Multigrid methods 2013

Authors


The First Copper Mountain Multigrid Conference on Multigrid Methods was organized in 1983 by Steve McCormick, who chaired the conference until 2003. Widely regarded as one of the premier international conferences on multigrid methods, it occurs every other year, alternating (as of 1990) with the equally successful Copper Mountain Conference on Iterative Methods. McCormick's successor as chairman, Van Henson, chaired the 2013 multigrid meeting, assisted by Program Cochairmen Ulrich Rüde and Irad Yavneh.

The conference began with two tutorial sessions given by Irad Yavneh and Van Henson. The sessions covered multigrid fundamentals as well as algebraic multigrid (AMG) and other advanced topics. The remaining 5 days of the conference were organized around a sequence of 25-min talks, with ample time for individual research discussions between colleagues. The winners of the student paper competition were Sigfried Cools from Universiteit Antwerpen, Stephanie Friedhoff from Tufts University, and Delyan Kalchev from Lawrence Livermore National Laboratory. They presented their papers in the session for students.

The Sixteenth Copper Mountain Multigrid Conference on Multigrid Methods was held in the Colorado Mountains on March 17–22, 2013. Since it was the 30th anniversary of the first multigrid conference, special as 30 is a multiple of 10, there was a display of memorabilia, including sacred texts and palimpsests. We look forward to the next conference, as 32 is a power of 2.This special issue contains six papers; other papers from the conference may appear in subsequent issues of this journal. The papers address a variety of applications and topics. Brief summaries of each paper follow, listed in order of submission.

In [1], the author considers a multigrid method for the solution of the highly indefinite Helmholtz equation with constant wave numbers. The method is a modification of the wave-ray method developed by Brandt and Livshits.

In [2], the authors discuss the existing theoretical bounds on performance of AMG, which are not as sharp as the bounds available on geometric multigrid, where Fourier analysis is available as a tool.

In [3], the authors derive a multigrid method for the solution of discretizations of partial differential equations. The method is novel in two aspects. First, the method is a hybrid geometric–algebraic method. Second, the smoother is based on Chebyshev–Jacobi methods.

In [4], the authors derive a multigrid method based on smoothed aggregation for the solution of discrete problems arising from stochastic partial differential equations.

In [5], the authors derive a multigrid method for the solution of a Fredholm integral equation of the first kind, arising from the 2D elastic frictional contact problem. The smoother is based on a preconditioner that approximate the inverse of the original coefficient matrix. The smoother reduces most of the components of the error but enlarges several smooth components. Two methods are studied to remedy this feature.

In [6], the authors consider new additive variants with improved convergence rates compared with classical additive multigrid. The application is to high performance computers, where a strategy for dealing with the increasing complexity of multiplicative multigrid on coarser grids is to replace it with additive multigrid.

In addition to the talks, there was an evening session ‘Multigrid Showcase’, organized by Harald Koestler, in which participants with multigrid packages presented demonstrations of applications that benefit form fast multigrid solvers.

The 2013 conference was organized by Front Range Scientific Computation, Inc. It was co-organized by the University of Colorado and the Center for Applied Scientific computing, Lawrence Livermore National Laboratory (LLNL) in cooperation with the Society for Industrial and Applied Mathematics. It was sponsored by the LLNL, Los Alamos National Laboratory, the Department of Energy, IBM Corporation, and the National Science Foundation. The Program Committee members for the conference were Susanne Brenner, Marian Brezina, Joel Dendy, Craig Douglas, Robert Falgout, Jim Jones, Kirk Jordan, Tom Manteuffel, Steve McCormick, David Moulton, Luke Olson, Kees Oosterlee, Joseph Pasciak, John Ruge, Klaus Stüben, Stefan Vandewalle, and Ulrike Yang. The Program Committee served as guest editors for the special issue.

ACKNOWLEDGEMENT

We thank the editors of Numerical Linear Algebra with Applications for hosting this special issue, with special thanks to Maya Neytcheva and Panayot Vassilevski. This work was performed under the auspices of the US Department of Energy by Los Alamos National Laboratory under Contract DE-AC52-06NA25396.

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