Kevin R. Long

An overview of the Trilinos project

The Trilinos Project is an effort to facilitate the design, development, integration, and ongoing support of mathematical software libraries within an object-oriented framework for the solution of large-scale, complex multiphysics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software.Trilinos uses a two-level software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common look-and-feel, including configuration, documentation, licensing, and bug-tracking.Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates high-quality software engineering practices that are increasingly required from simulation software.

A variational finite element method for source inversion for convective-diffusive transport

We consider the inverse problem of determining an arbitrary source in a time-dependent corvective-diffusive transport equation, given a velocity field and pointwise measurements of the concentration. Applications that give rise to such problems include determination of groundwater or airborne pollutant sources from measurements of concentrations, and identification of sources of chemical or biological attacks. To address ill-posedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first-order optimality system, which includes the initial-boundary value state problem, the final-boundary value adjoint problem, and the space-time boundary value source problem. We discretize in the space-time volume using Galerkin finite elements. Several examples demonstrate the influence of the density of the sensor array, the effectiveness of total variation regularization for discontinuous sources, the invertibility of the source as the transport becomes increasingly convection-dominated, the ability of the space-time inversion formulation to track moving sources, and the optimal convergence rate of the finite element approximation.

Unified Embedded Parallel Finite Element Computations via Software-Based Fréchet Differentiation

Computational analysis of systems governed by partial differential equations (PDEs) requires not only the calculation of a solution but the extraction of additional information such as the sensitivity of that solution with respect to input parameters or the inversion of the system in an optimization or design loop. Moving beyond the automation of discretization of PDEs by finite element methods, we present a mathematical framework that unifies the discretization of PDEs with these additional analysis requirements. In particular, Frechet differentiation on a class of functionals together with a high-performance finite element framework has led to a code, called Sundance, that provides high-level programming abstractions for the automatic, efficient evaluation of finite variational forms together with the derived operators required by engineering analysis.

Potential impacts of climate change on the ecology of dengue and its mosquito vector the Asian tiger mosquito (Aedes albopictus)

Shifts in temperature and precipitation patterns caused by global climate change may have profound impacts on the ecology of certain infectious diseases. We examine the potential impacts of climate change on the transmission and maintenance dynamics of dengue, a resurging mosquito-vectored infectious disease. In particular, we project changes in dengue season length for three cities: Atlanta, GA; Chicago, IL and Lubbock, TX. These cities are located on the edges of the range of the Asian tiger mosquito within the United States of America and were chosen as test cases. We use a disease model that explicitly incorporates mosquito population dynamics and high-resolution climate projections. Based on projected changes under the Special Report on Emissions Scenarios (SRES) A1fi (higher) and B1 (lower) emission scenarios as simulated by four global climate models, we found that the projected warming shortened mosquito lifespan, which in turn decreased the potential dengue season. These results illustrate the difficulty in predicting how climate change may alter complex systems.

Analytical potentials for barred galaxies

We present simple, analytical, prolate, and triaxial potential-density pairs in three dimensions. The potentials are convolutions of the density of a thin needle with a softened potential; the density is everywhere positive. Each choice of softening model and needle density generates a family of models that vary between the extremes of a needle and a sphere. With appropriate choices of the needle density and softening model, the potential can be found in terms of elementary functions. The softening potential need not be spherically symmetric; triaxial models can be constructed in this way

Sundance 2.0 tutorial.

Sundance is a system of software components that allows construction of an entire parallel simulator and its derivatives using a high-level symbolic language. With this high-level problem description, it is possible to specify a weak formulation of a PDE and its discretization method in a small amount of user-level code; furthermore, because derivatives are easily available, a simulation in Sundance is immediately suitable for accelerated PDE-constrained optimization algorithms. This paper is a tutorial for setting up and solving linear and nonlinear PDEs in Sundance. With several simple examples, we show how to set up mesh objects, geometric regions for BC application, the weak form of the PDE, and boundary conditions. Each example then illustrates use of an appropriate solver and solution visualization.

Large Scale Non-Linear Programming for PDE Constrained Optimization

Three years of large-scale PDE-constrained optimization research and development are summarized in this report. We have developed an optimization framework for 3 levels of SAND optimization and developed a powerful PDE prototyping tool. The optimization algorithms have been interfaced and tested on CVD problems using a chemically reacting fluid flow simulator resulting in an order of magnitude reduction in compute time over a black box method. Sandias simulation environment is reviewed by characterizing each discipline and identifying a possible target level of optimization. Because SAND algorithms are difficult to test on actual production codes, a symbolic simulator (Sundance) was developed and interfaced with a reduced-space sequential quadratic programming framework (rSQP++) to provide a PDE prototyping environment. The power of Sundance/rSQP++ is demonstrated by applying optimization to a series of different PDE-based problems. In addition, we show the merits of SAND methods by comparing seven levels of optimization for a source-inversion problem using Sundance and rSQP++. Algorithmic results are discussed for hierarchical control methods. The design of an interior point quadratic programming solver is presented.

Sundance Rapid Prototyping Tool for Parallel PDE Optimization

High-performance algorithms for PDE-constrained optimization often require application of operators and solution of systems of equations that are different from those used in a single solution of the PDE; consequently, exploration of a research idea entails startup costs for modification to the PDE solver. A software tool to enable rapid development of parallel codes for large-scale, complex PDEs on realistic problems would be a useful aid to research in this area. As part of Sandia’s research efforts in PDE-constrained optimization, we are developing Sundance, an environment in which a parallel PDE solver is accessed via a high-level problem description, using abstract concepts such as functions, operators, and regions. With this high-level problem description, it is possible to specify a variational formulation of a PDE and its discretization method in a small amount of user-level code. It is then straightforward to obtain operators such as Jacobians and Hessians for use in optimization algorithms.

Distribution of intensities of gamma-ray bursts

A spatial distribution of gamma-ray bursts which is based on a log N - log n(max) relation, where N is the number of bursts which have a maximum photon count rate in excess of n(max), is proposed. A relation of N of about n(max) exp -1.5 is predicted for the case of sources uniformly distributed in infinite, flat space. It is noted that the apparently isotropic distribution of about 100 bursts over the sky combined with the N of about n(max) exp -1.5 distribution does not permit a purely observational distance estimate for the sources. It is predicted that the data from the GRANAT and GRO missions will help distinguish between a Galactic and an extragalactic origin of the bursts. 22 references.

Sundance: High-level software for PDE-constrained optimization

Sundance is a package in the Trilinos suite designed to provide high-level components for the development of high-performance PDE simulators with built-in capabilities for PDE-constrained optimization. We review the implications of PDE-constrained optimization on simulator design requirements, then survey the architecture of the Sundance problem specification components. These components allow immediate extension of a forward simulator for use in an optimization context. We show examples of the use of these components to develop full-space and reduced-space codes for linear and nonlinear PDE-constrained inverse problems.