Descriptions of the Research Projects on kali

This page can be reached via kali's webpage at http://www.math.umbc.edu/kali.
Efficient Implementation of an Iterative Solver for Linear Systems of Equations on Beowulf Clusters
Kevin P. Allen and Matthias K. Gobbert, Department of Mathematics and Statistics, UMBC
July 2003 to July 2004
Description: The conjugate gradient method is applied to a large, sparse, highly structured linear system of equations obtained from a finite difference discretization of the Poisson equation. This prototype problem is used to analyze the performance of the parallel linear solver on a cluster of workstations. The matrix-free implementation of the matrix-vector product is shown to be optimal with respect to both memory usage and performance. The parallel implementation of the method can give excellent performance on a Beowulf cluster, a group of commodity workstations connected by a dedicated communication network. The optimal number of processors depends on the quality of the interconnect hardware. When only an ethernet interconnect is available, best performance is limited to up to 4 or 5 processors, since the conjugate gradient method necessarily involves several communications per iteration. Using a high-performance Myrinet interconnect, excellent speedup is possible for at least up to 32 processors. This justifies the use of the method as the computational kernel for the time-stepping in the numerical solution of a system of reaction-diffusion equations.
This work extended undergraduate research by Kevin Allen conducted until his graduation with a B.S. in May 2003.

Publications:

Parallel Simulations of the Linear Boltzmann Equation for Models in Microelectronics Manufacturing
Michael Reid, Matthias K. Gobbert, Department of Mathematics and Statistics, UMBC, and Timothy S. Cale, Process Evolution, Ltd. and School of Materials, Arizona State University
July 2003 to present
Description: Production steps in the manufacturing of microelectronic devices involve gas flow at a wide range of pressures. We develop a kinetic transport and reaction model based on a system of time-dependent linear Boltzmann equations. These kinetic equations have the property that velocity appears as an independent variable, in addition to position and time. A deterministic numerical solution for realistic three-dimensional application problems requires the discretization of the three-dimensional velocity space, the three-dimensional position space, and time.
We design a spectral Galerkin method to discretize the velocity space by specially chosen basis functions. The basis functions in the expansion lead to a system of hyperbolic conservation laws with constant diagonal coefficient matrices for each of the linear Boltzmann equations. These systems of conservation laws are solved using the discontinuous Galerkin finite element method. Stability and convergence of the method are verified analytically and demonstrated numerically. As an application example, we simulate chemical vapor deposition at the feature scale in two and three spatial dimensions and analyze the effect of pressure. Finally, we present parallel performance results which indicate that the implementation of the method possesses excellent scalability on a Beowulf cluster with a high-performance Myrinet interconnect.
Students involved: Michael Reid is currently performing undergraduate research. Samuel G. Webster graduated with a Ph.D. in May 2004. Mark L. Breitenbach graduated with a M.S. in December 2004.

Publications:

Solving Large-Scale Semidefinite Programs in Parallel
Madhu V. Nayakkankuppam, Department of Mathematics and Statistics, UMBC
July 2003 to September 2005
Description: Numerous applications of semidefinite programming, notabaly relaxations in combinatorial optimization, result in large-scale problems beyond the solution capabilities of interior-point algorithms. We implement a stand-alone, parallel and distributed version of the spectral bundle method that effectively handles sparse, large-scale instances of semidefinite programs with the usual block diagonal structure. Function and subgradient evaluations are handled by a Lanczos process specially tailored to exploit this block diagonal structure. Problem data is handled in a distributed manner to conserve memory and to hence allow solution of the largest instances. Near optimal speedup factors have been observed on typical problems from applications on up to 64 processors. Numerical experiments also show that this code significantly extends present solution capabilities. For instance, a Lovasz theta function SDP relaxation involving a 5000 x 5000 matrix variable coupling over 67000 constraints is solved to 3 digits of accuracy in under 12 minutes using 64 processors.

