Solving large sparse linear equations from discretizations of three-dimensional PDEs
Iain Duff (STFC Rutherford Appleton Laboratory, Oxfordshire, UK and CERFACS, Toulouse, France)
All slides are available at
http://ppam.pl/docs/duff.pdf
The most challenging problems for numerical linear algebra arguably arise from the discretization of partial differential equations from three-dimensional modelling. The structure of the resulting equations causes problems for direct methods inasmuch the factors are much denser than the original system while their complexity causes major problems for the convergence of iterative methods.
In this talk we show how direct and iterative methods can be combined to solve problems that are intractable by one class of methods alone. Examples of these hybrid methods include using a direct method as a coarse grid solver in multigrid or to solve subproblems in domain decomposition. We show examples of these approaches and indicate how they can be used to solve very large problems. We also examine hybrid methods that use a fast but potentially inaccurate sparse factorization.
Researchers in the Parallel Algorithms Group at CERFACS in Toulouse have recently solved three-dimensional Helmholtz problems in seismic modelling with over 65 billion unknowns. We briefly discuss how this has been done.