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Regularization by Discretization

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An Introduction to the Mathematical Theory of Inverse Problems

Part of the book series: Applied Mathematical Sciences ((AMS,volume 120))

Abstract

In this chapter, we study a different approach to regularizing operator equations of the form Kx = y, where x and y are elements of certain function spaces. This approach is motivated by the fact that for the numerical treatment of such equations one has to discretize the continuous problem and reduce it to a finite system of (linear or nonlinear) equations. We see in this chapter that the discretization schemes themselves are regularization strategies in the sense of Chap. 2.

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References

  1. R.A. Adams and J. Fournier. Sobolev Spaces. Academic Press, 2nd, repr. edition, 2005.

    Google Scholar 

  2. D.N. Arnold and W.L. Wendland. On the asymptotic convergence of collocation methods. Math. Comput., 41:349–381, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  3. D.N. Arnold and W.L. Wendland. The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math., 47:310–341, 1985.

    Article  MathSciNet  Google Scholar 

  4. K.E. Atkinson. A discrete Galerkin method for first kind integral equations. J. Integ. Equat. Appl., 1:343–363, 1988.

    Article  MATH  Google Scholar 

  5. K.E. Atkinson and I.H. Sloan. The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math. Comput., 56:119–139, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Backus and F. Gilbert. The resolving power of gross earth data. Geophys. J. R. Astron. Soc, 16:169–205, 1968.

    Article  MATH  Google Scholar 

  7. G. Backus and F. Gilbert. Uniqueness in the inversion of inaccurate gross earth data. Philos. Trans. R. Soc. London, 266:123–197, 1970.

    Article  MathSciNet  Google Scholar 

  8. J. Baumeister. Stable Solutions of Inverse Problems. Vieweg, Braunschweig, 1987.

    Book  Google Scholar 

  9. G. Bruckner. On the regularization of the ill-posed logarithmic kernel integral equation of the first kind. Inverse Problems, 11:65–78, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Caccin, C. Roberti, P. Russo, and L.A. Smaldone. The Backus-Gilbert inversion method and the processing of sampled data. IEEE Trans. Signal Process., 40:2823–2825, 1992.

    Article  Google Scholar 

  11. D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. Wiley-Interscience, New York, 1983.

    MATH  Google Scholar 

  12. M. Costabel. Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal., 19:613–626, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Costabel, V.J. Ervin, and E.P. Stephan. On the convergence of collocation methods for Symm’s integral equation on smooth open arcs. Math. Comput., 51:167–179, 1988.

    MathSciNet  MATH  Google Scholar 

  14. M. Costabel and E.P. Stephan. On the convergence of collocation methods for boundary integral equations on polygons. Math. Comput., 49:461–478, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Costabel and W. Wendland. Strong ellipticity of boundary integral operators. J. Reine Angew. Math., 372:39–63, 1986.

    MathSciNet  Google Scholar 

  16. L. Eldén. Algorithms for the regularization of ill-conditioned least squares problems. BIT, 17:134–145, 1977.

    Article  MATH  Google Scholar 

  17. L. Eldén. An algorithm for the regularization of ill-conditioned banded least squares problems. SIAM J. Sci. Stat. Comput., 5:237–254, 1984.

    Article  MATH  Google Scholar 

  18. J. Elschner. On spline approximation for a class of integral equations. I: Galerkin and collocation methods with piecewise polynomials. Math. Meth. Appl. Sci., 10:543–559, 1988.

    Google Scholar 

  19. H. Engl. On least-squares collocation for solving linear integral equations of the first kind with noisy right-hand-side. Boll. Geodesia Sc. Aff., 41:291–313, 1982.

    Google Scholar 

  20. H. Engl and W. Grever. Using the L-curve for determining optimal regularization parameters. Numer. Math., 69:25–31, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  21. B.G. Galerkin. Expansions in stability problems for elastic rods and plates. Vestnik Inzkenorov, 19:897–908, 1915. in Russian.

    Google Scholar 

  22. G.H. Golub and C. Reinsch. Singular value decomposition and least squares solutions. Numer. Math., 14:403–420, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Haario and E. Somersalo. The Backus–Gilbert method revisited: Background, implementation and examples. Numer. Funct. Anal. Optim., 9:917–943, 1985.

    Article  MathSciNet  Google Scholar 

  24. G. Hellwig. Partielle Differentialgleichungen. Teubner Verlag, Stuttgart, 1960.

    MATH  Google Scholar 

  25. G.C. Hsiao, P. Kopp, and W.L. Wendland. A Galerkin collocation method for some integral equations of the first kind. Computing, 25:89–113, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  26. G.C. Hsiao and R.C. MacCamy. Solution of boundary value problems by integral equations of the first kind. SIAM Review, 15:687–705, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  27. G.C. Hsiao and W.L. Wendland. A finite element method for some integral equations of the first kind. J. Math. Anal. Appl., 58:449–481, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  28. G.C. Hsiao and W.L. Wendland. The Aubin–Nitsche lemma for integral equations. J. Integ. Eq., 3:299–315, 1981.

    MathSciNet  MATH  Google Scholar 

  29. S.P. Huestis. The Backus–Gilbert problem for sampled band-limited functions. Inverse Problems, 8:873–887, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Joe and Y. Yan. A piecewise constant collocation method using cosine mesh grading for Symm’s equation. Numer. Math., 65:423–433, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  31. W.J. Kammerer and M.Z. Nashed. Iterative methods for best approximate solutions of integral equations of the first and second kinds. J. Math. Anal. Appl., 40:547–573, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Kirsch, B. Schomburg, and G. Berendt. The Backus–Gilbert method. Inverse Problems, 4:771–783, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Kirsch, B. Schomburg, and G. Berendt. Mathematical aspects of the Backus–Gilbert method. In B. Kummer, A. Vogel, R. Gorenflo, and C.O. Ofoegbu, editors, Inverse Modeling in Exploration Geophysics, Braunschweig, Wiesbaden, 1989. Vieweg–Verlag.

