Statistical inverse problem of partial differential equation: an example with stationary 1D heat conduction problem

Technical report


Chettapong Janya-anurak


Technical report IES-2011-15. In: Jürgen Beyerer, Alexey Pak (eds.), Proceedings of the 2011 Joint Workshop of Fraunhofer IOSB and Institute for Anthropomatics, Vision and Fusion Laboratory, Karlsruher Schriften zur Anthropomatik, vol. 11, KIT Scientific Publishing, 2012.



Local behaviour in a continuous system with spatially or temporally vari- able parameters is often described in terms of partial differential equations (PDEs). Given a system of PDEs, an inverse problem is to reconstruct parame- ters in every point given a limited number of observations or conditions. There exists a plethora of solution methods for various inverse problems, neverthe- less, this is still an active field of research. In particular, non-linear systems, such as heat transfer equation, pose the biggest challenge. In this report we present a novel method based on Bayesian statistics. The parameter fields are represented in terms of some basis functions with unknown coefficients, treated as random variables. Their posterior probability distribution is then computed using Markov Chain Monte-Carlo approach. Finally, the field is reconstructed using the values that maximize likelihood. We illustrate the method with the example of the one-dimensional heat transfer equation, and discuss various choices of the basis functions and the accuracy issues.