Transcript Technologies and Tools for High
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 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
OSTP Briefing, 4 May 2004
Three pillars of scientific understanding
Theory Experiment 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” OSTP Briefing, 4 May 2004
Example: turbulent combustion
Simulation models and methods:
Arrhenius kinetics with 84 reactions & 21 species Acoustically filtered hydrodynamics: 10 2
speedup Cartesian adaptive mesh refinement: 10 4
speedup Message-passing SIMD parallelism on 2048 procs
This simulation sits at the pinnacle of numerous prior achievements in experiment, theory,
applied mathematics,
and computer science 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
OSTP Briefing, 4 May 2004
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)
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
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%. 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
. OSTP Briefing, 4 May 2004
The imperative of terascale simulation
Experiments controversial Experiments dangerous
Environment
global climate contaminant transport
Biology
drug design genomics Experiments prohibited or impossible
Applied Physics
radiation transport supernovae
Scientific Simulation
Experiments difficult to instrument
Engineering
crash testing aerodynamics Experiments expensive
Lasers & Energy
combustion ICF
ITER: $5B
In these, and many other areas, simulation is an important complement to experiment.
OSTP Briefing, 4 May 2004
Gedanken experiment:
How to use a jar of peanut butter as 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…
OSTP Briefing, 4 May 2004
Gordon Bell Prize: “price performance”
Year Application
1989 1990 1992 1993 1994 1995 Reservoir modeling Electronic structure Polymer dynamics Image analysis Quant molecular dyn Comp fluid dynamics 1996 1997 Electronic structure Gravitation 1998 Quant chromodyn 1999 2000 2001 Gravitation Comp fluid dynamics Structural analysis
System
CM-2 IPSC cluster custom cluster cluster SGI cluster custom custom cluster cluster
$ per Mflops
1,250 1,000 154 333 278 159 56 12.5 6.9 1.9 0.24
Four orders of magnitude in 12 years
OSTP Briefing, 4 May 2004
Gordon Bell Prize: “peak performance”
Year Type Application No. Procs
1988 PDE Structures
System
8 Cray Y-MP 1989 PDE Seismic 2,048 CM-2 1990 1992 1993 1994 1995 1996 1997 PDE NB MC IE MC PDE NB 1998 MD 1999 PDE 2000 2001 2002 NB NB PDE Seismic Gravitation Boltzmann Structures QCD CFD Gravitation Magnetism CFD Gravitation Gravitation Climate 2,048 CM-2 512 Delta 1,024 CM-5 1,904 Paragon 128 NWT 160 NWT 4,096 ASCI Red 1,536 T3E-1200 5,832 ASCI BluePac 96 GRAPE-6 1,024 GRAPE-6 5,120 Earth Sim
Gflop/s
1.0 5.6 14 5.4 60
Four orders
170 1,020 627 1,349 11,550 26,500 OSTP Briefing, 4 May 2004
Gordon Bell Prize outpaces Moore’s Law
CONCUR RENCY!!!
Four orders of magnitude in 13 years
OSTP Briefing, 4 May 2004
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
Need: stability, optimality of representation & optimality of work Need adaptivity Need good colleagues OSTP Briefing, 4 May 2004
“Moore’s Law” for MHD simulations
Figure from “SCaLeS report,” Volume 2 “Semi-implicit”: All waves treated implicitly, but still stability-limited by transport “Partially implicit”: Fastest waves filtered, but still stability-limited by slower waves OSTP Briefing, 4 May 2004
“Moore’s Law” for combustion simulations
Combustion: “Effective speed” increases came from both faster hardware and improved algorithms.
10 6 5 4 9 8 7 3 2 1 0 1980
High Order Cray 2
1990
Low Mach ARK integrator complex chem AMR NERSC SP3
2000
Autocode NERSC RS/6000 Higher order AMR
2010
Calendar Year
Figure from “SCaLeS report,” Volume 2 OSTP Briefing, 4 May 2004
The power of optimal algorithms Advances in algorithmic efficiency rival advances in hardware architecture Consider Poisson’s equation on a cube of size N=n
3 Year Method Reference Storage Flops
1947 GE (banded) 1950 Optimal SOR 1971 CG 1984 Full MG Von Neumann & Goldstine Young Reid Brandt
n
5
n
3
n
3
n
3
n
7
n
4 log n
n
3.5
log n
n
3 64 64
2 u=f 64
If n=64, this implies an overall reduction in flops of ~16 million
*
*Six-months is reduced to 1 s
OSTP Briefing, 4 May 2004
Algorithms and Moore’s Law
This advance took place over a span of about 36 years, or 24 doubling times for Moore’s Law 2 24
16 million
the same as the factor from algorithms alone!
relative speedup year OSTP Briefing, 4 May 2004
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
Algorithm Born
Conjugate gradients Schwarz Alternating procedure Space-filling curves 1952 1869 1890
Why?
Reborn Why?
direct solver existence proof topological curiosity 1970s 1980s 1990s iterative solver parallel solver memory mapping function
OSTP Briefing, 4 May 2004
V&V loop
Designing a simulation code
(from 2001 SciDAC report)
Performance loop OSTP Briefing, 4 May 2004
A “perfect storm” for simulation
Hardware Infrastructure
1686
A R C H I T E C T U R E
(dates are somewhat symbolic) scientific models 1947 numerical algorithms computer architecture 1976 1992 “Computational science is undergoing a phase transition.” OSTP Briefing, 4 May 2004
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 Math CS OSTP Briefing, 4 May 2004
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)
OSTP Briefing, 4 May 2004
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:
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 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”
OSTP Briefing, 4 May 2004