We propose a preconditioning technique that is applicable in a “black box” fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation method which, for the targeted class of applica...

This paper presents a suucient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P(A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive deenite matrices with the same pattern P(A). This condition is also necessary in the s...

Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability, but need to be used with an appropriate preconditioner (e.g., ILU, AMG, Gauss-Seidel, etc.) for proper convergence. The choice of an effective preconditioner is highly problem dependent. We propose a novel fully algebraic...

In this paper, we discuss eecient implementation of a new class of preconditioners for linear systems arising from interior point methods. These new preconditioners give superior performancenear the solution of a linear programming problem where the linear systems are typically highly ill-conditioned. They rely upon the computation of an LU factorization of a subset of columns of the matrix of ...

Extended Abstract In this paper, we present a class of graph-based preconditioners for sparse linear systems arising from the finite element discretization of elliptic partial differential equations. A sparse matrix can be preconditioned by using a subgraph of its weighted adjacency matrix. The subgraph can be viewed as a " support graph " and the corresponding sparse matrix can be used as a pr...

The successive projection algorithm (SPA) can quickly solve a nonnegative matrix factorization problem under a separability assumption. Even if noise is added to the problem, SPA is robust as long as the perturbations caused by the noise are small. In particular, robustness against noise should be high when handling the problems arising from real applications. The preconditioner proposed by Gil...

Over the past several years, considerable effort has been placed on developing efficient solution algorithms for the incompressible Navier–Stokes equations. The effectiveness of these methods requires that the solution techniques for the linear subproblems generated by these algorithms exhibit robust and rapid convergence; These methods should be insensitive to problem parameters such as mesh s...

This paper discusses performance optimization on the dynamical core of global numerical weather prediction model in Global/Regional Assimilation and Prediction System (GRAPES). GRAPES is a new generation of numerical weather prediction system developed and currently used by Chinese Meteorology Administration. The computational performance of the dynamical core in GRAPES relies on the efficient ...

The purpose of this work is to provide a method which exploits the parallel blockwise algorithmic approach used in the framework of high performance sparse direct solvers in order to develop robust and efficient preconditioners based on a parallel incomplete factorization.

Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILU-based preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shift...