5951
eng
reportzib
0
--
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
T. J. 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
6023
eng
reportzib
0
--
2016-08-23
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Cameron--Martin theorems for sequences of Cauchy-distributed random variables
Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.
1438-0064
urn:nbn:de:0297-zib-60230
Han Cheng Lie
Han Cheng Lie
T. J. Sullivan
ZIB-Report
16-40
PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
Numerical Mathematics
Sullivan, Tim
no-project
Lie, Han
https://opus4.kobv.de/opus4-zib/files/6023/ZIB_report.pdf
5942
2016
2016
eng
reportzib
0
--
--
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Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors
This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.
urn:nbn:de:0297-zib-59422
10.3934/ipi.2017040
1438-0064
Appeared in: Inverse Problems and Imaging
no
T. J. Sullivan
T. J. Sullivan
ZIB-Report
16-30
eng
uncontrolled
Bayesian inverse problems
eng
uncontrolled
heavy-tailed distribution
eng
uncontrolled
Karhunen–Loève expansion
eng
uncontrolled
stable distribution
eng
uncontrolled
uncertainty quantification
eng
uncontrolled
well-posedness
MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX)
PARTIAL DIFFERENTIAL EQUATIONS
PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
STATISTICS
NUMERICAL ANALYSIS
Numerical Mathematics
Sullivan, Tim
no-project
Uncertainty Quantification
https://opus4.kobv.de/opus4-zib/files/5942/stable_bip.pdf