apma4301_NonlinearManifesto08

apma4301_NonlinearManifesto08 - Lecture #11 Part C 12...

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12 November 2008 Lecture #11 Part C Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2250@columbia.edu Applied Mathematics 4301: Numerical Methods for PDEs A Nonlinearly Implicit Manifesto
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David Keyes Applied Physics & Applied Mathematics Columbia University T ÇÉÇÄ|ÇxtÜÄç |ÅÑÄ|v|à ÅtÇ|yxáàÉ * * t ÑâuÄ|v wxvÄtÜtà|ÉÇ Éy ÑÜ|Çv|ÑÄxá tÇw |ÇàxÇà|ÉÇá
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got implicitness?
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Going implicit? z Why you would , if you could : 1. multiscale problems with good scale separation 2. coupled problems (“multiphysics”) 3. problems with uncertain or controllable inputs (optimization: design, control, inversion) z You can , so you should ! 1. optimal and scalable algorithms known 2. freely available software 3. reasonable learning curve that harvests legacy code
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Current focus on Jacobian-free implicit methods z Two stories to track in supercomputing ± raise the peak capability ± lower the entry threshold higher capability for hero users best practices for all users New York Blue at BNL (#10 on the Top 500) first frontier “new” frontier z Jacobian a steep price, in terms of coding ± very valuable to have, but not necessary ± approximations thereto often sufficient ± meanwhile, automatic differentiation tools are lowering the threshold
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Recent “E 3 ” report highlights limitations of explicit methods “The dominant computational solution strategy over the past 30 years has been the use of first-order- accurate operator-splitting, semi- implicit and explicit time integration methods, and decoupled nonlinear solution strategies . Such methods have not provided the stability properties needed to perform accurate simulations over the dynamical time-scales of interest. Moreover, in most cases, numerical errors and means for controlling such errors are understood heuristically at best.” 2007
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Recent E 3 report highlights opportunities for implicit methods “Research in linear and nonlinear solvers remains a critical focus area because the solvers provide the foundation for more advanced solution methods. In fact, as modeling becomes more sophisticated to include, increasingly, optimization, uncertainty quantification, perturbation analysis, and more, the speed and robustness of the linear and nonlinear solvers will directly determine the scope of feasible problems to be solved.” 2007
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2002 2003 2003-2004 (2 vol ) 2004 2006 2006 2007 Fusion Simulation Project June 2007 2007 Mathematical Challenges for the Department of Energy January 2008 2008 Some reports predicated on scalable implicit solvers … among many other developments
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Plan of presentation z Motivations for implicit solvers ± multi-scale, multi-physics, multi-solve (sensitivity, stability, uncertainty quantification, design, control, inversion) ± one-dimensional model problems, linear and nonlinear ± portfolio of real world problems z State-of-the-art for large-scale nonlinearly implicit solvers ± brief look at algorithms and software ± intuition about how they scale z Illustrative stories from the trenches ±
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apma4301_NonlinearManifesto08 - Lecture #11 Part C 12...

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