Skip to main content
SpringerLink
Log in

Inverse Problems

  • Chapter
Book cover Trends in Mathematical Optimization

  • 518 Accesses

Abstract

Following Keller [A6] we call two problems inverses of one another, if the formulations of each involves all or part of the solution of the other. From this definition it is arbitrary which one of the two problems we call the direct and which one the inverse problem. However, for historical - or other - reasons one of the two problems has been studied extensively for some time and is better understood than the other. This one we would call the direct problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

(A)General Aspects for Improperly-Posed and Inverese Problems

  1. Anger, G.: Einige Betrachtungen über inverse Probleme, Identifikationsprobleme und inkorrekt gestellte Probleme. In: Jahrbuch Uberblicke Mathemtik 1982, Springer, New York etc., 55–71.

    Google Scholar 

  2. Anger, G.(ed.): Inverse and improperly posed problems in differential equations. Akademie Verlag, Berlin, 1979.

    Google Scholar 

  3. Deuflhard, E. and Hairer, E.(eds.): Numerical treatment of inverse problems in differential and integral equations. Birkhäuser, Boston, etc., 1983.

    Google Scholar 

  4. Gorenflo, R.(ed.): Inkorrekt gestellte Probleme I, II. Conference Proceedings. Free University of Berlin, Berlin, 1977/78.

    Google Scholar 

  5. Hämmerlin, G. and Hoffmann, K.-H.(eds.): Improperly posed problems and their numerical treatment. ISNM 63, Birkhäuser, Basel etc., 1983.

    Google Scholar 

  6. Keller, J. B.: Inverse problems. Amer. Math. Monthly 83 (1976), 107–118.

    Article  Google Scholar 

  7. Lavrientiev, M. M. et al.: Some improperly posed problems of mathematical physics. Springer, New York etc., 1967.

    Google Scholar 

  8. Parker, R. L.: Understanding inverse theory. Ann. Rev. Earth. Planet. Sei. (1977), 35–64.

    Google Scholar 

  9. Payne, L. E.: Improperly posed problems in partial differential equations. Regional conference series in applied mathematics. SIAM, Philadelphia, 1975.

    Google Scholar 

  10. Sabatier, P. C.(ed.): Applied inverse problems. Lecture Notes in Physics 85, Springer, New York etc., 1978.

    Google Scholar 

  11. Tikhonov, A. M. and Arsenin, V. Y.: Solutions of ill-posed problems. Winston k Sons, Washington, 1977.

    Google Scholar 

(B) Inverse Potential Problem

  1. Anger, G.: Uniquely determined mass distributions in inverse problems. In: Veröffentlichungen des Zentralinstituts für Physik der Erde 52,2, Potsdam, 1977.

    Google Scholar 

  2. Brodsky, M. A.: On the uniqueness of the inverse potential problem for homogeneous polyhedrons. SIAM J. Appl. Math. 46 (1986), 345–350.

    Google Scholar 

  3. Weck, N.: Inverse Probleme der Potentialtheorie. Appl. Anal. 2 (1972), 195–204.

    Article  Google Scholar 

(C) Computer Tomography

  1. Herman, G. T.(ed.): Image reconstruction from projections. Topics in Applied Physics 32 (1979), Springer, New York etc.

    Google Scholar 

  2. Herman, G. T. and Lewitt, R. M.: Overview of image reconstruction from projections. In: Image reconstruction from projections (Hermann, G. T. ed.). Topics in Applied Physics 32 (1979), Springer, New York etc.

    Google Scholar 

  3. Herman, G. T. and Natterer, F.(eds.): Mathematical aspects of computerized tomography. Lecture Notes in Medical Informatics 8 (1981), Springer, New York etc.

    Google Scholar 

  4. Natterer, F.: Numerical inversion of the Radon transform. Numer. Math. 30 (1978), 81–91.

    Article  Google Scholar 

  5. Natterer, F.: A Sobolev space analysis of picture reconstruction. SIAM J. Appl. Math. 39 (1980), 402–411.

    Article  Google Scholar 

  6. Natterer, F.: Einige Beiträge der Mathematik zur Computer-Tomographie. ZAMM 64 (1984), T252–T260.

    Google Scholar 

  7. Roberts, A. J.: An introduction to the technique of reconstruction. SIAM J. Math. Anal. 16 (1985), 1243–1257.

    Google Scholar 

  8. Sheep, L. A. and Kruskal, J. B.: Computerized tomography: The new medical X-ray technology. Math. Monthly (1978), 420–439.

    Google Scholar 

  9. Schömberg, H. and Tasto, M.: Computer-Tomographie mit Ultraschall. Phys. Bl. 36 (1980), 311–313.

    Google Scholar 

(D) Backwards Heat Equation

  1. Buzbee, B. L. and Carasso, A.: On the numerical computation of parabolic problems for preceeding times. Math, of Comp. 27 (1973), 237–266.

