March 15, 2016 16:00-15:00
Maciej PASZYNSKI
, AGH University of Science and Technology, PolandFAST SOLVERS FOR MESH-BASED COMPUTATIONS
Direct solvers are the core part of many engineering analyses performed using different mesh-based methods, such as the finite difference method, the collocation method, the finite element method, and the isogeometric finite element or collocation methods. Existing direct solvers of linear equations (for example, MUMPS [1], SuperLU [2], PARDISO [3], or HSL [4]) 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 a new 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 [5]. 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 computational mesh allows for better decomposition of the computational probleminto 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 [6], shared-memory linux node [7] or for GPGPU [8]. Additionally it allows us to implement some special tricks such as the reuse of computations for identical sub-parts of the mesh [9], and the reutilization of LU factorizations over unrefined parts of the mesh [10]. 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 [11], that result in lower number of floating point operations than the one obtained from classical ordering algorithms analyzing only the sparsity of the matrix (for example nested-dissections from METIS [12], MD, AMD or AMF [13], or PORD [14]).
Mazarredo 14 , 48009 Bilbao, Basque Country, Spain www.bcamath.org
Scientific Seminar
[1] MUlti-frontal Massively Parallel Sparse direct solver, http://http://mumps.enseeiht.fr/
[2] S. Xiaoye, S. Li, An Overview of SuperLU: Algorithms, Implementation, and User Interface, Transactions on Mathematical Software, 31, 3 (2005) 302–325.
[3] PARDISO, Thread-safe solver of linear equations. http://www.pardisoproject.org/
[4] HSL, Harwell Subroutine Library. http://www.cse.scitech.ac.uk/nag/hsl/
[5] M. Paszyński, Fast solvers for mesh-based computations, CRC Press / Taylor & Francis, 2016
[6] M. Paszynski, D. Pardo, C. Torres-Verdin, L. Demkowicz, V. Calo, A Parallel Direct Solver for Self-Adaptive hp Finite Element Method, Journal of Parallel and Distributed Computing, 70, 3 (2010) 270–281
[7] A. Paszynska, M. Paszynski, K. Jopek, M. Wozniak, D. Goik, P. Gurgul, H. AbouEisha, M. Moshkov, V. Calo, A. Lenharth, D. Nguyen, K. Pingali, Quasi-Optimal Elimination Trees for 2D Grids with singularities, Scientiffic Programming, (2015) Article ID 303024
[8] M. Wozniak, K. Kuznik, M. Paszynski, V. Calo, D. Pardo Computational cost estimates for parallel shared memory isogeometric multi- frontal solvers, Computers and Mathematics with Applications, 67(10) (2014) 1864–1883
[9] M. Sieniek, M. Paszynski, Subtree reuse in multi-frontal solvers for regular grids in Step-and-Flash Imprint Nanolithography Modeling, Advanced Engineering Materials, 16(2) (2014) 231-240.
[10] M. Paszynski, V. Calo, D. Pardo, A direct solver with reutilization of previously-computed LU factorizations for h-adaptive finite element grids with point singularities, Computers and Mathematics with Applications, 65, 8 (2013) 1140–1151.
[11] H. AbouEisha, V. M. Calo, K. Jopek, M. Moshkov, A. Paszynska, M. Paszynski, M.Skotniczny, Optimization of Element Partition Trees for Two-Dimensional h Rened Meshes, submitted to Computers & Mathematics with Applications (2016)
[12] G. Karypis, V. Kumar, METIS - Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 2.0, Technical Report, (1995)
[13] P. Heggernes, S.C. Eisenstat, G. Kumfert, A. Pothen, The Computational Complexity of the Minimum Degree Algorithm, ICASE Report No. 2001–42, (2001).
[14] J. Schulze, Toward a tighter coupling of bottom-up and top-down sparse matrix ordering methods, BIT, 41, 4 (2001) 800.
