Zapraszamy na seminarium Prof. Macieja Paszyńskiego pod tyt.: "Fast Solvers for Mesh-Based Computations"

termin: 6.10.2017, godz. 12.15 s. 122 WETI A

AGH University of Science and Technology

home.agh.edu.pl/paszynsk

Abstract:

Direct solvers are the core part of many engineering analyses performed using different versions of Finite Element Method.

Existing direct solvers of linear equations (for example, MUMPS , SuperLU, PARDISO, or HSL) are based on

solving a linear system given by a global matrix and one or several right-hand sides.

The global matrix is provided either as a list of non-zero entries, or it is obtained from merging a sequence of element frontal matrices.

In both cases, the additional available knowledge about the structure of the computational mesh is lost.

In this talk I would like to present an alternative paradigm for designing direct solvers based on the structure of the computational mesh.

The construction of the solver algorithm is based on the additional available knowledge concerning the structure of the computational mesh.

The alternative method presented in this talk allows us to outperform traditional direct solver algorithms.

The construction of the direct solver algorithm based on the structure of the computational mesh allows for better decomposition

of the computational problem into sets of independent tasks. This in turn allows us to obtain a solver algorithm that delivers more efficient parallel implementation (for distributed memory linux cluster, shared-memory linux node or for GPGPU).

The mesh-based solvers deliver linear O(N) computational cost on h adaptive three-dimensional grids refined towards point or edge singularity,

as well as O(N^1.5) computational cost on three-dimensional grids refined towards face singularity.

Additionally it allows us to implement some special tricks such as the reuse of computations for identical sub-parts of the mesh,

and the reutilization of LU factorizations over unrefined parts of the mesh. These techniques are not easily available for classic direct solvers.

Additionally, by analyzing the structure of the computational mesh we can generate better ordering algorithms, that result in lower number of floating point operations than the one obtained from classical ordering algorithms analyzing only the sparsity of the matrix.

Finally, the mesh-based solvers allows for simple hybridization that improves the convergence of iterations for iterative solvers .