Luke Olson

Nathan Bell (PhD exp. spring 2008?)

Nathan's efforts are in the direction of discrete Laplacians. While applications are immense, optimal solvers are not well developed. Nathan is developing high-performance multilevel solvers to identify the target null-vectors and hodge decompositions in the problem.

Related work:

• [presentation] Algebraic Multigrid for Discrete Laplacians, Thirteenth Copper Mountain Conference on Multigrid Methods, March 2006.
• Bell, W., Olson, L., Algebraic Multigrid for k-form Laplacians, Numerical Linear Algebra with Applications, Volume 15, Issue 2-3, Pages 165-185, 2008.

Jacob Schroder (PhD exp. 2009?)

Jacob is researching robust multilevel solvers for high-order discontinuous Galerkin methods. Additional and related efforts include work with Sandia in constructing new approaches to smoothed aggregation based algebraic multigrid for larger classes of problems.

Related work:

• Schroder, J., Tuminaro, R., Olson, L., Generalized Strength of Connection in Algebraic Multigrid, CSRI Summer Proceedings 2007, Sandia National Laboratory, December, 2007
• [presentation] Defining a General Strength-of-Connection Concept in AMG, Tenth Copper Mountain Conference on Multigrid Methods, March 2006.

David Alber(PhD, spring 2007)

David's work is on parallel algebraic multigrid methods. His work has yeilded several new algorithms to improve scalability for a range of applications. In collaboration with LLNL, David's work overlaps into combinatorial scientific computing, high-performance computing, and iterative methods.

Related work:

• Alber, D., Olson, L., Parallel coarse grid selection, Numerical Linear Algebra with Applications,Volume 14, Issue 8, Pages 611-643, 2007
• [presentation] Bucket Sorted Independent Sets, Thirteenth Copper Mountain Conference on Multigrid Methods, March 2006.
• [poster] Evolution of a Coarse Grid Selection Algorithm,The SIAM Workshop on Combinatorial Scientific Computing, Feb 2006.
• [presentation] AMG Setup Phase Algorithms,SIAM Conference on Computational Science and Engineering, Feb 2006.
• [presentation] Parallel Coarse Grid Selection Strategies, Ninth Copper Mountain Conference on Iterative Methods, April 2006.

Xi Lin (Senior, Elec. Eng, UIUC)

Xi developed a high-order vector spectral elements in 3D for problems in electromagnetics.

Related work:

• Lin, X., Jin, J., Olson, L., An interpolatory spectral element method using curl-conforming vector basis functions on tetrahedra, , IEEE Antennas and Propagation International Symposium, June 2007, Feb 2006.
• A Nodal Spectral Element Method Using Curl-conforming Vector Basis Functions on Tetrahedra, SIAM Conference on Computational Science and Engineering, Feb 2006.

Parallel Coarsening Strategies for Algebraic Multigrid

Collaborator: David Alber

Setup times in multigrid algorithms are a large expense. We offer several new parallel algorithms to increase the efficiency in a parallel including , CLJP-c1, CLJP-c2, BSIS and others.

Preconditioning for High-Order DG in Electromagnetics

Collaborator: Jacob Schroder (UIUC)

We study multilevel (domain decomposition and multigrid) preconditioners for high-order discretizations of problems in electromagnetics. Stabilization terms enforced in the numerical fluxes of many Discontinuous Galerkin schemes greatly impact the (near) null-space of the resulting algebraic system. We investigate approaches to improve the conditioning of the problem.

Project supported by NSF DMS 06-12448.

Algebraic Multigrid for Discrete Laplacians

Collaborator: Nathan Bell

k-form Laplacians arise in numerous areas of application, yet fast solution methods to identify specific null-space vectors or hodge decompositions has received little attention. We develop optimal solvers for such cases.

Algebraic Multigrid high-order DG discretizations

New preconditioners are proposed to handle two problematic situations: very high order spectral elements in 3D and non-conforming discretizations throught LDG.

High-Order Vector Elements

High-order vector based spectral elements are studied as an extension to the popular Whitney form. The nontrivial elements are pursued in a simplex in 3D and offer natural form for high-order simulation.

Fast Solutions Methods for Multiscale Cellular Mechanics in Microcirculation

Collaborators: J. Freund (MIE at UIUC), A. Isfahani (TAM at UIUC)

Accurate and reliable simulations in the simulation rely on the solution to highly coupled systems of equations. Novel preconditioning strategies are being developed to address these complexities.

Project funded by UIUC CSE Fellowship (Isfahani)

Multiscale Modeling in Neutrophil Chemotaxis

Collaborators: C. Rao (Chemical Sciences at UIUC)

Analysis of neutrophil chemotaxis is complex due to the coupled physical models in the system. We are investigating new multiphysics/multiscale strategies to more accurately analyze the process computationally.

Matrix-Free Preconditioning for High-order LDG Methods

Collaborators: Jan Hesthaven (Brown University) Lucas Wilcox (Brown University)

One of the computational bottlenecks in a discontinuous high-order spectral method is the storage requirements of the the discretization operator and associated preconditioning operators. In this study we offer effective strategies toward an efficient matrix-free implementation.

Preconditioning for Adapted High-Order Discretizations of the Time Harmonic Maxwell's Equation

Collaborators: Jan Hesthaven (Brown University) Lucas Wilcox (Brown University)

The goal of this project is to develop successful hybrid Schwarz preconditioners for hp adapted discretizations of the time harmonic Maxwell's equation. Our focus is to acheive computationally efficient approaches to adjoint-based error estimation schemes.

Additive Schwarz Preconditioning for Indefinite Problems

Collaborators: Jan Hesthaven (Brown University) Lucas Wilcox (Brown University)

The effectiveness of several preconditioned additive domain decomposition methods is studied for indefinite problems arising in electromagnetics.

Project Supported by NSF VIGRE

Algebraic Multigrid for high-order finite elements

Collaborators: Jeff Heys (Arizona State University), Tom Manteuffel (University of Colorado at Boulder), Steve McCormick (University of Colorado at Boulder)

Classical approaches to multigrid methods for high-order discretizations have not scaled well with problem size. In this project, we outline an approach toward efficient algebraic multigrid (AMG) preconditioning. We rely on spectral equivalence of a low-order discretization on a related grid to maintain convergence while avoiding large computational complexities.

Project Supported by NSF VIGRE

The Parachute Problem Revisited

Collaborators: Manu Lohiya (undergrad, Brown University)

In this study, we revisit classic attempts to model a parachute drop with standard ODEs. Our approach improves the model by considering smoothness of the transitions in more detail and by adding elasticity of the suspension lines through a two body approach.

Project support by UTRA REU

Numerical Conservation Properties of H(div)-conforming Least-Squares Finite Element Methods for the Burgers Equation

Collaborators: Hans De Sterck (University of Waterloo) Tom Manteuffel (University of Colorado at Boulder), Steve McCormick (University of Colorado at Boulder)

Least-squares finite element methods for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector explicitly. Weak conservation theorems that are analogous to the Lax-Wendroff theorem for conservation finite difference methods are proved.

Project Supported by NSF VIGRE

Least--Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs

Collaborators: Hans De Sterck (University of Waterloo) Tom Manteuffel (University of Colorado at Boulder) Steve McCormick (University of Colorado at Boulder)

Least-squares finite element methods for scalar linear partial differential equations of hyperbolic type are studied. Theoretical considerations such as boundary data and well-posedness of the variational formulation are presented. Computational details are adressed in the context of an algebraic-based multigrid approach.

Project Supported by NSF VIGRE