788 research outputs found
A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements
A parallel fast direct solver for rank-compressible block tridiagonal linear
systems is presented. Algorithmic synergies between Cyclic Reduction and
Hierarchical matrix arithmetic operations result in a solver with arithmetic complexity and memory footprint. We provide a
baseline for performance and applicability by comparing with well known
implementations of the -LU factorization and algebraic multigrid
with a parallel implementation that leverages the concurrency features of the
method. Numerical experiments reveal that this method is comparable with other
fast direct solvers based on Hierarchical Matrices such as -LU and
that it can tackle problems where algebraic multigrid fails to converge
Using a multifrontal sparse solver in a high performance, finite element code
We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP
Additive Sweeping Preconditioner for the Helmholtz Equation
We introduce a new additive sweeping preconditioner for the Helmholtz
equation based on the perfect matched layer (PML). This method divides the
domain of interest into thin layers and proposes a new transmission condition
between the subdomains where the emphasis is on the boundary values of the
intermediate waves. This approach can be viewed as an effective approximation
of an additive decomposition of the solution operator. When combined with the
standard GMRES solver, the iteration number is essentially independent of the
frequency. Several numerical examples are tested to show the efficiency of this
new approach.Comment: 27 page
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