appl., 68 (1979) 580-594.
23. I.G. GRAHAM and I.H. SLOAN, Iterated Galerkin versus iterated collocation for integral equations of the second kind, IMA J. Numer. Anal., 5 (1985) 355-369.
24. C.W. GROETSCH, Comments on Morozov’s discrepancy principle, In: Improperly Posed Problems and Their Numerical Treatment, Eds. G.Hammerlin and K.H.Hoffmann, Birkhauser, Basel, 1983, pp. 97-104.
25. C.W. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman Publishing, Boston, London, Melbourne, 1984.
26. C.W. GROETSCH, Inverse Problems in Mathematical Sciences, Vieweg, Braunschweg, Wiesbaden, 1993.
27. C.W. GROETSCH, Convergence of a regularized degenerate kernel method for Fredholm integral equations of the first kind, Integr. Equat. Oper. Th., 13 (1990) 67-75.
28. C.W. GROETSCH and E. SCHOCK, Asymptotic convergence rate of Arcan-geli’s method for ill-posed problems, Applicable Analysis, 18 (1984) 175-182.
29. J. HADAMARD, Lectures on the Cauchy Problem in Linear Partial Differential equations Yale University Press, New Haven, 1923.
30. S. JOE, Collocation methods using piecewise polynomials for second kind integral equations, J. Comp. Appl. Math., 12&13 (1985) 391-400.
31. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo 1976.
32. S. KESAVAN, Topics in Functional Analysis, Wiley Eastern, New Delhi, 1989.
33. J.G. KIRKWOOD and J. RISEMAN, The intrinsic viscosities and diffusion constants of flexible macromolecules in solution, J. Chem. Phys., 16 (1948) 565-573.
34. A. KIRSCH, An introduction to the Mathematical Theory of Inverse Problems, Springer, New York, Berlin, Heidelberg, Tokyo 1996.
35. R. KRESS, Linear Integral Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo 1989.
36. J. LOCKER and P.M. PRENTER, Regularization with differential operators, J. Math. Anal. Appl., 74 (1980) 504-529.
37. S.H. KULKARNI and M.T. NAIR, A characterization of closed range operators, Indian J. Pure and Appl. Math. 31 (4) (2000) 353-361.
38. B.V. LIMAYE, Spectral Perturbation and Approximation with Numerical Experiments, Proceedings of the Centre for Mathematical Analysis, Australian National University, Vol. 13, 1986.
39. A.K. LOUIS, Inverse und Schlecht Gestellte Probleme, Teubner-Verlag, Stuttgart, 1989.
40. E.R. LOVE, The electrostatic field of two equal circular co-axial conducting discs, Quart. J. Mech. Appl. Math., 2 (1949) 428-451.
41. C.A. MICCHELLI and T.J. RIVLIN, A survey of optimal recovery, In: Optimal Estimation in Approximation Theory, Eds., C.A.Micchelli and T.J.Rivlin, Plenum press, 1977, pp. 1-53.
42. V.A. MOROZOV, On the solution of functional equations by the method of regularization, Soviet Math. Dokldy, 7 (1966) 414-417.
43. M.T. NAIR, A modified projection method for equations of the second kind, Bull. Austral. Math. Soc., 36 (1987) 485-492.
44. M.T. NAIR, On strongly stable approximations, Austral. Math. Soc. (Series A), 52 (1992) 251-260.
45. M.T. NAIR, A generalization of Arcangeli’s method for ill-posed problems leading to optimal rates, Integr. Equat. Oper., 15 (1992) 1042-1046.
46. M.T. NAIR, On Uniform convergence of approximation methods for operator equations of the second kind, Numer. Funct. Anal. and Optim., 13 (1992) 69-73.
47. M.T. NAIR, On accelerated refinement methods for operator equations of the second kind, J. Indian Math. Soc., 59 (1993) 135-140.
48. M.T. NAIR, A unified approach for regularized approximation methods for Fredholm integral equations of the first kind, Numer. Funct. Anal. and Optim., 15 (1994) 381-389.
49. M.T. NAIR, On Morozov’s method for Tikhonov regularization as an optimal order yielding algorithm, Zeit. Anal. und ihre Anwend., 18 (1999) 37-46.
50. M.T. NAIR, Optimal order results for a class of regularization methods using unbounded operators, Integr. Equat. Oper., 44 (2002) 79-92.
