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Applied Math Seminar
Modeling of physical systems underspecified by data
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Although it has long been recognized that simulations of most physical systems are fundamentally stochastic, this fact remains overlooked in most practical applications. Even essentially deterministic systems must be treated stochastically when their parameters, boundary and initial conditions, or forcing functions are under-specified by data. We concentrate on the effect of parameters because they induce multiplicative noise, and thus provide the opportunity for new applications. Parameter uncertainty, and resulting uncertainty in system states, have obvious effects on many important applications: manufacturing of composite materials, earthquakes and seismic monitoring, reservoir exploitation, and environmental remediation, to name a few. Our method of random domain decomposition provides a novel approach to dealing with the kinds of spatially heterogeneous random processes that typically appear in realistic simulations of physical systems. The method is based on a doubly stochastic model in which the problem domain is decomposed according to stochastic geometries into disjoint random fields. The stochastic decomposition is determined by variations in the parameter space based on additional (uncertain) geometric information that can be derived from new characterization techniques and also from expert knowledge. Previous work has tended to concentrate on spatially homogeneous parameterizations, or at most on heterogeneous parameter fields whose geometry is assumed known with certainty. This is almost never the case in natural systems. On the other hand, random domain decomposition allows us to estimate system states when heterogeneous parameterizations depend on realistic geometric uncertainty. |