Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

Hanne Kekkonen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

We consider the statistical non-linear inverse problem of recovering the absorption term f > 0 in the heat equation {∂tu-12Δu+fu=0onO×(0,T)u=gon∂ O×(0,T)u(·,0)=u0onO, where O ϵ ℝd is a bounded domain, T < ∞ is a fixed time, and g, u 0 are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u f on O×(0,T) . We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

Original languageEnglish
Article number035002
Number of pages29
JournalInverse Problems
Volume38
Issue number3
DOIs
Publication statusPublished - 2022

Keywords

  • Bayesian inference
  • frequentist consistency
  • Gaussian priors
  • nonlinear inverse problems
  • parabolic PDEs

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