Publication:

Globalization of Regional Models of the Atmosphere
Hai Zhang and Miodrag Rancic, Department of Physics, UMBC
July 2003 to August 2006
Description: Most of the global models of the atmosphere that are nowadays used in the climate prediction studies have been rarely (or not at all) systematically tested in the everyday weather forecasting mode, which is the most reliable way for their calibration and for understanding of their performance in the long-term climate simulations. There is also an increased interest in application of the high-resolution regional models in downscaling of the regional climate and for tracking of the hurricanes and local storms. In these experiments the standard one-way interaction paradigm, where the role of the global model is just to produces boundary conditions for the run of the high-resolution regional model, is often inadequate. The objective of this project is to develop a unified computational framework that will serve for expansion of the regional models of the atmosphere to global coverage. This "Globalization Framework" will be augmented by a capability for local enhancement of resolution over selected areas of interest and the movement of high-resolution domains over the globe. The globalization will be achieved through application of quasi-uniform spherical grids which avoid well-known problems of over-resolving in the polar regions typical for the standard longitude-latitude mesh.
For more information, see http://userpages.umbc.edu/~mrancic/GEF.html.
Account sponsor: Matthias K. Gobbert.

Publications:

The Standard Genetic Code Enhances Adaptive Evolution of Proteins
Wen Zhu and Stephen Freeland, Department of Biological Sciences, UMBC
March 2004
Description: The standard genetic code, by which most organisms translate genetic material into protein metabolism, is non-randomly organized. The Error Minimization hypothesis posits that natural selection produced a code that buffers genomes against the impact of mutations. However, previous studies supporting this hypothesis treat the code as an isolated trait, ignoring its influence on the evolution of the protein coding genome that it serves. Here we develop a population genetic model of molecular evolution to test the rate of adaptive gene evolution under different genetic codes. We show that the pattern of codon assignments has a profound effect on the speed of adaptive evolution, and offer a fundamental re-interpretation of the adaptive genetic code, from one that minimizes errors to one that enhances the efficacy of natural selection. The use of kali allowed the rapid completion of large numbers of simulations.
The simulations were conducted by Matthias K. Gobbert as part of the Scientific Computing and Statistical Data Analysis Laboratory SCSDAL.

Publications:

Numerical Simulation of Calcium Waves in Human Heart Cells
Michael Muscedere, Matthias K. Gobbert, Department of Mathematics and Statistics, UMBC
June 2004 to present
Description: The release of calcium ions in a human heart cell is modeled by a system of reaction-diffusion equations, which describe the interaction of the chemical species and the effects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposition of many Dirac delta functions in space; such a non-smooth right-hand side leads to divergence for many numerical methods. The calcium ions enter the cell at a large number of regularly spaced points throughout the cell; to resolve those points adequately for a cell with realistic three-dimensional dimensions, an extremely fine spatial mesh is needed. A finite element method is developed that addresses the two crucial issues for this and similar applications: Convergence of the method is demonstrated in extension of the classical theory that does not apply to non-smooth forcing functions like the Dirac delta function; and the memory usage of the method is optimal and thus allows for extremely fine three-dimensional meshes with many millions of degrees of freedom, already on a serial computer. Additionally, a parallel implementation of the algorithm allows for the solution on meshes with yet finer resolution than possible in serial.

Publications:

Gaussian Process Smoothing of Activation Pattern in Event Related FMRI Experiments
Anindya Roy, Department of Mathematics and Statistics, UMBC
August 2004
Description: The BOLD response in the task related sctive region is convoluted with noise from several sources. A gaussian process smoothing is applied to filter noise from the voxel specific response and create a gray image of activity pattern. The Gaussian process prior is a flexible prior where the smoothness can be controled by the covariance kernel. The difference image between two groups of subjects is created by the difference in the posterior estimates.
The simulations required a large amount of memory and were performed by Matthias K. Gobbert.

