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

: Chettapong, Janya-anurak

Postprint urn:nbn:de:0011-n-2143592 (1.3 MByte PDF)
MD5 Fingerprint: d1e29df320a56d835c022c08a94eea43
Erstellt am: 27.9.2012

Beyerer, Jürgen (Hrsg.); Pak, Alexey (Hrsg.) ; Fraunhofer-Institut für Optronik, Systemtechnik und Bildauswertung -IOSB-, Karlsruhe; Karlsruhe Institute of Technology -KIT-, Lehrstuhl für Interaktive Echtzeitsysteme -IES-:
Joint Workshop of Fraunhofer IOSB and Institute for Anthropomatics, Vision and Fusion Laboratory 2011. Proceedings : Triberg-Nussbach, Germany. From July, 17 to July, 22
Karlsruhe: KIT Scientific Publishing, 2012 (Karlsruher Schriften zur Anthropomatik 11)
ISBN: 978-3-86644-855-1
Fraunhofer Institute of Optronics, System Technologies and Image Exploitation and Institute for Anthropomatics, Vision and Fusion Laboratory (Joint Workshop) <2011, Triberg-Nussbach>
Konferenzbeitrag, Elektronische Publikation
Fraunhofer IOSB ()

Local behaviour in a continuous system with spatially or temporally variable parameters is often described in terms of partial differential equations (PDEs). Given a system of PDEs, an inverse problem is to reconstruct parameters in every point given a limited number of observations or conditions. There exists a plethora of solution methods for various inverse problems, nevertheless, 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.