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Linear Convergence of a Rearrangement Method for the One-dimensional Poisson Equation

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Abstract

In this paper, we study a rearrangement method for solving a maximization problem associated with Poisson’s equation with Dirichlet boundary conditions. The maximization problem is to find the forcing within a certain admissible set as to maximize the total displacement. The rearrangement method alternatively (i) solves the Poisson equation for a given forcing and (ii) defines a new forcing corresponding to a particular super-level-set of the solution. Rearrangement methods are frequently used for this problem and a wide variety of similar optimization problems due to their convergence guarantees and observed efficiency; however, the convergence rate for rearrangement methods has not generally been established. In this paper, for the one-dimensional problem, we establish linear convergence. We also discuss the higher dimensional problem and provide computational evidence for linear convergence of the rearrangement method in two dimensions.

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Acknowledgements

The authors would like to thank the Mathematics Division, National Center of Theoretical Sciences, Taipei, Taiwan for hosting a research pair program during June 15-June 30, 2019 to support this project.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA, 91711, USA

    Chiu-Yen Kao

  2. Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75918-74934, Iran

    Seyyed Abbas Mohammadi

  3. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Braxton Osting

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  1. Chiu-Yen Kao
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  2. Seyyed Abbas Mohammadi
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  3. Braxton Osting
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Correspondence to Braxton Osting.

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Chiu-Yen Kao is supported in part by an NSF Grant DMS-1818948. Braxton Osting acknowledges partial support from NSF DMS 16-19755 and 17-52202.

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Kao, CY., Mohammadi, S.A. & Osting, B. Linear Convergence of a Rearrangement Method for the One-dimensional Poisson Equation. J Sci Comput 86, 6 (2021). https://doi.org/10.1007/s10915-020-01389-5

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  • DOI: https://doi.org/10.1007/s10915-020-01389-5

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