Cellular Probabilistic Automata---A Novel Method for Uncertainty Propagation

Abstract

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.

A new network approach to Bayesian inference in partial differential equations

“…We consider a pipe consisting of six report locations, where node 1 is the stochastic boundary condition and node 6, the consumer's site; see Figure . Numerical experiments indicate that the choice of 5 × 15 symbols to code Ω and six nodes is sufficient to cover the essential structures of the dynamics of dissolved and adsorbed arsenate (see and discussion there). Note that – because of the construction – the matrix of transition probabilities is extremely sparse.…”

Cellular non-deterministic automata and partial differential equations

“…Unlike in the concept at hand, there the transitions are weighted with probabilities. In classic stochastic CA [6,7] one transition is chosen in every time step and at each site, and in cellular probabilistic automata for uncertainty propagation [32] all possible trajectories are followed at once to evolve a probability density.…”

“…Before pattern SA are introduced we first introduce the underlying de Bruijn state calculus which we developed on the basis of pattern ideas in deterministic CA theory [43,44], in the theory of de Bruijn graphs [45] and in pair approximation [46]. We note that it is also used in a probabilistic setting [24,32] Note that the Cantor metric can be introduced on X dB because P(E V ) is finite.…”

“…This is a consequence of the second step of simplification and information reduction. In [32] we add transition probabilities to the possible image states and thus develop a method for uncertainty propagation. In constrast, the focus of this work is on the basic non-deterministic aspect in the context of CA theory.…”

“…This allows to approximate their dynamical behavior with the standard theory of CA at the cost of losing -in a controlled way -further information. One of these SS, the pattern SA, can be extended to the probabilistic setup in [32]. However, the work at hand is more general and also introduces and categorizes other approximation ideas.…”