51. M.T. Nair, Functional Analysis: A first Course, Prentice-Hall of India, New Delhi, 2002.
52. M.T. NAIR and R.S. ANDERSSEN, Superconvergence of modified projection method for integral equations of the second kind, J. Integral Equations, 3 (1991) 255-269.
53. M.T. NAIR, M. HEGLAND and R.S. ANDERSSEN, The trade-off between regularity and stabilization in Tikhonov regularization, Math. Comp., 66 (1997) 193-206.
54. M.T. NAIR and S.V.PEREVERZEV, Regularized collocation method for Fredholm integral equations of the first kind, J. Complexity, 23 (2007) 454-467.
55. M.T. NAIR, S.V. PEREVERZEV and U. TAUTENHAHN, Regularization in Hilbert scales under general smoothing conditions, Inverse Problems, 21 (2005) 1851-1869.
56. M.T. NAIR and M.P. RAJAN, On improving accuracy for Arcangeli’s method for solving ill-posed equations, Integr. Equat. Oper. Th., 39 (2001) 496-501.
57. M.T. NAIR and M.P. RAJAN, Arcangeli’s discrepancy principle for a modified projection scheme for ill-posed problems, Numer. Funct. Anal. and Optim., 22 (1&2) (2001) 177-198.
58. M.T. NAIR and M.P. RAJAN, Arcangeli’s type discrepancy principles for a class of regularization methods using a modified projection scheme, Abstract and Appl. Anal., 6 (2001) 339-355.
59. M.T. NAIR and M.P. RAJAN, Generalized Arcangeli’s discrepancy principles for a class of regularization methods for solving ill-posed problems, J. Inv. Ill-Posed Problems, 10 (3) (2002) 281-294.
60. M.T. NAIR and E. SCHOCK, On the Ritz method and its generalization for ill-posed equations with non-self adjoint operators, J. International J. Pure and Appl. Math., 5 (2) (2003) 119-134.
61. M.T. NAIR, E. SCHOCK and U. TAUTENHAHN, Morozov’s discrepancy principle under general source conditions, Zeit. Anal. und ihere Anwend., 22 (1) (2003) 199-214.
62. A. NEUBAUER, An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error, Appl. Numer. Math., 14 (1988) 507-519.
63. R. PLATO and G. VAINIKKO, On the convergence of projection methods for solving ill-posed problems, Numer. Math., 57 (1990) 63-79.
64. S. PROSSDORF and B. SILBERMANN, Numerical Analysis for Integral Equations and Related Operator Equations, Birkhauser Verlag, Basel, 1991.
65. M.P. RAJAN, On Regularization and Approximation of Linear Ill-Posed Problems, Ph.D Thesis, IIT Madras, March 2001.
66. W. RUDIN, Real and Complex Analysis, 3rd Edition, McGraw-Hill International Editions, New York, 1098.
67. E. SCHOCK, Uber die Konvergengeschwindigkeit projektiver verfahren I, Math Z., 120 (1971) 148-156.
68. E. SCHOCK, Uber die Konvergengeschwindigkeit projektiver verfahren II, Math Z., 127 (1972) 191-198.
69. E. SCHOCK, Galerkin-like methods for equations of he second kind, J. Integral Equations, 4 (1982) 361-364.
70. E. SCHOCK, Parameter choice by discrepancy principles for the approximate solution of ill-posed problems, Integr. Equat. Oper. Th., 7 (1984) 895-898.
71. E. SCHOCK, Arbitrarily slowly convergence, uniform convergence and superconvergence of Galerkin-like methods, IMA J. Numer. Anal., 5 (1985) 153-160.
72. I.H. SLOAN, Four variants of Galerkin-like method for integral equations of the second kind, IMA J. Numer. Anal., 4 (1984) 9-17.
73. I.H. SLOAN and V. THOMEE, Superconvergence of the Galerkin iterates for for integral equations of the second kind, J. Integral Equations and Appl., 9 (1985) 1-23.
74. I.H. SLOAN, Superconvergence, In Numerical Solution of Integral Equations, Ed.: M.A. Golberg, Plenum Press, New York, Vol. 42 (1990) 35-70.
75. [75] G. VAINIKKO and P. UBA, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Ausutral. Math. Soc. Ser. B, 22 (1981) 431-438.
76. B. WENDROFF, Theoretical Numerical Analysis, Academic Press, New York, 1961.
77. A.G. WESCHULZ, What is the complexity of Fredholm problem of the second kind?, J. Integral Equations Appl. 9 (1985) 213-241.