Grant: Discovery Grants Program - Individual

Computational simulation of physical systems is a significant scientific and industrial tool. Recent improvements in simulation have come from the development of efficient parallel algorithms for heterogeneous systems, simulating problems with multiple materials, with varying material properties, and/or coupling to additional physical laws. Mathematically, these systems are modeled as coupled systems of partial differential equations representing physical conservation and energy laws, with variable and nonlinear coefficients reflecting the heterogeneity. Finite-element discretizations transform these continuum equations into finite-dimensional linear and non-linear systems; the solution of these systems is the core resource-intensive computational task in many simulation algorithms. My long-term research program focuses on the development and analysis of efficient parallel algorithms for solving the linear, linearized, and non-linear systems that result from these discretizations. My approach follows the multigrid methodology, where a hierarchical decomposition is used to ensure optimal complexity of the iterative solution process. In recent years, this has included a strong focus on structured-grid multigrid methods, which can naturally achieve high parallel efficiency. Of note, my research group has developed state-of-the-art simulation tools for two challenging classes of problems, flows of charged fluids (magnetohydrodynamics) and several phases of thermotropic liquid crystals. Concurrent with this work, I have undertaken the development of predictive algorithmic analysis tools, to help design and optimize solvers in this setting. An increasing focus of my group has been on structure-preserving numerical methods, developing discretizations and solvers that offer high-fidelity reflections of physical conservation laws and their geometric structure. The research goals of this proposal are the development of improved fundamental methodologies for high-performance scientific computing in these areas, as well as implementation and deployment of software tools that can be readily applied to challenging new physical applications. We will focus on two important extensions of our recent work, transitioning to higher-order spatial and temporal discretizations, leading to more complex (and denser) nonlinear systems to solve, and improving support for structure-preserving discretizations by incorporating structure-preserving ideas into our linear and nonlinear solvers. Furthermore, we will continue our development and use of advanced optimization tools, including those arising from Scientific Machine Learning, to improve the performance of our solvers in a broader range of settings. At all stages of this project, the training of HQP in algorithmic design and analysis, as well as programming in high-performance computing environments, will be a central theme, providing key skills that can be applied in both academia and industry.