Section: New Results

Parallel Low-Rank Linear System and Eigenvalue Solvers Using Tensor Decompositions

At the core of numerical simulations for scientific computing applications, one typically needs to solve an equation either in the form of a linear system (Ax = b) or an eigenvalue problem (Ax = ƛx) to determine the course of the simulation. A major breakthrough in this solution step is exploiting the inherent low-rank structure in the problem; an idea stemming from the observation that particles in the same spatial locality exhibit similar interactions with others in a distant cluster/region. This property has been exploited in many contexts such as fast multipole methods (FMM) and hierarchical matrices (H-matrices) in applications ranging from n-body simulations to electromagnetics, which amount to numerically compressing the matrix in order to reduce computational and memory costs. Recent theory along this direction involves employing tensor decomposition to quantize the matrix in the form of a tensor (through logical restructuring/reshaping) and use tensor decomposition to approximate it with a controllable global error. Once the matrix and vectors are compressed this way, one can similarly use the compressed tensor to carry out matrix-vector operations with significantly better compression rate than the H-matrix approach.

Despite these major recent breakthroughs in the theory and application of tensor-based methods, addressing large-scale real-world problems with these methods requires immense computational power, which necessitates highly optimized parallel algorithms and implementations. To this end, we have initiated the development of a tensor-based linear system and eigenvalue solver library called Celeste++(C++ library for Efficient low-rank Linear and Eigenvalue Solvers using Tensor decomposition) providing a complete framework for expressing a problem in tensor form, then effectuating all matrix-vector operations under this compressed form with tremendous computational and memory efficiency. The fruits of our preliminary studies led two project submissions at the national scale (ANR JCJC and CNRS PEPS JCJC, currently under evaluation) and one Severo Ochoa Mobility Grant for a collaboration visit to Barcelona Supercomputing Center (BSC). We also supervised an internship on the application of tensor solvers in the context of electromagnetic applications with very promising results for future work.