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Computational and Applied Mathematics in Scientific Discovery David Keyes Dept of Applied Physics & Applied Mathematics, Columbia University Office of Science & Technology Policy Briefing 4 May 2004

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OSTP Briefing, 4 May 2004 Presentation plan l Emergence of simulation a third modality for scientific and technological research l Applications drivers and trends in simulation infrastructure outstanding opportunities l Hurdles to simulation role of applied and computational mathematics l Success factors and recommendations current pathfinding U.S. programs

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OSTP Briefing, 4 May 2004 Three pillars of scientific understanding l Theory l Experiment l Simulation “theoretical experiments” Computational simulation : “a means of scientific discovery that employs a computer system to simulate a physical system according to laws derived from theory and experiment”

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OSTP Briefing, 4 May 2004 Example: turbulent combustion l Simulation models and methods: n Arrhenius kinetics with 84 reactions & 21 species n Acoustically filtered hydrodynamics: 10 2 speedup n Cartesian adaptive mesh refinement: 10 4 speedup n Message-passing SIMD parallelism on 2048 procs l Reaction zone location a delicate balance of fluxes of: species, momentum, internal energy l Directly relevant to: engines, turbines, furnaces, incinerators (energy efficiency, pollution mitigation) l Component model of other computational apps: firespread, stellar dynamics, chemical processing This simulation sits at the pinnacle of numerous prior achievements in experiment, theory, applied mathematics, and computer science

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OSTP Briefing, 4 May 2004 Theory, experiment and simulation check, spur and enrich each other! Images c/o R. Cheng (left), J. Bell (right) 2003 SIAM/ACM Prize in CS&E (J. Bell & P. Colella) Instantaneous flame front imaged by density of inert markerInstantaneous flame front imaged by fuel concentration

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OSTP Briefing, 4 May 2004 What would we do with 100-1000x more? Example: probe the structure of particles Constraints on the Standard Model parameters and . For the Standard Model to be correct, they must be restricted to the region of overlap of the solidly colored bands. The figure on the left shows the constraints as they exist today. The figure on the right shows the constraints as they would exist with no improvement in the experimental errors, but with lattice gauge theory uncertainties reduced to 3%.

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OSTP Briefing, 4 May 2004 What would we do with 100-1000x more? Example: predict future climates Resolution of Kuroshio Current: Simulations at various resolutions have demonstrated that, because equatorial meso-scale eddies have diameters ~10-200 km, the grid spacing must be < 10 km to adequately resolve the eddy spectrum. This is illustrated in four images of the sea-surface temperature. Figure (a) shows a snapshot from satellite observations, while the three other figures are snapshots from simulations at resolutions of (b) 2 , (c) 0.28 , and (d) 0.1 .

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OSTP Briefing, 4 May 2004 Environment global climate contaminant transport Lasers & Energy combustion ICF Engineering crash testing aerodynamics Biology drug design genomics Experiments controversial Applied Physics radiation transport supernovae Experiments prohibited or impossible Scientific Simulation Experiments dangerous In these, and many other areas, simulation is an important complement to experiment. Experiments difficult to instrument Experiments expensive ITER: $5B The imperative of terascale simulation

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OSTP Briefing, 4 May 2004 Gedanken experiment: How to use a jar of peanut butter as its price slides? l In 2004, at $3.19: make sandwiches l By 2007, at $0.80: make recipe substitutions l By 2010, at $0.20: use as feedstock for biopolymers, plastics, etc. l By 2113, at $0.05: heat homes l By 2116, at $0.012: pave roads The cost of computing has been on a curve like this for two decades and promises to continue for another one. Like everyone else, scientists should plan increasing uses for it…

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OSTP Briefing, 4 May 2004 Gordon Bell Prize: “price performance” Four orders of magnitude in 12 years

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OSTP Briefing, 4 May 2004 Gordon Bell Prize: “peak performance” Four orders of magnitude in 13 years

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OSTP Briefing, 4 May 2004 Gordon Bell Prize outpaces Moore’s Law Four orders of magnitude in 13 years Gordon Moore Gordon Bell “Demi” Moore CONCUR- RENCY!!!

