Project proposal

TOPICS IN ALGEBRAIC MULTIGRID ALGORITHMS
CARLOS E. SAUER∗

1. Project Proposal for 18.335 Many different real-life phenomena are modeled by Partial Differential Equations in order to reproduce thereal system in a mathematical framework. Often, numerical solutions for such models involve discretizations of the domain and the operators, reducing the problem to solve a system of linear equationsAx = b. As the size of the matrix A becomes larger, direct methods are no longer useful in practice. In such cases, iterative methods become very important and in this project I will focus on one ofthem: the Algebraic Multigrid Method. In contrast to Geometric Multigrid, which needs the operator and domain to be discretized in coarser grids, the Algebraic Multigrid (AMG) approach is only based onthe entries of A, like a black-box solver of Ax = b. While the Geometric Multigrid relies on a fixed hierarchy of grids, the Algebraic Multigrid adjust the hierarchy to maintain fixed and simplesmoothers on each step [1]. Additionally, the AMG algorithm has the advantage that it can be used for systems of linear equations that do not arise from differential equations [2]. In this project, I will beinvestigating the classical AMG algorithm, its performance, robustness and scalability [3]. Both complimentary processes: smoothing and coarse-grid correction will be analized in detail [4]. Toilustrate the procedure, an application modeled with the 2D-Poisson equation will be solved, assuming the matrix, obtained by discretization, is given as an input for the AMG algorithm. With this example,the competition between AMG and other iterative methods should be analyzed. Just if time allows, I am interested on studying also results for the case of A being real and symmetric. Moreover, forcomplex-valued matrices A, motivated by electromagnetic waves applications, a generalization of the AMG method can be considered [5].
REFERENCES [1] St¨ ben, K., A review of Algebraic Multigrid, J....