Computational and Applied Mathematics in Scientific Discovery

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

2. Presentation plan • Emergence of simulation a third modality for scientific and technological research • Applications drivers and trends in simulation infrastructure outstanding opportunities • Hurdles to simulation role of applied and computational mathematics • Success factors and recommendations current pathfinding U.S. programs

3. 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” Three pillars of scientific understanding • Theory • Experiment • Simulation “theoretical experiments”

4. This simulation sits at the pinnacle of numerous prior achievements in experiment, theory, applied mathematics, and computer science Example: turbulent combustion • Simulation models and methods: • Arrhenius kinetics with 84 reactions & 21 species • Acoustically filtered hydrodynamics: 102 speedup • Cartesian adaptive mesh refinement: 104 speedup • Message-passing SIMD parallelism on 2048 procs • Reaction zone location a delicate balance of fluxes of:species, momentum, internal energy • Directly relevant to: engines, turbines, furnaces, incinerators (energy efficiency, pollution mitigation) • Component model of other computational apps: firespread, stellar dynamics, chemical processing

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

6. What would we do with 100-1000x more? Example: probe the structure of particles Constraints on the Standard Model parameters r and h. 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%.

7. 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.

8. Engineeringcrash testingaerodynamics Lasers & Energycombustion ICF Biology drug design genomics ITER: $5B The imperative of terascale simulation Experiments prohibited or impossible Applied Physics radiation transport supernovae Experiments dangerous Experiments difficult to instrument Environment global climate contaminant transport Experiments controversial Experiments expensive Scientific Simulation In these, and many other areas, simulation is an important complement to experiment.

9. Gedanken experiment:How to use a jar of peanut butteras its price slides? • In 2004, at $3.19: make sandwiches • By 2007, at $0.80: make recipe substitutions • By 2010, at $0.20: use as feedstock for biopolymers, plastics, etc. • By 2113, at $0.05: heat homes • 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…

10. Four orders of magnitude in 12 years Gordon Bell Prize: “price performance”

11. Four orders of magnitude in 13 years Gordon Bell Prize: “peak performance”

12. Gordon Moore “Demi” Moore Four orders of magnitude in 13 years Gordon Bell Prize outpaces Moore’s Law Gordon Bell CONCUR-RENCY!!!

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

14. “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

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

16. 64 64 2u=f 64 * *Six-months is reduced to 1 s The power of optimal algorithms • Advances in algorithmic efficiency rival advances in hardware architecture • Consider Poisson’s equation on a cube of size N=n3 • If n=64, this implies an overall reduction in flops of ~16 million

17. relative speedup year Algorithms and Moore’s Law • This advance took place over a span of about 36 years, or 24 doubling times for Moore’s Law • 22416 million  the same as the factor from algorithms alone!

18. Algorithm Born Why? Reborn Why? Conjugate gradients 1952 direct solver 1970s iterative solver Schwarz Alternating procedure 1869 existence proof 1980s parallel solver Space-filling curves 1890 topological curiosity 1990s memory mapping function Whence new algorithms? • Algorithms arise to fill the gap between architectures that are available and applications that must be executed • 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 • Many algorithms are mined from the literature, rather than invented–underlining the importance of basic research

19. Performance loop V&V loop Designing a simulation code (from 2001 SciDAC report)

20. 1686 1947 1976 1992 A “perfect storm” for simulation (dates are somewhat symbolic) Hardware Infrastructure scientific models A R C H I T E C T U R E S numerical algorithms computer architecture scientific software engineering “Computational science is undergoing a phase transition.”

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

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

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

24. 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

25. 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”