    Google Scholar 

  34. R. Kress, 1994. personal communication.

    Google Scholar 

  35. R. Kress. Linear Integral Equations. Springer, New York, 2nd edition, 1999.

    Book  MATH  Google Scholar 

  36. R. Kress and I.H. Sloan. On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numer. Math., 66:199–214, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  37. A.K. Louis. Inverse und schlecht gestellte Probleme. Teubner–Verlag, Stuttgart, 1989.

    Google Scholar 

  38. A.K. Louis and P. Maass. Smoothed projection methods for the moment problem. Numer. Math., 59:277–294, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  39. J.T. Marti. An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind. SIAM J. Numer. Anal., 15:1071–1076, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  40. J.T. Marti. On the convergence of an algorithm computing minimum-norm solutions to ill-posed problems. Math. Comput., 34:521–527, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  41. M.Z. Nashed. On moment discretization and least-squares solution of linear integral equations of the first kind. J. Math. Anal. Appl., 53:359–366, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  42. M.Z. Nashed and G. Wahba. Convergence rates of approximate least squares solution of linear integral and operator equations of the first kind. Math. Comput., 28:69–80, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  43. I.P. Natanson. Constructive Function Theory. Frederick Ungar, New York, 1965.

    Google Scholar 

  44. F. Natterer. Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math., 28:329–341, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  45. G.I. Petrov. Application of Galerkin’s method to a problem of the stability of the flow of a viscous fluid. Priklad. Matem. Mekh., 4:3–12, 1940. (In Russian).

    Google Scholar 

  46. Lord Rayleigh. On the dynamical theory of gratings. Proc. R. Soc. Lon. A, 79:399–416, 1907.

    Google Scholar 

  47. G.R. Richter. Numerical solution of integral equations of the first kind with nonsmooth kernels. SIAM J. Numer. Anal., 15:511–522, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Ritz. Über lineare Funktionalgleichungen. Acta Math., 41:71–98, 1918.

    Google Scholar 

  49. G. Rodriguez and S. Seatzu. Numerical solution of the finite moment problem in a reproducing kernel Hilbert space. J. Comput. Appl. Math., 33:233–244, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Saranen. The modified quadrature method for logarithmic-kernel integral equations on closed curves. J. Integ. Eq. Appl., 3:575–600, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Saranen and I.H. Sloan. Quadrature methods for logarithmic-kernel integral equations on closed curves. IMA J. Numer. Anal., 12:167–187, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Saranen and W.L. Wendland. On the asymptotic convergence of collocation methods with spline functions of even degree. Math. Comput., 45:91–108, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  53. G. Schmidt. On spline collocation methods for boundary integral equations in the plane. Math. Meth. Appl. Sci., 7:74–89, 1985.

    Article  MATH  Google Scholar 

  54. E. Schock. What are the proper condition numbers of discretized ill-posed problems? Lin. Alg. Appl., 81:129–136, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  55. B. Schomburg and G. Berendt. On the convergence of the Backus–Gilbert algorithm. Inverse Problems, 3:341–346, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  56. T.I. Seidman. Nonconvergence results for the application of least squares estimation to ill-posed problems. J. Optim. Theory Appl., 30:535–547, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  57. I.H. Sloan. Error analysis of boundary integral methods. Acta Numer., 1:287–339, 1992.

    Article  MathSciNet  Google Scholar 

  58. I.H. Sloan and B.J. Burn. An unconventional quadrature method for logarithmic-kernel integral equations on closed curves. J. Integ. Eq. Appl., 4:117–151, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  59. I.H. Sloan and W.L. Wendland. A quadrature-based approach to improving the collocation method for splines of even degree. Z. Anal. Anw., 8:362–376, 1989.

    MathSciNet  Google Scholar 

  60. W. Wendland. On Galerkin collocation methods for integral equations of elliptic boundary value problems. In R. Leis, editor, Numerical Treatment of Integral Equations, volume ISNM 53, pages 244–275, Basel, 1979. Birkhäuser–Verlag.

    Google Scholar 

  61. J. Werner. Optimization Theory and Applications. Vieweg–Verlag, Braunschweig, Wiesbaden, 1984.

    Google Scholar 

  62. X.G. Xia and M.Z. Nashed. The Backus–Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces. Inverse Problems, 10:785–804, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  63. X.G. Xia and Z. Zhang. A note on ‘the Backus–Gilbert inversion method and the processing of sampled data’. IEEE Trans. Signal Process., 43:776–778, 1995.

    Article  Google Scholar 

  64. Y. Yan and I.H. Sloan. On integral equations of the first kind with logarithmic kernels. J. Integ. Eq. Appl., 1:549–579, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  65. Y. Yan and I.H. Sloan. Mesh grading for integral equations of the first kind with logarithmic kernel. SIAM J. Numer. Anal., 26:574–587, 1989.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Karlsruhe Institute of Technology (KIT), Kaiserstrasse 89, 76133, Karlsruhe, Germany

    Andreas Kirsch

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  1. Andreas Kirsch
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Kirsch, A. (2011). Regularization by Discretization. In: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8474-6_3

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