A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrix-vector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system, and the proximity of the approximate to the true inverse is estimated. An extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach.
The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second-order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit timestepping scheme. Optimal a priori error bounds are derived in the energy norm and the L 2-norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order O(h min{s, }) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L 2error is shown to converge with the optimal order O(h +1). Numerical results confirm the expected convergence rates and illustrate the versatility of the method.
Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete numerical scheme is explicit, is inherently parallel, and exactly conserves a discrete energy. Starting from the standard second-order "leap-frog" scheme, time-stepping methods of arbitrary order of accuracy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory. Introduction.The efficient and accurate numerical solution of the wave equation is of fundamental importance for the simulation of time dependent acoustic, electromagnetic, or elastic wave phenomena. Finite difference methods are commonly used for the simulation of time dependent waves because of their simplicity and their efficiency on structured Cartesian meshes [27,28,34]. However, in the presence of complex geometry or small geometric features that require locally refined meshes, their usefulness is somewhat limited. In contrast, finite element methods (FEMs) easily handle locally refined unstructured meshes; moreover, their extension to high order is straightforward, even in the presence of curved boundaries or material interfaces.The finite element Galerkin discretization of second-order hyperbolic problems typically leads to a second-order system of ordinary differential equations. Even if explicit time stepping is employed, the mass matrix arising from the spatial discretization by standard continuous finite elements must be inverted at each time step-a major drawback in terms of efficiency. To overcome that problem, various "mass lumping" techniques have been proposed, which effectively replace the mass matrix by a diagonal approximation. While straightforward for piecewise linear elements [8,32], mass lumping techniques require particular quadrature or cubature rules at higher order to preserve the accuracy and guarantee numerical stability [14,22].Alternatively, discontinuous Galerkin (DG) methods offer even greater flexibility for local mesh refinement by accommodating nonconforming grids and hanging nodes.
Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and physical instabilities on both large and small scales. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Here, explicit off-line test criteria for stable accurate discrete filtering are developed for use in the above context and mimic the classical stability analysis for finite difference schemes. First, constant coefficient partial differential equations, which are randomly forced and damped to mimic mesh scale energy spectra in the above problems are developed as off-line filtering test problems. Then mathematical analysis is used to show that under natural suitable hypothesis the time filtering algorithms for general finite difference discrete approximations to an s ؋ s partial differential equation system with suitable observations decompose into much simpler independent s-dimensional filtering problems for each spatial wave number separately; in other test problems, such block diagonal models rigorously provide upper and lower bounds on the filtering algorithm. In this fashion, elementary off-line filtering criteria can be developed for complex spatially extended systems. The theory is illustrated for time filters by using both unstable and implicit difference scheme approximations to the stochastically forced heat equation where the combined effects of filter stability and model error are analyzed through the simpler off-line criteria.M any contemporary problems in science ranging from protein folding in molecular dynamics to scale up of smallscale effects in nanotechnology to making accurate predictions of the coupled atmosphere-ocean system involve partial observations of extremely complicated systems with many degrees of freedom. Novel mathematical issues arise in the attempt to quantify the behavior of such complex multiscale systems (1, 2). For example, in the coupled atmosphere-ocean system, the current practical models for prediction of both weather and climate involve general circulation models where the physical equations for these extremely complex flows are discretized in space and time and the effects of unresolved processes are parametrized according to various recipes; the result of this process involves a model for the prediction of weather and climate from partial observations of an extremely unstable, chaotic dynamical system with several billion degrees of freedom. Bayesian hierarchical modeling (3) and reduced order filtering strategies (4-12) have been developed with some success in these extremely complex systems. The basis for such dynamic prediction strategies for the complex spatially extended systems is the classical Kalman filtering algorithm (13-16). Mathematical issues arise in the practical application of these filtering strategies to complex spatially extended systems and are the focus of this article.One mathematical issu...
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