Approximation of Functions with Parameter-Dependent or Stochastic Shock Locations Arising from Hyperbolic Partial Differential Equations

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Many mathematical models of scientific or engineering problems are influenced by design parameters and random parameters. An example is the modeling of flow around an airfoil. A design parameter is its shape, and random parameters include, for example, perturbations in the air before a plane travels through it. These parameters pose some natural questions for engineers and the mathematical models they use: What is the best possible shape? Can we compute safety tolerances of the wing and limit the probability of failure? With increasing computational power, such questions attracted considerable attention in recent years. However, for physical phenomena with sharp transitions, only brute-force approaches are available, which quickly become challenging even with advanced computer hardware. In the wing example, one such transition is the sonic boom, a rapid change in the pressure around the wing at high speed. This project aims to develop new algorithms that can handle such rapid transitions in parametric models, far more efficiently than currently. The method under development will be applicable to a range of other engineering problems as well, such as environmental questions in groundwater flows or the simulation of bio-molecules.The goal of the project is the development of new approximation schemes for functions with parameter-dependent jumps or kinks, motivated by solutions of parametric and stochastic hyperbolic partial differential equations. These singularities pose serious challenges by deteriorating the convergence rates for established methods such as reduced basis methods, proper orthogonal decomposition, or polynomial chaos expansions. Recent work offers a new method to approximate functions with jump discontinuities and achieves super-polynomial convergence rates for many problems. It serves as a proof of principle that high-order methods are advantageous, but it needs to be developed further to make it practical: In most realistic problems, jumps not only move but they also interact, and parameter spaces are typically high-dimensional. Addressing these issues is the goal of this project. The new algorithms will be tested numerically, in particular regarding the convergence rates that can be achieved. In addition, the investigator plans to prove convergence rates for model classes of parametric functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.