Tensor-train Methods for Partial Differential Equations and its application to a Neutron Transport Problem
Tensor network techniques are known for their ability to approximate low-rank structures and beat the curse of dimensionality. They are also increasingly acknowledged as fundamental mathematical tools for efficiently solving high-dimensional Partial Differential Equations (PDEs). In this talk, we present a novel method that incorporates the Tensor Train (TT) and Quantized Tensor Train (QTT) formats for the computational resolution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian coordinates.
Overview
Abstract
Tensor network techniques are known for their ability to approximate low-rank structures and beat the curse of dimensionality. They are also increasingly acknowledged as fundamental mathematical tools for efficiently solving high-dimensional Partial Differential Equations (PDEs). In this talk, we present a novel method that incorporates the Tensor Train (TT) and Quantized Tensor Train (QTT) formats for the computational resolution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian coordinates.
Our "plain" discretization of a three-dimensional (3D) BNTE requires three distinct methods: diamond differencing, multigroup-in-energy, and discrete ordinate collocation. These techniques lead to the formulation of large, generalized, multi-dimensional eigenvalue problems, which typically require a matrix-free approach and substantial computational resources, such as massively parallel computer clusters. In our tensor network approach, we use the tensor-train format to represent the unknown fields and discrete operators associated with the partial differential equation (PDE). Then, we decompose the TT cores in the QTT format, and, finally, we combine a fixed-point iteration method with tensor network optimization strategies to solve the tensorized eigenvalue problem.
Compared with the PARallel TIme-dependent SN (PARTISN) solver currently used at the Los Alamos National Laboratory, our method results in a Yottabite compression of the memory storage requirements. In addition, the computational performance increases over 7500 times with a maximum discrepancy of less than 1e-5 with the PARTSN solutions. These improvements allow our MATLAB implementation of the method to compute the ground-state eigenvalue in about 30 seconds of elapsed time on a desktop computer.
Brief Biography
Gianmarco Manzini is a senior researcher in the Theoretical Division of the Los Alamos National Laboratory in Los Alamos, NM, and a research director at IMATI-CNR in Pavia, Italy. He holds a Ph.D. in Fluid Mechanics and Astrophysics from the University of Toulouse, France. His research interests include numerical methods for partial differential equations (finite elements, mimetic/compatible methods, virtual elements, and tensor-based methods).
