Section: Scientific Foundations
High Performance Scientific Computing
This research is in the area of high performance scientific computing, and in particular in parallel matrix algorithms. This is a subject of crucial importance for numerical simulations as well as other scientific and industrial applications, in which linear algebra problems arise frequently. The modern numerical simulations coupled with ever growing and more powerful computational platforms have been a major driving force behind a progress in numerous areas as different as fundamental science, technical/technological applications, life sciences.
The main focus of this research is on the design of efficient, portable linear algebra algorithms, such that solving a large set of linear equations or computing eigenvalues and eigenvectors. The characteristics of the matrices commonly encountered in this situations can vary significantly, as are the computational platforms used for the calculations.
Nonetheless two common trends are easily discernible. First, the problems to solve are larger and larger, since the numerical simulations are using higher resolution. Second, the architecture of today's supercomputers is getting very complex, and so the developed algorithms need to be adapted to these new achitectures.
A number of methods and solvers exist for solving linear systems. They can be divided into three classes: direct, iterative or semi-iterative. Direct methods (LU factorization for solving linear systems and QR factorization for solving least squares problems) are often preferred because of their robustness. The methods differ significantly depending on whether the matrices are dense (all nonzero entries) or sparse (very few nonzero entries, common in matrices arising from physical modelling). Iterative methods as Krylov subspace iterations are less robust, but they are widely used because of their limited memory requirements and good scalability properties on sparse matrices. Preconditioners are used to accelerate the convergence of iterative methods. Semi-iterative methods such as subdomain methods are hybrid direct/iterative methods which can be good tradeoffs.
Efficient linear algebra algorithms
For the last several years, we have worked on a novel approach to dense and sparse linear algebra algorithms, which aims at minimizing the communication, in terms of both its volume and a number of transferred messages. This research is motivated by technological trends showing an increasing communication cost. Its main goal is to reformulate and redesign linear algebra algorithms so that they are optimal in an amount of the communication they perform, while retaining the numerical stability. The work here involves both theoretical investigation and practical coding on diverse computational platforms.
The theoretical investigation focuses on identifying lower bounds on communication for different operations in linear algebra, where communication refers to data movement between processors in the parallel case, and to data movement between different levels of memory hierarchy in the sequential case. The lower bounds are used to study the existing algorithms, understand their communication bottlenecks, and design new algorithms that attain them. The results obtained to date concern the LU, QR and rank revealing QR factorizations of dense matrices.
This research focuses on the design of linear algebra algorithms that minimize the cost of communication. Communication costs include both latency and bandwidth, whether between processors on a parallel computer or between memory hierarchy levels on a sequential machine. The stability of the new algorithms represents an important part of this work.
Preconditioning techniques
Solving a sparse linear system of equations is the most time consuming operation at the heart of many scientific applications, and therefore it has received a lot of attention over the years. While direct methods are robust, they are often prohibitive because of their time and memory requirements. Iterative methods are widely used because of their limited memory requirements, but they need an efficient preconditioner to accelerate their convergence. In this direction of research we focus on preconditioning techniques for solving large sparse systems.
