Abstract: A system and method for preconditioning or smoothing (e.g., multi-color DILU preconditioning) for iterative solving of a system of equations. The method includes accessing a matrix comprising a plurality of coefficients of a system of equations and accessing coloring information corresponding to the matrix. The method further includes determining a diagonal matrix based on the matrix and the coloring information corresponding to the matrix. The determining of the diagonal matrix may be determined in parallel on a per color basis. The method may further include determining an updated solution to the system of equations where the updated solution is determined in parallel on a per color basis using the diagonal matrix.

Abstract: A preconditioner processor and a method of computing a preconditioning matrix. In one embodiment, the preconditioner processor has parallel computing pipelines including: (1) a graph coloring circuit operable to identify parallelisms in a sparse linear system, (2) an incomplete lower triangle, upper triangle factorization (ILU) computer configured to employ the parallel computing pipelines according to the parallelisms to: (2a) determine a sparsity pattern for an ILU preconditioning matrix, and (2b) compute non-zero elements of the ILU preconditioning matrix according to the sparsity pattern, and (3) a memory communicably couplable to the parallel computing pipelines and configured to store the ILU preconditioning matrix.

Abstract: A system and method for re-factorizing a square input matrix on a parallel processor. In one embodiment, the system includes: (1) a matrix generator operable to generate an intermediate matrix by embedding a permuted form of the input matrix in a zeroed-out sparsity pattern of a combination of lower and upper triangular matrices resulting from a prior LU factorization of a previous matrix having a same sparsity pattern, reordering to minimize fill-in and pivoting strategy as the input matrix and (2) a re-factorizer associated with the matrix generator and operable to use parallel threads to apply an incomplete-LU factorization with zero fill-in on the intermediate matrix.

Abstract: A parallel multicolor ILU factorization preconditioner processor and a method of computing an ILU preconditioning matrix. One embodiment of the preconditioning processor having parallel computing pipelines includes: (1) a graph coloring circuit operable to identify parallelisms in a sparse linear system, (2) an ILU computer configured to employ the parallel computing pipelines according to the parallelisms to: (2a) determine a sparsity pattern for an ILU preconditioning matrix, and (2b) compute non-zero elements of the ILU preconditioning matrix according to the sparsity pattern, and (3) a memory communicably couplable to the parallel computing pipelines and configured to store the ILU preconditioning matrix.

Abstract: A system and method for re-factorizing a square input matrix on a parallel processor. In one embodiment, the system includes: (1) a matrix generator operable to generate an intermediate matrix by embedding a permuted form of the input matrix in a zeroed-out sparsity pattern of a combination of lower and upper triangular matrices resulting from a prior LU factorization of a previous matrix having a same sparsity pattern, reordering to minimize fill-in and pivoting strategy as the input matrix and (2) a re-factorizer associated with the matrix generator and operable to use parallel threads to apply an incomplete-LU factorization with zero fill-in on the intermediate matrix.

Abstract: A system and method for preconditioning or smoothing (e.g., multi-color DILU preconditioning) for iterative solving of a system of equations. The method includes accessing a matrix comprising a plurality of coefficients of a system of equations and accessing coloring information corresponding to the matrix. The method further includes determining a diagonal matrix based on the matrix and the coloring information corresponding to the matrix. The determining of the diagonal matrix may be determined in parallel on a per color basis. The method may further include determining an updated solution to the system of equations where the updated solution is determined in parallel on a per color basis using the diagonal matrix.

Abstract: One embodiment sets forth a method for solving linear equation systems that include the same matrix A coupled with multiple right-hand-side vectors. For each new right-hand-side vector, a solver expands an existing Krylov subspace based on the Krylov subspace and data associated with the previous right-hand-side vector. The solver then uses the expanded Krylov subspace to approximately solve the linear equation system for the new right-hand-side vector. By expanding the Krylov subspace for each new right-hand-side vector, the solver continually leverages the information from the preceding right-hand-side vectors. Advantageously, expanding the Krylov subspace is typically computationally quicker than prior art-techniques, such as creating a new Krylov subspace or transforming an existing Krylov subspace. Consequently, by implementing the disclosed techniques, the likelihood of exceeding time constraints associated with algorithms that include solving certain classes of linear equation systems may be decreased.