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OSTP Briefing, 4 May 2004 Hurdles to simulation l “Triple finiteness” of computers n finite precision n finite number of words n finite processing rate l Curse of dimensionality n Moore’s Law quickly eaten up in 3 space dimensions plus time l Curse of knowledge explosion n no one scientist can track all necessary developments Need: stability, optimality of representation & optimality of work Need adaptivity Need good colleagues

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OSTP Briefing, 4 May 2004 “Moore’s Law” for MHD simulations “Semi-implicit”: All waves treated implicitly, but still stability-limited by transport “Partially implicit”: Fastest waves filtered, but still stability-limited by slower waves Figure from “SCaLeS report,” Volume 2

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OSTP Briefing, 4 May 2004 “Moore’s Law” for combustion simulations Figure from “SCaLeS report,” Volume 2 Combustion: “Effective speed” increases came from both faster hardware and improved algorithms.

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OSTP Briefing, 4 May 2004 The power of optimal algorithms l Advances in algorithmic efficiency rival advances in hardware architecture l Consider Poisson’s equation on a cube of size N=n 3 l If n=64, this implies an overall reduction in flops of ~16 million YearMethodReferenceStorageFlops 1947GE (banded)Von Neumann & Goldstinen5n5 n7n7 1950Optimal SORYoungn3n3 n 4 log n 1971CGReidn3n3 n 3.5 log n 1984Full MGBrandtn3n3 n3n3 2u=f2u=f 64 *Six-months is reduced to 1 s *

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OSTP Briefing, 4 May 2004 year relative speedup Algorithms and Moore’s Law l This advance took place over a span of about 36 years, or 24 doubling times for Moore’s Law l 2 24 16 million the same as the factor from algorithms alone!

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OSTP Briefing, 4 May 2004 Whence new algorithms? l Algorithms arise to fill the gap between architectures that are available and applications that must be executed l Many algorithmic advances are oriented towards particular physical problems that defy the assumptions of today’s optimal methods – e.g., anisotropy, inhomogeneity, geometrical irregularity, mathematical singularity – underlining the importance of applied research l Many algorithms are mined from the literature, rather than invented – underlining the importance of basic research memory mapping function 1990stopological curiosity 1890Space-filling curves parallel solver1980sexistence proof 1869Schwarz Alternating procedure iterative solver1970sdirect solver1952Conjugate gradients Why?RebornWhy?BornAlgorithm

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OSTP Briefing, 4 May 2004 Designing a simulation code (from 2001 SciDAC report) V&V loop Performance loop

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OSTP Briefing, 4 May 2004 Hardware Infrastructure ARCHITECTURESARCHITECTURES A “perfect storm” for simulation scientific models numerical algorithms computer architecture scientific software engineering “Computational science is undergoing a phase transition.” (dates are somewhat symbolic) 1686 1947 1976 1992

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OSTP Briefing, 4 May 2004 How large-scale simulation is structured l Applications-driven n flow is from applications to enabling technologies n applications expose challenges, enabling technologies respond l Enabling technologies- intensive n in many cases, the application agenda is well-defined n architecture, algorithms, and software represent bottlenecks l Most worthwhile development may be at the interface CS Math Applications

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OSTP Briefing, 4 May 2004 Positive features for simulation initiative l Bold expectations for simulation n for new scientific discovery, not just for “fitting” experiments l Recognition that leading-edge simulation is interdisciplinary n physicists and chemists not supported to write their own software infrastructure; deliverables intertwined with those of math & CS experts l Fostering of lab-university collaborations n complementary strengths l Commitment to distributed hierarchical memory computers n new code must target this architecture type n commitment to maintenance of software infrastructure (rare to find this)

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l Chapter 1. Introduction l Chapter 2. Scientific Discovery through Advanced Computing: a Successful Pilot Program l Chapter 3. Anatomy of a Large- scale Simulation l Chapter 4. Opportunities at the Scientific Horizon l Chapter 5. Enabling Mathematics and Computer Science Tools l Chapter 6. Recommendations and Discussion Volume 2 (due out 2004): l 11 chapters on applications l 8 chapters on mathematical methods l 8 chapters on computer science and infrastructure First fruits

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OSTP Briefing, 4 May 2004 SCaLeS made eight recommendations: Major new investments in computational science Multidisciplinary teams New computational facilities Research in software infrastructure Research in algorithms Recruitment of computational scientists Network infrastructure Examination of innovative, high-risk computer architecture

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OSTP Briefing, 4 May 2004 On “Experimental Mathematics” “There will be opened a gateway and a road to a large and excellent science into which minds more piercing than mine shall penetrate to recesses still deeper.” Galileo (1564-1642) on “experimental mathematics”

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