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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. Presentation plan. Emergence of simulation

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computational and applied mathematics in scientific discovery

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

presentation plan
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

three pillars of scientific understanding

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”

example turbulent combustion

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
theory experiment and simulation check spur and enrich each other
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)

what would we do with 100 1000x more example probe the structure of particles
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%.

what would we do with 100 1000x more example predict future climates
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.

the imperative of terascale simulation

Engineeringcrash testingaerodynamics

Lasers & Energycombustion ICF


drug design




The imperative of terascale simulation

Experiments prohibited or impossible



radiation transport


Experiments dangerous

Experiments difficult to instrument


global climate

contaminant transport

Experiments controversial

Experiments expensive



In these, and many other areas, simulation is an important complement to experiment.

gedanken experiment how to use a jar of peanut butter as its price slides
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…

gordon bell prize outpaces moore s law

Gordon Moore

“Demi” Moore

Four orders of magnitude in 13 years

Gordon Bell Prize outpaces Moore’s Law

Gordon Bell


hurdles to simulation

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
moore s law for mhd simulations
“Moore’s Law” for MHD simulations


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

moore s law for combustion simulations
“Moore’s Law” for combustion simulations

Combustion: “Effective speed” increases came from both faster hardware and improved algorithms.

Figure from “SCaLeS report,” Volume 2

the power of optimal algorithms






*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

relative speedup


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!
whence new algorithms






Conjugate gradients


direct solver


iterative solver

Schwarz Alternating procedure


existence proof


parallel solver

Space-filling curves


topological curiosity


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
designing a simulation code

Performance loop

V&V loop

Designing a simulation code

(from 2001 SciDAC report)






A “perfect storm” for simulation

(dates are somewhat symbolic)

Hardware Infrastructure

scientific models














numerical algorithms

computer architecture

scientific software engineering

“Computational science is undergoing a phase transition.”

how large scale simulation is structured



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


positive features for simulation initiative
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)

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
scales made eight recommendations
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
on experimental mathematics
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”