A new network approach to Bayesian inference in partial differential equations

• Published in: International journal for numerical methods in engineering Vol. 104; no. 5; pp. 313 - 329
• Main Authors: Kohler, Dominic, Marzouk, Youssef M., Müller, Johannes, Wever, Utz
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 02.11.2015; Wiley Subscription Services, Inc
• Subjects: Algorithms; Arsenates; Bayesian analysis; Boyen-Koller algorithm; cellular probabilistic automata; dynamic Bayesian networks; hyperbolic; Inference; inverse; Mathematical models; Networks; Partial differential equations; probabilistic methods; Probability theory
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4928

Summary We introduce a novel numerical approach to parameter estimation in partial differential equations in a Bayesian inference context. The main idea is to translate the equation into a state‐discrete dynamic Bayesian network with the discretization of cellular probabilistic automata. There exists a vast pool of inference algorithms in the probabilistic graphical models field, which can be applied to the network. In particular, we reformulate the parameter estimation as a filtering problem, discuss requirements for according tools in our specific setup, and choose the Boyen–Koller algorithm. To demonstrate our ideas, the scheme is applied to the problem of arsenate advection and adsorption in a water pipe: from measurements of the concentration of dissolved arsenate at the outflow boundary condition, we infer the strength of an arsenate source at the inflow boundary condition. Copyright © 2015 John Wiley & Sons, Ltd.