Approximation of functions with jumps in the simulation of parametric PDEs
Dr. Gerrit Welper
Department of Mathematics
University of Southern California
Abstract: Mathematical models typically depend on a large number of parameters that can be deterministic, such as design parameters controlled by engineers, or random such as measurement uncertainties or uncontrolled influences from the environment. With increasing computational power came a considerable interest to incorporate these parametric influences into numerical simulations. In recent years significant progress has been made for elliptic and parabolic PDE models, but problems with jumps and kinks, ubiquitous for hyperbolic PDEs, elliptic PDEs with jumping diffusion and in the realm of level-set methods, are still poorly understood. In particular, these jumps and kinks diminish the effectiveness of the known approximation schemes underlying the numerical simulations.
To overcome these issues, we introduce transformations of the physical domain that align the jumps and kinks in parameter. This extra step ensures that the jumps and kinks do not hurt the approximation schemes known form the elliptic regime. The talk discusses the method itself and two challenges that come with it: First, these transforms are highly problem dependent and automatically found via an optimization problem similar neural network training. Second, a global alignment of the jumps is usually not possible, so that we discuss strategies to do so locally, without loosing its benefits.
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