We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen Loève expansion of a stochastic PDE posed in a onedimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tenso...

We study a parallel Newton-Krylov-Schwarz (NKS) based algorithm for solving large sparse systems resulting from a fully implicit discretization of partial differential equations arising from petroleum reservoir simulations. Our NKS algorithm is designed by combining an inexact Newton method with a rank-2 updated quasi-Newton method. In order to improve the computational efficiency, both DDM and...

Work on generalizing the deflated, restarted GMRES algorithm, useful in lattice studies using stochastic noise methods, is reported. We first show how the multi-mass extension of deflated GMRES can be implemented. We then give a deflated GMRES method that can be used on multiple right-hand sides of Ax = b in an efficient manner. We also discuss and give numerical results on the possibilty of co...

Through the research of the parallel computational model based on the principal and subordinate mode and the basic theory of Gmres Algorithm in Krylov subspace, this essay raises a improvement parallel Predict-Correct Gmres(m) algorithm which posses Predict-Correct pattern, and shows the computing examples for linear equations. After the comparison with the result from the new parallel Predict-...

This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in three dimensions. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers. The key idea is to apply the sweeping preconditioner recursively to the quasi-two-dimensional auxiliary problems introduced in the three-dimensional (3D) sw...

Traditional semi-implicit formulations of nonhydrostatic compressible models may not be stable in the presence of steep terrain when pressure gradient terms are split and lagged in time. If all pressure gradient terms and the divergence are treated implicitly, the resulting wave equation for the pressure contains o -diagonal cross-derivative terms leading to a highly nonsymmetric linear system ...

A numerical method for solving the equations modeling acoustic scattering in three dimensions is presented. The method is capable of handling several dozen scatterers, each of which is several wave-lengths long, on a personal work station. Even for geometries involving cavities, solutions accurate to seven digits or better were obtained. The method relies on a Boundary Integral Equation formula...

A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton’s method. Newton–Krylov methods are often implemented in “matrix-free” form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approxim...

A preconditioned directional-implicit agglomeration algorithm is developed for solving twoand three-dimensional viscous flows on highly anisotropic unstructured meshes of mixed-element types. The multigrid smoother consists of a pre-conditioned pointor line-implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm. Directional coarsening or ag...

A recent proof-of-principle study proposes an energy-and charge-conserving, nonlinearly implicit elec-trostatic particle-in-cell (PIC) algorithm in one dimension [Chen et al, The algorithm in the reference employs an unpreconditioned Jacobian-free Newton-Krylov method, which ensures nonlinear convergence at every timestep (resolving the dynamical timescale of interest). Kinetic enslavement, whi...