A. Louise Perkins

A domain decomposition method for solving a two-dimensional viscous Burgers' equation

Abstract We present an efficient finite difference numerical solution for a convection equation that includes diffusion terms with small coefficients. The equation is first advanced on a coarse mesh. Regions of significant diffusion activity are identified using a threshold criterion. The coarse mesh is dynamically decomposed and the mesh is refined in these regions. The decomposition is heterogeneous in the sense that the problem formulations as well as the solution methods may vary from mesh to mesh. In particular, a mixed Euler-Lagrange method is developed that explicitly advances the coarse mesh relative to an Eulerian reference frame, and implicitly advances the refined meshes relative to moving Lagrangian reference frames. On the refined meshes the method is second-order accurate in space. Asynchronous Schwarz iterations are used between overlapping refined meshes, when needed, to communicate the data dependence of the implicit refined solution. The computation is distributed across the four processors of the shared memory CRAY-2.

A Frequency Accurate Temporal Derivative Finite Difference Approximation

A method is presented for designing temporal derivative finite difference approximations that achieve specified accuracy in the frequency domain. A general average value approximation with undetermined coefficients is fitted in the spatial frequency domain to attain the desired properties of the approximation. A set of constraints to insure that the approximation convergences as the grid spacing approaches zero and satisfies the Lax Equivalence Theorem are imposed on the fitted coefficients. The specification of the underlying partial differential equation is required in order to replace the temporal frequency domain dependence of the approximation with an explicit spatial frequency domain relation based on the dispersion relation of the PDE. A practical design of the approximations is pursued using an heuristic zero placement method which results in a linear matrix formulation.

A frequency accuraterth order spatial derivative finite difference approximation

A method for the specification and design of finite difference spatial derivative approximations of general order r is presented. The method uses a difference polynomial with undetermined coefficients. Spatial frequency domain-based criteria, which include phase velocity, group velocity, and dissipation requirements at a priori selected spatial frequencies, are used to find the appropriate coefficient values. The method is formulated as an optimal design problem but is pursued heuristically. The general derivative approximation and the design method are suitable for use in more general design problems involving finite difference schemes for linear and nonlinear partial differential equations.

P-Bar theory

In this paper we propose a method for identifying the semantic context of segments of text within a larger document. Our method is based on an extension of Chomskys x-bar theory. We adapt the x-bar concept of headedness to a coarser granularity of text, such as paragraphs. Using this method, which we call P-bar, we map a set of vocabulary domains to a unique semantic context. Using a rule-based error-driven algorithm, we show that this approach has significant context identification skill within a Rayleigh-Ritz like approximation framework.

A Frequency Accurate Finite Difference Scheme for Burgers Equation

A method is presented for designing a one-step, explicit finite difference scheme for solving the inviscid Burgers equation based on an a priori specification of dissipation and phase accuracy requirements. Frequency accurate temporal and spatial approximations with undetermined coefficients are used, together with a set of constraints that ensure that the approximations converge as the spatial and temporal grid sizes approach zero and satisfy the Lax Equivalence Theorem. A practical design of the difference scheme using a heuristic zero placement method, combined with a stability requirement, results in a linear matrix problem which is solved to obtain the undetermined coefficients. The partial differential equation itself provides the relationship between the temporal and spatial frequency dependence in the numerical approximation.

Internal Boundary Conditions and the Feedback Loop for Nested Two-Way Communication Schemes

We address the relationship between the interior boundary conditions and the feedback mechanism in nested two-way domain decomposition. In particular, we compare an averaging feedback loop with a relaxation feedback loop, using Dirichlet boundary conditions. We construct a two-way nested communication scheme that couples the internal boundary conditions with a feedback loop, maintaining consistency as defined herein. This coupled consistency requirement guarantees that the aggregate solution makes a smooth transition across an internal interface. Both feedback methods were tested by using a barotropic modon for the initial condition. For strong relaxation values combined with a limited refinement ratio, the relaxation feedback method is shown to be more accurate than the averaging feedback method.

A mixed directed-undirected data structure for a parallel implementation of a domain decomposition algorithm

The choice of data structures influences the parallelization, efficiency and the manageability of a mesh refinement program. We introduce a mixed directed-undirected graph that combines both communication and scheduling needs. An inverted index is maintained for the directed graph to improve code performance and readability.

Tailored domain decomposition

Abstract We developed a biased domain decomposition clustering algorithm designed for use in parallel processing for the numerical solution of partial differential equations. The biased clustering algorithm is designed to work on a distributed memory parallel computer. To date, the algorithm has been on the simulated Cray-2 shared memory multiprocessor.

Locating parallel numerical tasks in the solution of viscous fluid flow

We present a method to solve the heat equation that couples mesh refinement with explicit time steps greater than the Courant condition limit. The method is implemented in parallel and executes efficiently.

A Parameterization Study of Short Read Assembly Using the Velvet Assembler

In this study, we examine approaches to the problem of assembling large, contiguous sections of genetic code from short reads generated from laboratory techniques. We explore the Eulerian Path approach in detail, utilizing a de Bruijn Graph, and demonstrate current software technologies and algorithms using a sample genome. We investigate the input parameters of Velvet and discuss their implications.