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The Bayesian approach to inverse Robin problems

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  • Rasmussen, Aksel Kaastrup
  • Seizilles, Fanny
  • Girolami, Mark
  • Kazlauskaite, Ieva

Abstract

In this paper, we investigate the Bayesian approach to inverse Robin problems. These are problems for certain elliptic boundary value problems of determining a Robin coefficient on a hidden part of the boundary from Cauchy data on the observable part. Such a nonlinear inverse problem arises naturally in the initialization of large-scale ice sheet models that are crucial in climate and sea-level predictions. We motivate the Bayesian approach for a prototypical Robin inverse problem by showing that the posterior mean converges in probability to the data-generating ground truth as the number of observations increases. Related to the stability theory for inverse Robin problems, we establish a logarithmic convergence rate for Sobolev-regular Robin coefficients, whereas for analytic coefficients we can attain an algebraic rate. The use of rescaled analytic Gaussian priors in posterior consistency for nonlinear inverse problems is new and may be of separate interest in other inverse problems. Our numerical results illustrate the convergence property in two observation settings.

Suggested Citation

  • Rasmussen, Aksel Kaastrup & Seizilles, Fanny & Girolami, Mark & Kazlauskaite, Ieva, 2024. "The Bayesian approach to inverse Robin problems," LSE Research Online Documents on Economics 126262, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:126262
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    File URL: http://eprints.lse.ac.uk/126262/
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    References listed on IDEAS

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    1. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, April.
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    More about this item

    Keywords

    nonlinear inverse problems; Bayesian inference; posterior consistency; Gaussian processes; MCMC;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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