Parallel solution strategies for triangular systems arising from oil reservoir simulations

Abstract

The results so far appear to indicate that promising alternatives to standard backward and forward eliminations in a parallel programming environment do exist. At present a lack of available hardware prohibits the author from proving their worth further here, but at least the timing models themselves reflect how comparatively inefficient orthodox parallel triangular solvers are. Even though not every banded matrix of this type can be expected to reduce to just one sparse factor, solutions based on several sparse factors would still appear to be competetive with the existing elimination process. The 2 colour, or red/black reordering strategy also merits further investigation. This method is particularly appealing because formulating the actual reordering is fairly trivial once the nx, ny, nz parameters of the problem are known. The main disadvantage comes with the effect both reorderings may have on the convergence rate of the problem [4]. Initial investigations have found a three fold increase in the number of iterations when preconditioned TFQMR is applied to a the 2 colour reordered problem. Whether such a degredation in performance is acceptable within the context of the overall solution, or is even advisable regarding the quality of the solution, is yet to be seen. With this in mind, we may find ourselves limited to strategies (5) and (8), which only reorder when solving the triangular factorized systems in the preconditioning stage.

This work was funded by the U.K.'s Department of Trade and Industry as part of the EUREKA Project EU 638, PARSIM.