SpringerLink
Log in

Mathematische Lösungspotentiale, Strategien und Dilemmata

  • Chapter
  • First Online:
Handbuch Oberflächennahe Geothermie

Zusammenfassung

Das Kapitel stellt eine integrative Zusammenschau aller wesentlichen in der oberflächennahen Exploration verfügbaren Datensysteme und Methoden aus mathematischer Sicht bereit, um auf diese Weise zur Risikoreduzierung in einem geothermischen Projekt beizutragen. Die in der Exploration auftretenden physikalisch motivierten Grundgleichungen werden beschrieben und klassifiziert. Sie führen zu sogenannten „Inversen Problemen“. Dieser Begriff wird in seiner mathematischen Bedeutung erläutert und die den Inversen Problemen innewohnende Eigenschaft der Schlecht-Gestelltheit im Hinblick auf die daraus resultierenden Schwierigkeiten bei der Lösung dem Leser verdeutlicht. Auf die Behandlung dieser Art von Gleichungen durch Regularisierung wird ausführlich eingegangen.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Similar content being viewed by others

Literatur

  • Anger, G.: A characterization of the inverse gravimetric source problem through extremal measures. Rev. Geophys. Space Phys. 19, 299–306 (1981)

    Article  Google Scholar 

  • Anger, G.: Inverse problems in differentail equation. Akademie Verlag, Berlin und Plenum Press, London (1990)

    Google Scholar 

  • Bauer, M., Freeden, W., Jacobi, H., Neu, T. (Hrsg.): Handbuch Tiefe Geothermie: Prospektion, Exploration, Realisierung, Nutzung. Springer, Berlin Heidelberg (2014)

    Google Scholar 

  • Baumeister, J.: Stable solution of inverse problems. Vieweg Braunschweig (1987)

    Google Scholar 

  • Biondi, B. L.: Three-dimensional seismic imaging. Society of Exploration Geophysicists, Tulsa (2006)

    Google Scholar 

  • Björck, Å. Elden, L.: Ill-Posed Problems: Theory and Practice, Kap. Methods in Numerical Algebra for Ill-Posed Problems. Reidel, Dodrecht and Boston (1981)

    Google Scholar 

  • Blakely, R.: Potential theory in gravity and magnetic application. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  • Blick, C.: Multiscale Potential Methodes in Geothermal Research. Dissertation, AG Geomathematik, TU Kaiserslautern (2015)

    Google Scholar 

  • Blick, C., Freeden, W., Nutz, H.: Gesteigerte Investmentsicherheit durch geomathematische Verfahren. Sonderveröffentlichung des GDMB Verlag GmbH S. 1–22 (2015)

    Google Scholar 

  • Carlslaw, H. S., Jaeger, J. C.: Conduction of heat in solids, 2. Aufl. Oxford University Press, Oxford (1959)

    Google Scholar 

  • Engl, H.: Integralgleichungen. Springer, Berlin (1997)

    Chapter  Google Scholar 

  • Engl, H. W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

  • Engl, H., Louis, A.K., Rundell, W. (Hrsg.): Inverse problems in geophysical applications. SIAM, Philadelphia (1997)

    Google Scholar 

  • Freeden, W.: Multiscale modelling of speaceborne geodata. Teubner, Stuttgart (1999)

    Google Scholar 

  • Freeden, W.: Geomathematics: Its role, its aim, and its potential. In: Freeden, W., Nashed, M. Z., Sonar, T. (Hrsg.) Handbook of Geomathematics, Bd. 1, 2. Aufl., S. 3–78. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  • Freeden, W., Blick, C.: Signal decorrelation by means of multiscale methods. World of Mining. 65(5), 304–317 (2013)

    Google Scholar 

  • Freeden, W., Gerhards, C.: Geomathematically oriented potential theory. CRC Press, Taylor & Francis, Boca Raton (2013)

    Google Scholar 

  • Freeden, W., Mayer, C.: Stokes problem, layer potentials and regularizations, and multiscale applications. In: Freeden, W., Nashed, M. Z., Sonar, T. (Hrsg.) Handbook of geomathematic}, Bd. 1, 2. Aufl., S. 1155–1254. Springer, Heidelberg (2015)