    Article  Google Scholar 

  2. Carasso, A.: Error bounds in the final value problem for the heat equation. SIAM J. Math. Anal. 7 (1976), 195–199.

    Google Scholar 

  3. Colton, D. S.: The approximation of solutions to the backwards heat equation in a nonhomogeneous medium. J. Math. Anal. Appl. 72 (1979), 418–429.

    Article  Google Scholar 

  4. Elden, L.: Regularization of the backwards solution of parabolic problems. In: Inverse and improperly posed problems in differential equations ( Anger, G. ed.). Berlin, 1979.

    Google Scholar 

  5. Ewing, R. E.: The approximation of certain parabolic equations backwards in time by Sobolev equations. SIAM J. Math. Anal. 6 (1975), 283–294.

    Article  Google Scholar 

  6. Höhn, W.: Finite elements for parabolic equations backwards in time. Preprint no. 556, Technische Hochschule Darmstadt, 1980.

    Google Scholar 

  7. Kobayashi, T.: Initial state determination for distributed parameter systems. SIAM J. Control and Optim. 14 (1976), 934–944.

    Article  Google Scholar 

  8. Miller, K.: Efficient numerical methods for backward solution of parabolic problems with variable coefficients. In: Improperly posed problems ( Carasso, A. ed.), Pitman, Boston etc., 1975.

    Google Scholar 

  9. Showalter, R. E.: The final value problem for evolution equations. J. Math. Anal. Appi. 47 (1974), 563–572.

    Article  Google Scholar 

(E) Abel’s Equation

  1. Brunner, H.: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations. J. Comput. Appi. Math. 8 (1982), 213–229.

    Article  Google Scholar 

  2. Gorenflo, R.: Abel integral equations: Applications-motivated solution concepts. Preprint No. 209, Freie Universität Berlin, 1986.

    Google Scholar 

  3. Gorenflo, R. and Vessella, S.: Basic theory and some applications of Abel integral equations. Preprint Consiglio Nazionale delle Ricerche, Istituto di Analisi Globale e Applicazioni, Firenze 1984.

    Google Scholar 

  4. Krasnov, M. L., Kissélev, A. I. and Makarenko, G. I.: Equations intégrales: problèmes et exércices. Editions Mir, Moscow, 1977.

    Google Scholar 

  5. Tsalyuk, Z. B.: Volterra integral equations. J. Soviet Mathematics 12 (1979), 715–758.

    Article  Google Scholar 

  6. Zabreyko, P. P. et al: Integral equations - A reference text. Noordhoff, Leiden, 1975.

    Google Scholar 

(F) Inverse Problem for the Helmsholtz Equation

  1. Colton, D.: The inverse scattering problem for time-harmonic acoustic waves. SIAM Review 26 (1984), 323–350.

    Article  Google Scholar 

  2. Colton, D. and Kirsch, A.: Dense sets and far Held pattern in acoustic wave propagation. SIAM J. Math. Anal. 15 (1984), 996–1006.

    Article  Google Scholar 

  3. Colton, D. and Kress, R.: Integral equation methods in scattering theory. Wiley, New York etc., 1983.

    Google Scholar 

  4. Colton, D. and Monk, P.: A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. SIAM J. Appi Math. 45 (1985), 1039–1053.

    Article  Google Scholar 

  5. Colton, D. and Monk, P.: A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II. SIAM J. Appi. Math. 46 (1986), 506–523.

    Article  Google Scholar 

  6. Colton, D. and Monk, P.: The numerical solution of the three dimensional inverse scattering problem for time-harmonic acoustic waves. University of Delaware, Technical Report.

    Google Scholar 

  7. Kirsch, A.: Generalized boundary value and control problems for the Helmholtz equation. Göttingen, Habilitationsschrift 1984.

    Google Scholar 

  8. Lax, P. D. and Phillips, R. S.: Scattering theory. Academic Press, New York etc., 1967.

    Google Scholar 

  9. Leis, R.: Zur Dinchletschen Randwertaufgabe des Aussenraums der Schwingungsgleichung. Math. Zeitschrift 90 (1965), 205–211.

    Article  Google Scholar 

  10. Roger, A.: Newton Kantorovich algorithm applied to an electromagnetic inverse problem. IEEE Trans. Ant. Prop. AP-29 (1981), 232–238.

    Article  Google Scholar 

  11. Sleeman, B. D.: The inverse problem of acoustic scattering. IMA J. Appl. Math. 29 (1982), 113–142.

    Article  Google Scholar 

(G) Identification of Parameters in Parabolic and Elliptic Equations

  1. Astrom, K. J. and Eykhoff, P.: Systems identification - A survey. Automatica 7 (1971), 123–162.

    Article  Google Scholar 

  2. Banks, H. T.: A survey of some problems and recent results for parameter estimation and optimal control in delay and distributed parameter systems. In: Volterra and Functional differential Equations, ( Hannsgen, K. B. et al. eds.), Marcel Dekker, New York, 1982, 3–24.

