Multi-Grid at Scale?

Mark Ainsworth 

Brown University

Wednesday, July 19, 9:00 am

Abstract:

Multigrid and multilevel iterative algorithms are often the method of choice for the solution of large-scale systems of linear equations arising from discretisation of partial differential equations using finite element or finite difference methods. The multigrid method is non-trivial in that it involves a number of components including smoothing relaxation, coarse grid solve, prolongation and restriction operators between grids in the multilevel hierarchy, and the convergence behaviour of the method has been extensively analysed in the context of standard computer architectures. However, comparatively little is known about the resilience or fault-tolerance of the algorithm on next generation hardware architectures which are expected to suffer from frequent data corruption and hardware failures.

In this talk, we will present a mathematical model for the occurrence of failures and analyse the resiliency, or otherwise, of the standard multigrid method under the model. Numerical examples are presented to illustrate the theoretical developments. 

This is joint work with Christian Glusa (Brown University).

Biography:

Professor Ainsworth obtained his PhD from the University of Durham, United Kingdom in 1989. He is currently Professor of Applied Mathematics at Brown University, and also holds a Joint Faculty Appointment with the Mathematics and Computer Science Group at Oak Ridge National Laboratory, Tennessee.

Ainsworth's original research interests are in the numerical approximation of partial differential equations. He is, together with J.T. Oden, the co-author of a monograph on,  A Posteriori Error Estimation in Finite Element Analysis.  

Ainsworth has worked in a range of areas relating to the numerical solution of partial differential equations including a posteriori error estimation and adaptive solution of PDEs using high order finite element methods, multiigrid and domain decomposition methods for the solution of large scale linear algebra problems, the analysis of dispersive and dissipative behaviour of numerical methods for wave propagation, hierarchical and multiscale modelling.  

Ainsworth is currently researching in several new areas: 

• resiliency of state of the art numerical algorithms such as the multigrid method on emerging architectures such Exascale machines
• numerical analysis and modelling using fractional partial differential equations
• compression of scientific data arising from very large scale simulations on leadership computing facilities
• the use of techniques from Computer Aided Geometric Design and in particular, Bernstein-Bezier polynomials for the analysis and efficient implementation of high order finite element methods