Steve McCormick
Research Projects
Relevant publications can be found on Amath and ResearchGate.
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Mixed- and Progressive-Precision Multigrid
- Together with Rasmus Tamstorf and Joe Benzaken at Disney Animations, we are continuing to develop theory and algorithms for dealing with finite precision in the development of iterative refinement, multigrid V-cycles, and FMG.
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Multigrid Methods for Cloth Simulation
- Together with Rasmus Tamstorf at Disney Animations, we are continuing to develop theory and multigrid algorithms for cloth simulation.
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Adaptive Algebraic Multigrid (αAMG) & Smoothed Aggregation (αSA)
- Begun in the early 1980s with John Ruge and later including several others, we developed the first adaptive multigrid algorithms that aims to provide effective coarsening when it is difficult or even infeasible to do in practice. Its first significant success was to provide a solver for Quantum Chromodynamics (QCD) that for the first time eliminates so-called critical slowing down.
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Algebraic Multigrid (AMG)
- In the early 1980s together with Achi Brandt and John Ruge, we developed the original algebraic multigrid (AMG) method and applied it successfully to the purely algebraic form of the North American Datum problem. Further development continued throughout several subsequent decades.
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Fast Adaptive Composite-Grid (FAC)
- Together with Jim Thomas in 1986, and later with several others, we developed the Fast Adaptive Composite Grid (FAC) method for multigrid solution of problems on locally refined grids. The benefit of the method stems from it being fully in the spirit of conventional multigrid on uniform grids, thereby ensuring its optimality as well as other multigrid properties. We also developed a parallelizable asynchronous version of FAC, denoted aFAC, that allows for simultaneous processing of all levels of refinement.
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Fast Solver for Dirac-Wilson Quantum Chromodynamics (QCD)
- In 2007, together with James Brannick, Marian Brezina, David Keyes, Oren Livne, Ira Livshits, Scott MacLachlan, Tom Manteuffel, John Ruge, and Ludmil Zikatanov, we developed the first solver for lattice QCD that eliminates so-called critical slowing down. It uses the level hierarchy that the uniform grids of QCD allows, but it develops the interpolation coefficients by way of adaptive SA that enabled the method to achieve optimal performance.
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First-Order System Least Squares (FOSLS)
- Together with Tom Manteuffel, Zhiqiang Cai, and Raytcho Lazarov, in 1994 and later with several others, we developed FOSLS for transforming PDEs into H^1 equivalent systems that allow for standard finite element discretization and straightforward optimal multigrid solvers. For the first time, formulations were developed with these properties for Stokes, Navier-Stokes, linear elasticity (especially in or near the incompressible limit), and several other applications. We also developed a dual-like method denoted FOSLL* that provides many of the same benefits but does not require increased regularity.
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Rayleigh Quotient Multigrid (RQMG)
- In the late 1980s together with Jan Mandel, we developed RQMG as a fully variational eigenproblem solver for SPD matrices. It provides an alternative to inverse-iteration-muiltigrid and FAS eigensolvers in that RQMG exhibits similar performance characteristics but has the advantage of being globally convergent in practice.
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Unigrid
- In the early 1980s together with John Ruge, we developed a new algorithm called unigrid that only operates on the finest grid but is nevertheless equivalent to standard multigrid. It is much more expensive than multigrid, but it is an effective prototype to test multigrid ideas. Its other advantages are that it is very simple to implement, it requires very few lines of code, it drastically reduces the chances for mistakes, and it is easy to make major changes such as adding local refinement.
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V-Cycle Theory for General Smoothers
- In 1985, we developed the first theory for the V-cycle that applies to general smoothers, including Gauss-Seidel and conjugate gradients. Additional multilevel theory was developed in later years.