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Luke Olson

Research

Research history     ::     Ongoing projects     ::     Related interests

Summary...

Areas of interest:    numerical analysis, scientific computing, high performance computing
Keywords: numerical PDEs, finite elements, spectral elements, multigrid, iterative methods

Research directions

Group...

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Jacob Schroder   (PhD expected May 2010)

Jacob works on extending algebraic multigrid (AMG) robustness. This includes new strength-of-connection methods for the setup algorithm, extending AMG to complex and non-symmetric problems, AMG for electromagnetics by enriching the coarse space with plane-waves, and developing for the PyAMG project. Jacob collaborates with Sandia National Laboratories and will be going to Colorado-Boulder for a postdoc in the Fall.

  • [presentation] A General Interpolation Strategy for Algebraic Multigrid Using Energy-Minimization, Eleventh Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. April 4, 2010
  • [presentation] Smoothed Aggregation Multigrid for Helmholtz Problems, Fourteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March 23, 2009.
  • [presentation] A General Strength-of-Connection Concept in AMG, Tenth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. April 7, 2008.

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Jehanzeb Hameed Chaudhry   (PhD expected May 2011)

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James Lai  (PhD expected May 2012)

James has been working on algebraic multigrid (AMG) methods for high order H curl problems which arise in electromagnetics. He is also working on combining ideas from discontinuous Galerkin (DG) methods and first order system of least squares (FOSLS) for a new approach to solving Stoke's flow equations.

  • [presentation] Algebraic multigrid for high order hierarchical H curl elements. 11th Copper Mountain Conference on Iterative Methods.

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Matt Wrobel

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Steven Dalton

Group Affiliates...

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Yuki Kimura  (Chemical and Biomolecular Engineering)

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John Fettig  (Mechanical Science and Engineering)

Group Alumni...

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Nathan Bell, PhD, 2008  (now at Nvidia)

Nathan's thesis was on discrete Laplacians. While applications are immense, optimal solvers are not well developed for these problems. Nathan developed high-performance multilevel solvers to identify the target null-vectors and hodge decompositions.

  • [presentation] Algebraic Multigrid for Discrete Laplacians, Thirteenth Copper Mountain Conference on Multigrid Methods, March 2006.

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David Alber, PhD, spring 2007  (now at Microsoft)

David worked on parallel algebraic multigrid methods. His work 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.

The Projects...

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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.

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Preconditioning for High-Order DG in Electromagnetics

Collaborator: Jacob Schroder (Illinois)

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.
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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.

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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.

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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.

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Fast Solutions Methods for Multiscale Cellular Mechanics in Microcirculation

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

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 Illinois CSE Fellowship (Isfahani)

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Multiscale Modeling in Neutrophil Chemotaxis

Collaborators: C. Rao (Chemical Sciences at Illinois)

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.

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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.

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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.

The Older Projects...

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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
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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
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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

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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
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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

Publications

pdf Bell, W., Olson, L., Algebraic Multigrid for k-form Laplacians, Numerical Linear Algebra with Applications, Volume 15, Issue 2-3, Pages 165-185, 2008.

pdf D. M. Alber and L. N. Olson, Parallel Coarse Grid Selection, Numerical Linear Algebra with Applications, Volume 14, Issue 8, Pages 611-643, 2007

pdf L. N. Olson, Algebraic Multigrid Preconditioning of High-Order Spectral Elements for Elliptic Problems on a Simplicial Mesh SIAM J. Sci. Comput., Volume 29, Issue 5, pages 2189-2209, September 2007

pdf Schroder, J., Tuminaro, R., Olson, L., Generalized Strength of Connection in Algebraic Multigrid, CSRI Summer Proceedings 2007, Sandia National Laboratory, December, 2007

pdf Lin, X., Olson, L., Jin, J., An interpolatory spectral element method using curl-conforming vector basis functions on tetrahedra, IEEE Antennas and Propagation International Symposium, June 2007

pdf J. J. Heys, T. A. Manteuffel, S. F. McCormick, and L. N. Olson, Algebraic Multigrid for Higher-Order Finite Elements, J. Comp. Phys., 26, 2005, no. 2, 520-532.

