A Unified Framework for Large Scale Scientific Computing

Principal Investigator: Jun Zhang
Graduate Research Assistant: TBD

Funding Sources: National Science Foundation
Funding Division: CCF
Funding Program: CISE
Program Director:
Contract Number: CCF-0727600
Estimated Budget: $190,000
Duration: 10/01/2007 - 09/30/2010 (36 months)


We propose to build a computational framework for scalable and high efficiency solution of elliptic partial differential equations (PEDs). We will develop a novel high-order multiscale multigrid computation methodology, to embed high accuracy computation in fast computing methods in a seamless way. In particular, we will study a class of two-scale grid sixth-order compact difference schemes for elliptic PDEs; design geometric two-scale (multiscale) multigrid methods and robust scalable block semialgebraic multigrid methods to solve the resulting large sparse linear systems, and implement and test the scalable high resolution computational framework on high performance parallel computers.

Intellectual Merit: The developed high performance scalable high accuracy computational techniques will simultaneously advance the numerical solution of PDEs in two fronts. One is to compute high accuracy solution by using high-order discretization methods, another is to compute the discrete solution in a minimum amount of computer time by using the fastest sparse linear system solvers. These two fronts have previously been pushed forward separately by two different camps of researchers. The novelty of the proposed work is not just to advance these two areas by advancing the two fronts simultaneously but separately. Instead, the proposed new computational framework will advance the two fronts collectively by fusing the ideas and advantages of multiscale discretization and multigrid computations, to achieve the ultimate goal of computing accurate numerical solution of PDEs at the minimum computer costs. The proposed research work is the convergence of years of research work by many researchers in several different areas. The proposed computational algorithms combine, for the first time, the high-order discretization of the governing equations and the fast solution of the resulting sparse linear systems in a seamless multiscale multigrid computational framework. This computational framework will possess high accuracy, high speed, high scalability, and will deliver optimal efficiency for computing the numerical solution of systems of elliptic PDEs.

Broader Impact: Fast numerical solution methods for systems of PDEs can impact many computational science and engineering and industry modeling and simulation applications. They are routinely used in the U.S. national laboratories to study complex science problems using advanced computer simulation technologies. As U.S. high-tech industry moves from experiment-based design and development to computer-assisted design and development, higher performance numerical methods and faster computer simulation techniques will benefit U.S. industry by enabling design and development engineers to conduct quick verification to test their new ideas on computers, before committing to expensive experiments. These technologies are essential for the U.S. industry to maintain its leadership position in the competitive world market. Graduate and undergraduate students will be trained in this project to be the next generation researchers and educators with solid scientific computing skills. We will involve members from underrepresented groups in this research project, and disseminate research results as fast and as widely as possible both in traditional ways and with internet technologies.

Technical Reports and Computer Software:
  1. Fortran Program for Multiscale Multigrid Computation of 2D Convection Diffusion Equation (Yin Wang, and Jun Zhang), Download the software code mgsix2D.f.
  2. High Accuracy and Scalable Multiscale Multigrid Computation for 3D Convection Diffusion Equation (Yin Wang, and Jun Zhang), Technical Report CMIDA-HiPSCCS 016-09, Department of Computer Science, University of Kentucky, KY, 2009.
    Abstract. Download the PDF file yinwang5.pdf.
  3. Integrated Fast and High Accuracy Computation of Convection Diffusion Equations Using Multiscale Multigrid Method (Yin Wang, and Jun Zhang), Technical Report CMIDA-HiPSCCS 015-09, Department of Computer Science, University of Kentucky, KY, 2009.
    Abstract. Download the PDF file yinwang4.pdf.
  4. Sixth Order Compact Scheme Combined with Multigrid Method and Extrapolation Technique for 2D Poisson Equation (Yin Wang, and Jun Zhang), Technical Report CMIDA-HiPSCCS 003-08, Department of Computer Science, University of Kentucky, KY, 2008.
    Abstract. Download the PDF file yinwang3.pdf.

Conference, Workshop, and Seminar Presentations:

  1. Yin Wang, and Jun Zhang, "A Sixth Order Finite Difference Computation with Multigrid Method and Extrapolation for 2D Poisson Equation", Poster presentation at the 4th Kentucky Innovation and Enterprise Conference, Lexington, Kentucky, April 17, 2008.
  2. Yin Wang, and Jun Zhang, "Block Diagonal Preconditioners using Diagonal Shifting and Stabilized SVD for Solving Dense Linear Systems in Electromagnetics", presented at the Intensive International Workshop on High Performance Computing for Informatics and Biosciences, Bowling Green, KY, October 5-6, 2007.


  1. The Ph.D. student supported by the award has won the University of Kentucky Presidential Fellowship for the year 2009-2010.

This page is supported by the U.S. National Science Foundation. However, any opinions, findings, and conclusions or recommendations expressed in this documents are those of the author and do not necessarily reflect the views of the U.S. National Science Foundation.

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This page was created on Friday, September 21, 2007, by
Jun Zhang
Last modified on Wednesday, September 23, 2009.