    Google Scholar 

  • Freeden, W., Nutz, H.: Mathematik als Schlüsseltechnologie zum Verständnis des Systems “Tiefe Geothermie“. Jahresber Dtsch Math-Ver., Springer, Berlin. Heidelber. 117(1), 45–84 (2014a)

    Google Scholar 

  • Freeden, W., Nutz, H.: Mathematische Methoden. In: Bauer, M., Freeden, W., Jacobi, H., Neu, T. (Hrsg.) Handbuch Tiefe Geothermie, S. 125–221. Springer Spektrum, Heidelberg (2014b)

    Google Scholar 

  • Gehlin, S.: Thermal response test method development and evalutation. Dissertation, Luleå tekniska universitet (2002)

    Google Scholar 

  • Grasmair, M., Haltmeier, M., Scherzer, O.: Sparsity in inverse geophysical problems. In: Freeden, W., Nashed, M. Z., Sonar, T. (Hrsg.) Handbook of Geomathematics, 2. Aufl., S. 1659–1687. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  • Groetsch, C.: The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston (1984)

    Google Scholar 

  • Hadamard, J.: Sur les problèmes aux dérivés partielles et leur signification physique. Princeton Univ. Bull. 13, 49–52 (1902)

    Google Scholar 

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

    Google Scholar 

  • Jacobs, F., Meyer, H.: Geophysik – Signale aus der Erde. Teubner, Stuttgart (1992)

    Book  Google Scholar 

  • Kammerer, W., Nashed, M. Z.: Iterative methods for best approximate solutions of linear integral equation of the first and second kind. J. Math. Anal. Appl. 40, 547–573 (1972)

    Article  Google Scholar 

  • Kellogg, O.: Foundations of potential theory. Frederick Ungar Publishing Company, New York (1929)

    Chapter  Google Scholar 

  • Kirsch, A.: An introduction to the mathematical theory of inverse problems. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  • Lieusternik, L. A., Sobolev, V. J.: Elements of Functional Analysis. Ungar, New York (1961)

    Google Scholar 

  • Louis, A. K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart (1989)

    Chapter  Google Scholar 

  • Marti, J. T.: On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems. Math. Comp. 34, 521–527 (1980)

    Article  Google Scholar 

  • Menke, W.: Geophysical data analysis. Discrete inverse theory. Academic Press, Orlando (1984)

    Chapter  Google Scholar 

  • Morozov, V. A.: Methods for solving incorrectly posed problems. Springer, New York (1984)

    Book  Google Scholar 

  • Müller, C.: Foundations of the Mathemaical Theory of Electromagnetic Waves. Springer, Berlin, Heidelberg, New York (1969)

    Book  Google Scholar 

  • Möhringer, S.: Deccorrelation of gravimetric data. Dissertation, AG Geomathematik. TU Kaiserslautern (2014)

    Google Scholar 

  • Nashed, M. Z.: Generalized inverse, normal solvability and iteration for singular operator equations. In: Rall, L. B. (Hrsg.) Nonlinear functional analysis and applications, S. 311–359. Academic, New York (1971)

    Chapter  Google Scholar 

  • Nashed, M. Z. (Hrsg.): Generalized Inverses and Applications. Academic Press, New York (1976a)

    Google Scholar 

  • Nashed, M. Z.: Generalized Inverses and Applications, Kap. Aspects of Generalized Inverses in Analysis and Regularization, S. 193–244. Academic Press, New York (1976b)

    Chapter  Google Scholar 

  • Nashed, M. Z.: On moment-discretization and least-squares solutions of linear integration equations of the first kind. J. Math. Anal. Appl. 53, 359–366 (1976c)

    Article  Google Scholar 

  • Nashed, M. Z.: Operator-theoretic and computational approaches to ill-posed problems with application to antenna theory. IEEE Trans. Antennas Propagation. 29, 220–231 (1981a)

