5951
eng
reportzib
0
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2016-05-25
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Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems
This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.
1438-0064
urn:nbn:de:0297-zib-59513
urn:nbn:de:0297-zib-59513
Tim Sullivan
Jon Cockayne
Chris Oates
T. J. Sullivan
Mark Girolami
ZIB-Report
16-31
eng
uncontrolled
Probabilistic Numerics
eng
uncontrolled
Partial Differential Equations
eng
uncontrolled
Inverse Problems
eng
uncontrolled
Meshless Methods
eng
uncontrolled
Gaussian Processes
eng
uncontrolled
Pseudo-Marginal MCMC
Numerical Mathematics
Sullivan, Tim
no-project
Uncertainty Quantification
https://opus4.kobv.de/opus4-zib/files/5951/pmm_pde_bip.pdf