Innovative Numerical Methods for High-Dimensional Applications

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This project aims to develop efficient numerical algorithms for a class of important systems in science and engineering that are modeled with high-dimensional partial differential equations (PDE) such as the many-body Schrödinger equation. Examples of such systems include many-body quantum mechanics, dynamics of chemical systems, learning and control of complex systems, and spectral methods for high-dimensional data. The numerical solution of high-dimensional PDE has been one of the greatest challenges in computational science and remains a formidable task even with today's computational power and algorithmic advances. Efficient numerical simulations present opportunities for major breakthroughs in scientific understanding. This project will employ modern techniques to develop novel efficient numerical algorithms for such important systems. Graduate students will be trained through involvement in the research.The research project combines mathematical analysis and algorithmic design to make progress in numerical methods for high-dimensional PDE. The research will draw from and further develop ideas and tools from recent advances in computational physics, quantum chemistry, and machine learning. In particular, the project will use modern techniques for large-scale optimization and nonlinear parameterization of high-dimensional functions. Specifically, the PI will (1) develop novel highly efficient coordinate algorithms for large-scale eigenvalue problems, and (2) develop and analyze efficient methods based on neural-network parameterization of the solutions for high-dimensional PDE.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.