A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension

Abstract

A parallel fast direct solver based on the Divide & Conquer method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the ve{point nite di erence scheme or the piecewise linear nite elements on a triangulated rectangular mesh. The Divide & Conquer method has the arithmetical complexity O(N log N), and it is closely related to the cyclic reduction, but instead of using the matrix polynomial factorization the so{called partial solution technique is employed. The method is presented and analyzed in a general base q framework and based on this analysis, the base four variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential e ciency and numerical stability of the considered method compared to the well{known BLKTRI{implementation of the generalized cyclic reduction. The good scalability properties of the parallel Divide & Conquer method are demonstrated in a distributed memory Cray T3E computer.

Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

“…Recently, it has also been brought to our attention that the properties of the Sommerfeld eigenvectors established in Section 3, in particular, their linear independence (Proposition 7), as well as the easy way to invert the transformation W according to formula (53), have been observed experimentally in the context of building the direct solve preconditioners for large-scale scattering problems, see, e.g., [38,39]. However, to the best of our knowledge no justification of these properties similar to the one given in Section 3 has ever been provided, and we therefore expect that our results may appear useful not only for solving the NLH, but for other applications as well.…”

“…Besides (35), (38), or equivalently (39), (40), there are many alternative approaches to treating the evanescent waves q m 2 and q Àm 2 near the boundaries m = ±M. 4 In fact, the component c 2 q m 2 in formula (33) can be replaced with almost any linear combination of q m 2 and q Àm 2 , and the same is true regarding the component c À2 q Àm 2 in formula (36), as long as the chosen linear combinations are linearly independent.…”

“…Proof. Linear independence of boundary conditions (39), (40) is apparent, as due to (33) and (36) the homogeneous system (25) supplemented by (35) and (38) (25) supplemented by boundary conditions (39), (45) may only have a trivial solution. h Sufficient conditions for stability are somewhat more delicate, see [21].…”

“…Formulae (64) are analogous to (35), and formulae (65) are analogous to (38). In the capacity of boundary conditions, Eqs.…”

Fast Poisson Solvers for Graphics Processing Units

“…Later, the idea of the partial fraction expansions was applied to the matrix rational functions occurring in the formulas, thus leading to the discovery of a parallel variant [13]. The radix-q PSCR (Partial Solution variant of the Cyclic Reduction) method [14][15][16][17] represents a different kind of approach based on the partial solution technique [18,19]. Excellent surveys on these kind of methods can be found in [20] and [21].…”