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A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions
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A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions
Bibliography:
Bibliography
F. Ruggeri, Z. Sawlan, M. Scavino, R. Tempone,
A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions
, accepted for publication in Bayesian Analysis, April 2016
Authors:
F. Ruggeri, Z. Sawlan, M. Scavino, R. Tempone
Keywords:
Linear Parabolic PDEs, Noisy Boundary Parameters, Bayesian Inference, Heat Equation, Thermal Diffusivity
Year:
2016
Abstract:
In this work we develop a Bayesian setting to infer unknown parameters in initialboundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the timedependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a spacedependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the nonnormalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.
ISSN:
2016
http://projecteuclid.org/euclid.ba/1463078272
No
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