pdf J. S. Hesthaven, L. N. Olson, and L. Wilcox, L.N. Olson, J.S. Hesthaven, and L.C. Wilcox, 2007, Developments in Overlapping High-Order Nodal Discontinuous Galerkin Methods, Lect. Notes Comput. Sci. Eng. 55, 325-332.

pdf H. De Sterck, T. A. Manteuffel, S. F. McCormick, and L. N. Olson, Numerical Conservation Properties of H(div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation, SIAM J. Sci. Comput. 26 (2005), no. 5, 1573-1597

pdf H. De Sterck, T. J. Manteuffel, S. F. McCormick, and L. N. Olson, Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs, SIAM J. Sci. Comput. 26 (2004), no. 1, 31-54

pdf L. N. Olson, A Dual Least-Squares Finite Element Method for Hyperbolic PDEs, Eleventh Copper Mountain Conference On Multigrid Methods Student Paper Competition, 2003.

pdf L. N. Olson, Multilevel Least-Squares Finite Element Methods for Hyperbolic PDEs Ph.D. Thesis, 2003.

Presentations

pdf 2008 January Advances in AMG for Topological Problems: A smoothed aggregation approach to k-form Laplacians, Applied Mathematics Departmental Colloquium, Boulder, Colorado.

pdf 2007 July Algebraic Multigrid (AMG) for High-Order Spectral Elements, 6th International Congress on Industrial and Applied Mathematics, Zurich, Switzerland. Minisymposium: Developments in algebraic multigrid methods for real-world applications

pdf 2007 July Using Algebraic Multigrid (AMG) for High-Order Spectral Elements, Argonne National Laboratory, Argonne, IL.

pdf 2006 Apr. Algebraic Multigrid Preconditioning for Spectral Elements: Strategies for unstructured, 3D, DG problems, Ninth Copper Mountain Conference on Iterative Methods, Copper Mountain, CO.

pdf 2005 Nov. A High-Order Mess, Graduate Student Seminar, Department of Computer Science, University of Illinois at Urbana-Champaign.

pdf 2005 Nov. Developments in Multigrid for High-Order Spectral Elements, Computer Science Seminar, Department of Computer Science and Engineering, University of Minnesota.

pdf 2005 May Algebraic Multigrid (AMG) for Higher-Order Methods, 7th IMACS International Symposium on Iterative Methods in Scientific Computing,Toronto, ON.

pdf 2005 Apr. Algebraic Multigrid (AMG) for Higher-Order Methods, Twelfth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO.

pdf 2005 Jan. Overlapping Schwarz-type Preconditioning and High-Order Elements, 16th International Conference on Domain Decomposition Methods.

pdf 2004 June Preconditioning High-Order Discontinuous Galerkin Discretizations of the Helmholtz Equation, International Conference On Spectral and High-Order Methods (ICOSAHOM) 2004, Brown University, Providence, RI.

pdf 2004 Apr. Solution Methods for High-Order Discontinuous Galerkin Discretizations of the Helmholtz Equation, Eighth Copper Mountain Conference on Iterative Methods, Coper Mountain, CO.

pdf 2003 Oct. Least-Squares Finite Element Methods:An Overview of the Methodology and Extensions to Hyperbolic Conservation Laws, Scientific Computing and Numerical Analysis Seminar, Brown University, Providence, RI.

pdf 2003 July Least-Squares Finite Element Methods for Hyperbolic PDEs, International Council for Industrial and Applied Mathematics Conference, Sydney, AUS.

pdf 2003 Apr. Least-Squares Finite Element Methods for Linear Hyperbolic PDEs: A study of AMG, Eleventh Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO.

pdf 2003 Feb. Least-Squares Finite Element Method for Linear Hyperbolic PDEs, SIAM 2003 Computational Science and Engineering Conference, San Diego, CA.

pdf 2002 Nov. De Rham Cohomology and Finite Elements Applied to Hyperbolic PDEs, Computational Mathematics Seminar, University of Colorado, Boulder, CO.

pdf 2002 Apr. Developments in Multilevel Methods for Time-Dependent Hyperbolic PDEs, Seventh Copper Mountain Conference on Iterative Methods, Copper Mountain, CO.

Past Work...

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