    Article  Google Scholar 

  • Nashed, M. Z.: Ill-Posed Problems: Theory and Practice, Kap. Analysis and Implementation of Projection-Regularization Methods for Ill-Posed Problems. Reidel, Dordrecht and Boston (1981b)

    Google Scholar 

  • Nashed, M. Z., Scherzer, O.: Inverse problems, image analysis and medical imaging (contemporary mathematics). Amer. Math. Soc. 313 (2002)

    Google Scholar 

  • Nashed, M. Z., Wahba, G.: Convergence rate of approximate solutions to linear operator equations of the first kind. Math. Comp. 28, 69–80 (1974a)

    Article  Google Scholar 

  • Nashed, M. Z., Wahba, G.: Regularization and approximation of linear operator equations in reproducing kernel spaces. Bull. Amer. Math. Soc. 80, 1213–1218 (1974b)

    Article  Google Scholar 

  • Natterer, F.: The finite element method for ill-posed problems. RAIRO Analyse Numerique. 11, 271–278 (1977)

    Article  Google Scholar 

  • Nettleton, L.: Gravity and magnetics in oil prospecting. McGraw-Hill, New York (1976)

    Google Scholar 

  • Scherzer, O. (Hrsg.): Handbook of mathematical methods in imaging, 2. Aufl., Springer Science & Business Media, New York (2015)

    Google Scholar 

  • Snieder, R.: The perturbation method in elastic wave scattering, scattering and Inverse scattering in pure and applied science, Kap. General theory of elastic wave scattering, S. 528–542. Academic Press, San Diago (2002)

    Chapter  Google Scholar 

  • Song, M.: Regularization-Projection Methods and Finite Element Approximations for Ill-Posed Linear Operator Equations. Dissertation, Univ. Michigan, Ann Arbor (1978)

    Google Scholar 

  • 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, 798–825 (1974)

    Article  Google Scholar 

  • Symes, W. W.: Kinematics of reverse time s-g migration. Technical report, Technical report, Rice University, Houston (2003)

    Google Scholar 

  • Symes, W. W.: Reverse time migration with optimal checkpointing. Geophys., 72, SM213–SM221 (2007)

    Article  Google Scholar 

  • Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation. Geophys. 49, 1259–1266 (1984)

    Article  Google Scholar 

  • Tikhonov, A. N.: On the stability of inverse problems. Dokl. Akad. Nauk SSSR. 39, 195–198 (1943)

    Google Scholar 

  • Tikhonov, A. N.: On the solution of incorrectly formulated problems and the regularization method. Dokl. Akad. Nauk SSSR. 151, 501–504 (1963)

    Google Scholar 

  • Varah, J.: On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems. SIAM J. Numer. Anal. 10, 257–267 (1973)

    Article  Google Scholar 

  • Yilmaz, O.: Seismic data analysis: processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists, Tulsa (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehr- und Forschungsgebiet „Geomathematik“, TU Kaiserslautern, Paul-Ehrlich-Straße 26, 67653, Kaiserslautern, Deutschland

    Willi Freeden

  2. CBM – Gesellschaft für Consulting, Business and Management mbH, Niederbexbacher Straße 67, 66450, Bexbach, Deutschland

    Helga Nutz

Authors
  1. Willi Freeden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helga Nutz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Willi Freeden .

Editor information

Editors and Affiliations

  1. Lehr- und Forschungsgebiet „Arbeits- und Gesundheitsschutz“, RWTH Aachen, Aachen, Germany

    Mathias Bauer

  2. Lehr- und Forschungsgebiet „Geomathematik“, TU Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  3. Jacobi & Partner Industrieberatung, Essen, Germany

    Hans Jacobi

  4. Bergbau-Energie-Umwelt-Infrastruktur, proG.E.O. Ingenieurgesellschaft mbH, Saarbrücken, Germany

    Thomas Neu

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Freeden, W., Nutz, H. (2018). Mathematische Lösungspotentiale, Strategien und Dilemmata. In: Bauer, M., Freeden, W., Jacobi, H., Neu, T. (eds) Handbuch Oberflächennahe Geothermie. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-50307-2_5

Download citation

Publish with us

Policies and ethics

Search

Navigation