Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation
Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity func...
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MDPI AG
2025-04-01
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| Series: | Foundations |
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| Online Access: | https://www.mdpi.com/2673-9321/5/2/14 |
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| author | Matteo Giordano |
| author_facet | Matteo Giordano |
| author_sort | Matteo Giordano |
| collection | DOAJ |
| description | Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet–Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained. |
| format | Article |
| id | doaj-art-97b9f3d1402745b7b4c5e3b26a6a414d |
| institution | Kabale University |
| issn | 2673-9321 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Foundations |
| spelling | doaj-art-97b9f3d1402745b7b4c5e3b26a6a414d2025-08-20T03:27:17ZengMDPI AGFoundations2673-93212025-04-01521410.3390/foundations5020014Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and ImplementationMatteo Giordano0ESOMAS Department, University of Turin, Corso Unione Sovietica 218/bis, 10137 Turin, ItalyParameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet–Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained.https://www.mdpi.com/2673-9321/5/2/14inverse problemsGaussian priorsfrequentist consistencyposterior meanMarkov chain Monte Carlo |
| spellingShingle | Matteo Giordano Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation Foundations inverse problems Gaussian priors frequentist consistency posterior mean Markov chain Monte Carlo |
| title | Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation |
| title_full | Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation |
| title_fullStr | Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation |
| title_full_unstemmed | Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation |
| title_short | Bayesian Nonparametric Inference in Elliptic PDEs: Convergence Rates and Implementation |
| title_sort | bayesian nonparametric inference in elliptic pdes convergence rates and implementation |
| topic | inverse problems Gaussian priors frequentist consistency posterior mean Markov chain Monte Carlo |
| url | https://www.mdpi.com/2673-9321/5/2/14 |
| work_keys_str_mv | AT matteogiordano bayesiannonparametricinferenceinellipticpdesconvergenceratesandimplementation |