Skip to main content
SpringerLink
Log in

On level sets and an approximation problem for the numerical solution of a free boundary problem

Über Niveaumengen und ein Approximationsproblem für die numerische Lösung eines freien Randwertproblems

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In optimization theory an essential assumption for the existence of solutions as well as the convergence of algorithms is the compactness of the so-called level sets. Here some general tools for the proof of this property in the case of minimum norm problems are provided. Finally for an approximation problem suggested in [3] for the numerical solution of a class of free boundary problems, compactness of the corresponding sets is shown. Therewith for this approximation problem, existence of a solution and in addition global convergence of the family of algorithms presented in [4] is guaranteed.

Zusammenfassung

Eine wesentliche Voraussetzung sowohl für die Existenz von Lösungen als auch für die Konvergenz von Algorithmen in der Optimierungstheorie ist die Kompaktheit der sogenannten Niveaumengen. Wir stellen hier einige Hilfsmittel für den Nachweis dieser Eigenschaft im Fall von Normminimierungsproblemen bereit. Schließlich kann Kompaktheit der entsprechenden Mengen für ein Approximationsproblem, das in [3] für die numerische Lösung einer Klasse von freien Randwertproblemen vorgeschlagen wurde, gezeigt werden. Damit ist für dieses Approximationsproblem einmal Existenz einer Lösung und zum anderen globale Konvergenz der in [4] beschriebenen Verfahren gesichert.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bernfeld, S. R., Lakshmikantham, V.: An introduction to nonlinear boundary value problems. New York-London: Academic Press 1974.

    Google Scholar 

  2. Krabs, W.: Optimierung und Approximation. Stuttgart: Teubner 1975.

    Google Scholar 

  3. Reemtsen, R., Lozano, C.: An approximation technique for the numerical solution of a Stefan problem. AMI, Technical Report No. 57 A, University of Delaware, Newark, Del. June 1979.

    Google Scholar 

  4. Reemtsen, R.: On the convergence of a class of nonlinear approximation methods. Technische Hochschule Darmstadt, Preprint-Nr. 536, März 1980.

  5. Rosenbloom, P. C., Widder, D. V.: Expansion in terms of heat polynomials and associated functions. Trans. Amer. Math. Soc.92, 220–266 (1959).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik Arbeitsgruppe 10 Kontroll- und Approximationstheorie, Technische Hochschule Darmstadt, Schlossgartenstrasse 7, D-6100, Darmstadt, Federal Republic of Germany

    R. Reemtsen

Authors
  1. R. Reemtsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reemtsen, R. On level sets and an approximation problem for the numerical solution of a free boundary problem. Computing 27, 27–35 (1981). https://doi.org/10.1007/BF02243436

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02243436

Subject classifications

Search

Navigation