    Google Scholar 

  3. Banks, H. T. and Kunisch, K.: An approximation theory for nonlinear partial dif¬ferential equations with applications to identification and control. SIAM J. Control and Optim. 20 (1982), 815–849.

    Article  Google Scholar 

  4. Chavent, G.: Optimization problems in the inversion of the 2-D wave equation. (To appear).

    Google Scholar 

  5. Hoffmann, K.-H.: Identifikationsprobleme bei partiellen Differentialgleichungen. In: Numerische Behandlung von Differentialgleichungen, Vol. 3 (Albrecht and Collatz eds.), ISNM 56 (1981), 97–116.

    Google Scholar 

  6. Kubrusly, C. S.: Distributed parameter system identifications. Int. J. Control 26 (1977), 509–535.

    Article  Google Scholar 

  7. Nguyen, V. V. and Wood, E. F.: Review and unification of linear identifiability concepts. SIAM Rev. 24 (1982), 34–51.

    Article  Google Scholar 

  8. Polis, M. P. and Goodson, R. E.: Parameter identification in distributed systems: A synthesizing overview. Proc. IEEE 64 (1976), 43–61.

    Article  Google Scholar 

  9. Ray, W. H.: Some recent applications of distributed parameter systems theory - A survey. Automatica 14 (1978), 281–287.

    Article  Google Scholar 

(H) General Theory for Equations of the First Kind

  1. Engl, H. and Neubauer, A.: Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind. In: Constructive Methods for the Practical Treatment of Integral Equations (Hämmerlin, G. and Hoffmann, K.-H. eds.), ISNM 73, Birkhäuser, Basel etc., 1985.

    Google Scholar 

  2. Friedman, V.: Method of successive approximations for Fredholm integral equations of the first kind. Uspehi Mat. Nauk 11 (1956), 233–234 (in Russian).

    Google Scholar 

  3. Groetsch, C. W.: The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston etc., 1984.

    Google Scholar 

  4. Ivanov, V. K.: Approximate solution of operator equations of the first kind. U.S.S.R. Comput. Math, and Math. Phys. 6 (1966), 197–205.

    Article  Google Scholar 

  5. Landweber, L.: An iteration formula for Fredholm integral equations of the first kind. Amer. J. Math. 73 (1951), 615–624.

    Article  Google Scholar 

  6. Morozov, V. A.: On the solutions of functional equations by the method of regularization. Soviet Math. Doklady 7 (1966), 414–417.

    Google Scholar 

  7. Morozov, V. A.: The error principle in the solution of operational equations by the regularization method. U.S.S.R. Comput. Math, and Math. Phys. 8 (1968), 63–87.

    Article  Google Scholar 

  8. Strand, O. N.: Theory and methods related to the singular function expansion and Landweber’s iteration for integral equations of the first kind. SIAM J. Numer. Anal. 11 (1974), 789–825.

    Article  Google Scholar 

  9. Tikhonov, A. N.: Regularization of incorrectly posed problems. Soviet Doklady 4 (1963), 1624–1627.

    Google Scholar 

(I) Other References

  1. Hadamard, J.: Lectures on the Cauchy problem in linear partial differential equations. Yale University Press, New Haven, 1923.

    Google Scholar 

  2. Herbst, O.: Konstruktion von Quasi-Newton Verfahren mit Hilfe konjugierter Richtungen. PhD Thesis, Göttingen, 1984.

    Google Scholar 

  3. Kerz, W.: Einführung in die Geophysik I, II. Bibliographisches Institut Mannheim, 1985.

    Google Scholar 

  4. Lanczos, C.: Linear differential operators. Van Nostrand, New York, 1961.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestr. 16 - 18, 3400, Göttingen, West Germany

    Andreas Kirsch

Authors
  1. Andreas Kirsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    Karl-Heinz Hoffmann

  2. Mathematisches Institut, Universität Bayreuth, Universitätsstrasse 30, D-8580, Bayreuth, Germany

    Jochem Zowe

  3. U.F.R. Mathématiques, Informatique, Université Paul Sabatier, 118, Route de Narbonne, F-31062, Toulouse Cédex, France

    Jean-Baptiste Hiriart-Urruty

  4. INRIA, Rocquencourt, F-78153, Le Chesnay, France

    Claude Lemarechal

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Birkhäuser Verlag Basel

About this chapter

Cite this chapter

Kirsch, A. (1988). Inverse Problems. In: Hoffmann, KH., Zowe, J., Hiriart-Urruty, JB., Lemarechal, C. (eds) Trends in Mathematical Optimization. International Series of Numerical Mathematics/Internationale Schriftenreihe zur Numerischen Mathematik/Série internationale d’Analyse numérique, vol 84. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9297-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9297-1_9

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9984-0

  • Online ISBN: 978-3-0348-9297-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation