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 (PDEs) 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 flows of charged fluids (magnetohydrodynamics) and nematic and chiral 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. Furthermore, I have contributed fundamental algorithms and analysis tools to the rapidly growing field of parallel-in-time simulation.******The research goals of this proposal are the development of improved methodologies for high-performance scientific computing in these areas. A challenging physical system, smectic liquid crystals, will drive this research, providing new challenges from the dependence of free energy on an auxiliary variable. Concurrently, we will develop a robust optimization viewpoint on local Fourier analysis, the best-practices tool for optimizing algorithmic parameters for monolithic multigrid methods. This will allow us to design and analyse systems of increasing complexity, freed from the traditional high CPU times required for brute-force analysis of monolithic algorithms for coupled finite-element discretizations. Finally, I will continue to develop both algorithms and analysis tools for space-time systems, with a focus on the multigrid reduction-in-time algorithm and corresponding semi-algebraic mode analysis tool. 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 in computational science and engineering that can be applied in both academia and industry.