DYNAMICALLY ORTHOGONAL TENSOR METHODS FOR HIGH-DIMENSIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Abstract
In this proposal we address the challenging problem of computing the numerical solution to high-dimensional time-dependent nonlinear partial differential equations (PDEs). This subject is currently on the verge of becoming central to many new application areas such as feedback control of stochastic dynamical systems, mean field games, machine learning, and optimal transport. The key idea of the proposed method relies on a hierarchical decomposition of the PDE solution space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently visualized by binary trees or more general graphs, yields series expansions of the PDE solution that can effectively address high dimensionality. By enforcing dynamic orthogonality conditions at each level of binary tree, we convert the PDE into a system of low-dimensional coupled evolution equations for the hierarchical tensor modes. This allows us to represent and compute the temporal evolution of the PDE solution on tensor manifolds with constant rank, with no need for computationally expensive rank reduction methods. We will also develop new algorithms for dynamic addition and removal of tensor modes and address the problem of solving high-dimensional nonlinear PDEs in complex geometries. The proposed work will have a significant and broad impact as it will set the foundations of a new computational paradigm that allows for simulation, control and optimization several challenging systems of interest to AFOSR and AFRL, such as non-neutral plasma dynamics (Maxwell-Vlasov-Boltzmann equations), feedback control of stochastic dynamical systems (Hamilton-Jacobi-Bellman equations), mean field control, and hypersonic dynamics in rarefied gases (Boltzmann equation).
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 12, 2021
- Source ID
- FA95502010174
Entities
People
- Daniele Venturi
Organizations
- Air Force Office of Scientific Research
- United States Air Force
- University of California, Santa Cruz