Nonlinear Splitting for Modeling and Control

As scientific and engineering applications continue to grow in complexity, there is a pressing need for novel, scalable, and efficient algorithms for both the simulation and optimization of physical systems. When combined with experimental data, these tools have become indispensable for predicting the behavior of complex systems and optimizing the design of technologies whose physics are governed by complex processes. The mathematical foundation of many such algorithms are partial differential equations (PDEs). However, many large-scale PDE simulation and optimization problems remain computationally intractable due to high dimensionality, complex geometries, widely varying spatial and temporal scales, and intricately coupled physics. This project will develop new mathematical and computational techniques for solving and optimizing systems that depend on high-dimensional systems of partial differential equations. In particular, the focus will be on equations arising from inertial confinement fusion, a promising approach to achieving fusion energy. Furthermore, these new methods will provide broadly applicable tools for simulation and optimization across a wide range of scientific and engineering disciplines. The project provides research training opportunities for graduate students. The specific aim of this work is to generalize a technique known as splitting. Splitting reduces the computational cost of a simulation by dividing the problem into different subproblems that can each be solved more efficiently. Commonly employed splitting techniques assume that a system can be decomposed as an additive sum of components. However, in many realistic systems, including the equations governing inertial confinement fusion, physical processes interact nonlinearly, and the additive splittings are unable to separate nonlinear couplings between solution components, leading to degradation in accuracy, stability, and efficiency. This proposal develops a more general and flexible splitting framework that allows one to split nonlinear terms to achieve improved efficiency with mathematical guarantees of accuracy and stability. The proposed methods focus on evolution equations, specifically forward simulations using time integration methods, and parameter optimization problems that employ gradient-based optimization methods. 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.