Colloquium Mathematics - Yiming Bu (PhD student)

Title:

A class of linear solvers based on multilevel and supernodal factorization

Abstract:

The solution of large and sparse linear systems is a critical component of modern science and technical simulations. Iterative methods, namely the class of modern Krylov subspace methods, are often used to solve large-scale linear systems. In order to improve the robustness of the iterative methods and to reduce the number of iterations needed to achieve convergence, preconditioning techniques are often regarded as crucial components of the linear system solution.

In the thesis, a class of algebraic multilevel solvers is presented for conditioning general linear system equations arising from computational science and technical applications. They can produce sparse patterns and save memory costs by applying recursive combinatorial algorithms. Robustness is improved by combining factorization with recently developed overlapping and compression strategies and by using efficient local solvers. We have demonstrated the good performance of the proposed strategies with numerical experiments on realistic matrix problems, also in comparison with some of the most popular algebraic preconditioners used today.

The new conditioning strategies and ideas have a large degree of generality. They can be included in other existing preconditioning and iterative solution packages that are used today. They also offer inherent parallelism, making them very attractive for solving large linear systems on massively parallel computers and hardware accelerators such as GPUs.