Validation of Deterministic Methods for the Computation of Bit Error Ratios
Walter Pellegrini and Curtis R. Menyuk, Department of Computer Science and Electrical Engineering, UMBC, and John Zweck, Department of Mathematics and Statistics, UMBC
August to November 2004
Description: In order to accurately compute the bit error ratios for an optical transmission system it is necessary to numerically calculate the probability density function (pdf) of the received noisy signal down to values that are typically on the order of 10-15. We compare two methods for comuputing this pdf: A deterministic, computationally efficient method called the Covariance Matrix Method that is based on a linearization of the nonlinear Schroedinger equation, and a statistical method based on recursive importance sampling called the Multicanonical Monte Carlo Method which enables the the tails of the pdf to be adepquately sampled in a reasonable amount of time. By comparing the results of the two methods, we will mutually validate them.
The results were used as part of Walter Pellegrini's M.S. thesis (December 2004).
Account sponsor: Matthias K. Gobbert.

Publications:

Operator-Based Upscaling for the Acoustic Wave Equation
Tetyana Vdovina, Oksana Korostyshevskaya, and Susan E. Minkoff, Department of Mathematics and Statistics, UMBC
November 2004 to March 2005
Description: Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different temporal scales. Operator-based upscaling allows us to capture the effect of fine scales on the coarse grid without solving the original full fine-scale problem. Operator-based upscaling applied to the constant density variable sound velocity acoustic wave equation consists of two stages. First we solve for the fine-scale information internal to each coarse block. Than we use the subgrid solutions to define an upscaled operator on the coarse grid. The original boundary conditions and fine-grid velocity field are used in both stages. The equivalence between the variational form and staggered finite-difference scheme allows us to use finite differences to solve the subgrid problems. Due to the homogeneous Neumann boundary conditions imposed on each coarse block the subgrid problems decouple, which leads to the natural parallelization of the first stage of the method. The only communication in the algorithm occurs during the transition between solving the subgrid problems and solving the coarse problem. Timing studies indicate that the parallel algorithm has near optimal speedup. Three numerical experiments with variable velocity models compare the upscaled and full finite-difference solutions. The results show that operator-based upscaling captures the essential fine-scale information and models wave propagation accurately.
Account sponsor: Matthias K. Gobbert.

Publications:

Calculation of the Anharmonic Vibrational Spectrum for Large Biomolecular Systems
Susan K. Gregurick, Rad Balu, Hailiang Zhang, and Eli Zukowski, Department of Chemistry and Biochemistry, UMBC
November 2005 to present
Description: THz vibrational spectroscopy of biomolecular systems is widely recognized for its sensitivity to force constants influencing the global nuclear motions that extend over a large fraction of the biomolecular framework. We have previously calculated the low frequency THz spectra for the crystaline amino acids of Serine and Cystine and found good aggrement with the corresponding experimental THz spectroscopy [Balu et al., Chem. Phys. Letts., accepted (2005)]. We then extended our methodology for crystalline peptides in differing configurations and found that the inclusing of water molecules greatly effected the accuracy of our calculation [Balu et al., J. Phys. Chem. A, submitted (2005)]. The proposed research effort will directly address this shortcomming by exploring both the harmonic approximations made in the previous calculations and the limited inclusion of polarization in the empirical water force field. From this work, we shall be able to extend crystalline vibrational calculations to include anharmonic effects and we shall improve the polarization functions used in standard emperical force fields.
Account sponsor: Matthias K. Gobbert.

Publications:

Solving an Inverse Problem In Acoustic Wave Equation Using Upscaling Operation
Sean Griffith, Shiming Yang, and Susan E. Minkoff, Department of Mathematics and Statistics, UMBC
November 2004 to present
Description: Given some observed data, we are interested in finding out parameters that specify models of physics systems. Such inverse process generally requires to minimize an error functional. Some optimization methods require the gradient of such functional. In this project, we need to determine the sound velocity as one parameter in an acoustic wave equation. The adjoint-state mehod is applied to compute the gradient of this functional. Besides, the operator-based upscaling is used to solve the adjoint problem derived from the adjoint-state method. Our current work is to implement these methods in a parallel computing environment. The correctness of solution will be checked, and speedup is desired through timing studies.
Account sponsor: Matthias K. Gobbert.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0215373.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Copyright © 2003-2008 by Matthias K. Gobbert. All Rights Reserved.
This page version 3.8, July 2008.