Loyce M. Adams

Additive polynomial preconditioners for parallel computers

Abstract We present a polynomial preconditioner that can be used with the conjugate gradient method to solve symmetric and positive definite systems of linear equations. Each step of the preconditioning is achieved by simultaneously taking an iteration of the SOR method and an iteration of the reverse SOR method (equations taken in reverse order) and averaging the results. This yields a symmetric preconditioner that can be implemented on parallel computers by performing the forward and reverse SOR iterations simultaneously. We give necessary and sufficient conditions for additive preconditioners to be positive definite. We find an optimal parameter, ω, for the SOR-Additive linear stationary iterative method applied to 2-cyclic matrices. We show this method is asymptotically twice as fast as SSOR when the optimal ω is used. We compare our preconditioner to the SSOR polynomial preconditioner for a model problem. With the optimal ω, our preconditioner was found to be as effective as the SSOR polynomial preconditioner in reducing the number of conjugate gradient iterations. Parallel implementations of both methods are discussed for vector and multiple processors. Results show that if the same number of processors are used for both preconditioners, the SSOR preconditioner is more effective. If twice as many processors are used for the SOR-Additive preconditioner, it becomes more efficient than the SSOR preconditioner when the number of equations assigned to a processor is small. These results are confirmed by the Blue Chip emulator at the University of Washington.

The Pattern Method for incorporating tidal uncertainty into probabilistic tsunami hazard assessment (PTHA)

In this paper, we describe a general framework for incorporating tidal uncertainty into probabilistic tsunami hazard assessment and propose the Pattern Method and a simpler special case called the

Fourier analysis of two-level hierarchical basis preconditioners

\Delta t

Tsunami Hazard Assessment of the Ocosta School Site in Westport, WA

Δt Method as effective approaches. The general framework also covers the method developed by Mofjeld et al. (J Atmos Ocean Technol 24(1):117–123, 2007) that was used for the 2009 Seaside, Oregon probabilistic study by González et al. (J Geophys Res 114(C11):023, 2009). We show that the Pattern Method is superior to past approaches because it takes advantage of our ability to run the tsunami simulation at multiple tide stages and uses the time history of flow depth at strategic gauge locations to infer the temporal pattern of waves that is unique to each tsunami source. Combining these patterns with knowledge of the tide cycle at a particular location improves the ability to estimate the probability that a wave will arrive at a time when the tidal stage is sufficiently large that a quantity of interest such as the maximum flow depth exceeds a specified level. Python scripts to accompany this paper are available at DOI 10.5281/zenodo.12406.

Karachi tides during the 1945 Makran tsunami

Cartesian grid methods encompass a wide variety of techniques used to solve partial differential equations in more than one space dimension on uniform Cartesian grids even when the underlying geometry is complex and not aligned with the grid. The authors` groups work on Immersed Interface Methods (IIM) was originally motivated by the desire to understand and improve the ``Immersed Boundary Method``, developed by Charles Peskin to solve incompressible Navier-Stokes equations in complicated geometries with moving elastic boundaries. This report briefly discusses the development of the Immersed Interface Methods and gives examples of application of the method in solving several partial differential equations.

Tsunami Hazard Assessment of the Elementary School Berm Site in Long Beach, WA

Abstract We use Fourier analysis techniques to find the condition number of one, two, and three dimensional model problems that are preconditioned with a two-level hierarchical basis matrix. The results show that all these problems have constant condition numbers as the mesh size becomes small provided a block diagonal scaling is applied. Limitations and extensions of the Fourier procedure are discussed.