TY - GEN
A1 - Cockayne, Jon
A1 - Oates, Chris
A1 - Sullivan, T. J.
A1 - Girolami, Mark
T1 - Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems
N2 - 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.
T3 - ZIB-Report - 16-31
KW - Probabilistic Numerics
KW - Partial Differential Equations
KW - Inverse Problems
KW - Meshless Methods
KW - Gaussian Processes
KW - Pseudo-Marginal MCMC
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59513
SN - 1438-0064
ER -