Ali Dorostkar

CPU and GPU Performance of Large Scale Numerical Simulations in Geophysics

In this work we benchmark the performance of a preconditioned iterative method, used in large scale computer simulations of a geophysical application, namely, the elastic Glacial Isostatic Adjustment model. The model is discretized using the finite element method that gives raise to algebraic systems of equations with matrices that are large, sparse, nonsymmetric, indefinite and with a saddle point structure. The efficiency of solving systems of the latter type is crucial as it is to be embedded in a time-evolution procedure, where systems with matrices of similar type have to be solved repeatedly many times. The implementation is based on available open source software packages - Deal.II, Trilinos, PARALUTION and AGMG. These packages provide toolboxes with state-of-the-art implementations of iterative solution methods and preconditioners for multicore computer platforms and GPU. We present performance results in terms of numerical and the computational efficiency, number of iterations and execution time, and compare the timing results against a sparse direct solver from a commercial finite element package, that is often used by applied scientists in their simulations.

Numerical and computational aspects of some block-preconditioners for saddle point systems

Element-wise approximation of the Schur complement (EWS) for saddle point matrices.Spectral bounds are shown when the Schur complement is symmetric negative definite.EWS leads to a numerically and computationally optimal iterative solver.Numerical simulations on various computer architectures. Linear systems with two-by-two block matrices are usually preconditioned by block lower- or upper-triangular systems that require an approximation of the related Schur complement. In this work, in the finite element framework, we consider one special such approximation, namely, the element-wise Schur complement. It is sparse and its construction is perfectly parallelizable, making it an appropriate ingredient when building preconditioners for iterative solvers executed on both distributed and shared memory computer architectures. For saddle point matrices with symmetric positive (semi-)definite blocks we show that the Schur complement is spectrally equivalent to the so-constructed approximation and derive spectral equivalence bounds. We also illustrate the quality of the approximation for nonsymmetric problems, where we observe the same good numerical efficiency.Furthermore, we demonstrate the computational and numerical performance of the corresponding preconditioned iterative solution method on a large scale model benchmark problem originating from the elastic glacial isostatic adjustment model discretized using the finite element method.

Schur Complement Matrix and Its (Elementwise) Approximation: A Spectral Analysis Based on GLT Sequences

Schur complement matrix and its (elementwise) approximation : A spectral analysis based on GLT sequences

Multidimensional Performance and Scalability Analysis for Diverse Applications Based on System Monitoring Data

Multidimensional performance and scalability analysis for diverse applications based on system monitoring data

Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem