Adaptive High Order Low-Rank Tensor Methods for High-Dimensional Partial Differential Equations with Application to Kinetic Simulations

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Kinetic models find a wide range of applications in science and engineering. One celebrated example is the Vlasov-Maxwell system which plays a fundamental role in plasma physics. Among many existing challenges for deterministic kinetic simulations, the curse of dimensionality and the associated huge computational cost have been a key obstacle for realistic high-dimensional simulations. The project focuses on the development of a novel computational framework for kinetic simulations by integrating the emerging low-rank tensor decompositions with advanced high order discretization schemes. The framework can lead to a class of accurate and efficient algorithms that are resistant to the curse of dimensionality for solving challenging high-dimensional problems. This project will support one graduate student during each of the three years of the project. The main objective of the project is to develop a novel computational framework for simulating high-dimensional kinetic models with high order accuracy, numerical stability, and manageable cost. The algorithms incorporate the emerging tensor approach and advanced high order discretization schemes, aiming to accurately and efficiently extract the low-rank structure of the solution data with significantly reduced complexity and potentially break the curse of dimensionality. Another project focus is to develop a novel low-rank tensor approach that can mitigate possible severe rank increase for transport-dominated simulations, which is known as a major limitation for the existing low-rank tensor methods. The key component in the algorithm design is to consider the underlying flow map system. Several computational and theoretical aspects of the proposed algorithms will be investigated, including structure preservation, complexity, stability and accuracy analysis, application to the rotating Vlasov model, and beyond. Through this research, both numerical and analytical techniques for solving challenging high-dimensional problems will be advanced.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.