Abstract. We propose a novel density based numerical method for uncertainty propagation under distinct partial differential equation dynamics. The main idea is to translate them into objects that we call cellular probabilistic automata and to evolve the latter. The translation is achieved by state discretization as in set oriented numerics and the use of the locality concept from cellular automata theory. We develop the method using the example of initial value uncertainties under deterministic dynamics and show that it is consistent. As an application we discuss arsenate transportation and adsorption in drinking water pipes and compare our results to Monte Carlo computations.

SUMMARYWe 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.

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