Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems

Anne Poot, Iuri Rocha, Pierre Kerfriden, Frans van der Meer
Proceedings of the First International Conference on Probabilistic Numerics, PMLR 271:95-102, 2025.

Abstract

Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model the error and propagate it to the posterior. In this work, the performance of the random mesh finite element method (RM-FEM) is investigated in a Bayesian inverse setting. We show how interpolation error negatively affects the RM-FEM posterior, and how these negative effects can be diminished. In scenarios where FEM is biased for a quantity of interest, we find that RM-FEM struggles to accurately model this bias.

Cite this Paper

BibTeX
@InProceedings{pmlr-v271-poot25a, title = {Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems}, author = {Poot, Anne and Rocha, Iuri and Kerfriden, Pierre and Meer, Frans van der}, booktitle = {Proceedings of the First International Conference on Probabilistic Numerics}, pages = {95--102}, year = {2025}, editor = {Kanagawa, Motonobu and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp}, volume = {271}, series = {Proceedings of Machine Learning Research}, month = {01--03 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v271/main/assets/poot25a/poot25a.pdf}, url = {https://proceedings.mlr.press/v271/poot25a.html}, abstract = {Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model the error and propagate it to the posterior. In this work, the performance of the random mesh finite element method (RM-FEM) is investigated in a Bayesian inverse setting. We show how interpolation error negatively affects the RM-FEM posterior, and how these negative effects can be diminished. In scenarios where FEM is biased for a quantity of interest, we find that RM-FEM struggles to accurately model this bias.} }
Endnote
%0 Conference Paper %T Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems %A Anne Poot %A Iuri Rocha %A Pierre Kerfriden %A Frans van der Meer %B Proceedings of the First International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2025 %E Motonobu Kanagawa %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %F pmlr-v271-poot25a %I PMLR %P 95--102 %U https://proceedings.mlr.press/v271/poot25a.html %V 271 %X Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model the error and propagate it to the posterior. In this work, the performance of the random mesh finite element method (RM-FEM) is investigated in a Bayesian inverse setting. We show how interpolation error negatively affects the RM-FEM posterior, and how these negative effects can be diminished. In scenarios where FEM is biased for a quantity of interest, we find that RM-FEM struggles to accurately model this bias.
APA
Poot, A., Rocha, I., Kerfriden, P. & Meer, F.v.d.. (2025). Effects of Interpolation Error and Bias on the Random Mesh Finite Element Method for Inverse Problems. Proceedings of the First International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 271:95-102 Available from https://proceedings.mlr.press/v271/poot25a.html.

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