Robert White

Multisplitting with different weighting schemes

Parallel algorithms generated by multisplittings are considered. A parallel algorithm may be formed, first, by concurrently executing the iteration associated with each splitting, and second, by forming a weighted sum of these computations. However, it is not imperative that the weighting be done last. Convergence results are obtained for a variety of other weighting schemes. In particular, it is shown that preweighting is in some cases more desirable than the traditional postweighting. Furthermore, we indicate how one can use a symmetric weighting scheme to obtain a good multisplitting version of the SSOR preconditioner. These algorithms are illustrated by computations done on an Alliant

Populations with impulsive culling: control and identification

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Identification of hazards with impulsive sources

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Nonlinear least squares algorithm for identification of hazards

This paper is a numerical study of populations with dispersion (Fickian diffusion) in one or two directions and with a finite number of impulsive culling sites. The intensities and locations of the culling sites are used for optimal control of the population density. The identifications of the model parameters and location of the culling sites are determined from the given population density data. The Levenberg–Marquardt, variation of the simulated-annealing and semilinear-SOR algorithms are used.

Multisplitting Methods: Optimal Schemes for the Unknowns in a Given Overlap

Given some observations downstream can one determine the location and intensities of point sources of a hazard (pollutant chemical or biological)? The unknown concentrations are governed by the diffusion-advection partial differential equation. The corresponding algebraic system is studied. The fixed location problem is considered using reordering, the Schur complement and nonnegative least squares. A nonlinear problem is proposed, and an iterative method is formulated based on nonnegative least squares and Newtons method. The variable location problem is tackled with simulated annealing. The complexities of controlling aquatic populations, which are nonlinear, time-dependent and have multiple sources, will be illustrated.

Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI

Given observations of selected concentrations one wishes to determine unknown intensities and locations of the sources for a hazard. The concentration of the hazard is governed by a steady-state nonlinear diffusion–advection partial differential equation and the best fit of the data. The discretized version leads to a coupled nonlinear algebraic system and a nonlinear least squares problem. The coefficient matrix is a nonsingular M-matrix and is not symmetric. Iterative methods are compositions of nonnegative least squares and Picard/Newton methods, and a convergence proof is given. The singular values of the associated least squares matrix are important in the convergence proof, the sensitivity to the parameters of the model, and the location of the observation sites.

High Performance Computing

Consider a linear algebraic problem where the set of unknowns is a union of subsets. Let the coefficient matrix have a splitting associated with each subset. The traditional multisplitting method forms a weighted sum, over the overlapping unknowns, of the iterates for each such splitting to obtain a single parallel iterative method. An optimal alternative to the weighted sums will be presented. Convergence of this new form of multisplitting (MS) method can be studied for both symmetric positive definite (SPD) matrices and M-matrices. Applications to PDEs and the equilibrium equations for fluid flow in a driven cavity will be presented.