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The Role of Analysis in Modern High Performance Computing

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The Role of Analysis in Modern High Performance Computing J. L. Schwarzmeier, Cray Inc, July 6, 2007 – PowerPoint PPT presentation

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Title: The Role of Analysis in Modern High Performance Computing


1
The Role of Analysis in Modern High Performance
Computing
  • J. L. Schwarzmeier, Cray Inc, July 6, 2007

2
Outline
  • Where I am coming from
  • Describe process of going from problem to be
    solved to solution on computer and role of
    analysis throughout
  • Conclusions

3
Where I am coming from
  • As much as I have studied and enjoyed math, I am
    not a researcher in analysis ? a personal view of
    analysis and applied math
  • However, did see lots of use of analysis for
    theoretical and numerical purposes in my own work
    and work of others
  • 20 years since I left LANL for Cray
  • Cray personnel generally do not work with
    customers at the level of their fundamental
    equations. But continue to see sophisticated uses
    of analysis by customers
  • Sometimes we do recommend alternative numerical
    schemes to improve performance
  • Will describe journey of identifying problem to
    be solved ? to obtaining solution on a
    computer, role of analysis, sprinkled with
    examples I have encountered.

4
Role of analysis in the sciences
  • Is the use of analysis is science/engineering
    relegated to a bygone era before computers and
    before all science had been discovered?
    NO!!
  • Myth all science has already been discovered
  • Reality no, laws of physics known, but focus
    has changed to solving more realistic problems
  • wave function of the universe,
    , where N particles in universe,
    satisfies

  • approximately true but totally useless
    equation. Since 1920s about the only quantum
    mechanical problems that have been solved
    exactly are hydrogen atom, single particle in
    harmonic oscillator potential, or other similarly
    idealized situations. Even on most powerful
    supercomputers, quantum chemistry codes struggle
    mightily to approximately solve N-atom
    Schroedinger equation for N O(10) atoms.
  • Even on todays supercomputers, important
    problems cannot be solved by brute force clever
    algorithms and implementations are required and
    knowing what approximations to make

5
Circle of dependency
  • Scientific progress today often requires
    multi-disciplinary collaboration
  • Team of scientists
  • Mathematics, numerical methods analysis
  • Computer specialists
  • Do we have enough students entering
    analysis/applied math? Will their training allow
    them to bridge the gap from scientist/engineer to
    computer programmer?

2. Analysis to understand properties, guide
solution
1. Scientists define problem
3. Numerics and computer implementation
6
Step 1 of the Journey role of scientists/engineer
s
  • A big opportunity for analysis and computing
    today is solving new, realistic, specific, often
    multi-disciplinary problems. Specific model
    equations are derived from general equations by
    physical and mathematical intuition
  • Focus on narrowed range of problems of interest
  • Of fundamental equations which terms are
    important how are multi-disciplinary PDEs
    coupled together? How to deal with greatly
    disparate space/time scales? ? modified set of
    equations
  • In climate studies current model couples
    atmosphere, ocean, land, sea ice. Seek to improve
    model by including 100 chemical species in
    atmosphere full carbon cycle modeling with
    interaction of vegetation, plant decay, release
    of CO2 full fresh water hydrology with river
    basin modeling and drainage into oceans higher
    spatial resolution to model cloud physics and
    effects of narrow land formations Florida on
    oceanic currents better modeling of man-made
    interactions such as fires, deforestation,
    development, irrigation, pollution, etc. This is
    a much expanded set of equations and unknowns
    needs mathematical foundation

7
Another multi-disciplinary example
  • Researchers at Rice University Use Cray
    Supercomputer to Unlock Biomedical Mysteries and
    Aid Future Diagnostics
  • Members of the Team for Advanced Flow
    Simulation and Modeling at Rice are collaborating
    with colleagues from other institutions to create
    computational fluid dynamics models that mimic
    how blood courses through the brains arteries
    and interacts with an aneurysm on a vessel wall.
    An aneurysm is a balloon-like protrusion of an
    artery that could be fatal if it bursts. The team
    uses the Cray system to simulate numerically how
    the blood, artery and aneurysm interact with each
    other. The data is then loaded into a program
    from Computational Engineering International
    called EnSight, which provides visualization and
    analytical capabilities.
  • Accurate blood-flow simulations are
    extremely complex because an artery wall isnt
    rigid and blood pressure fluctuates with the
    beating of the heart, says Tayfun Tezduyar,
    professor of mechanical engineering at Rice. We
    want to understand how much a cerebral artery
    wall deforms, how blood flow is affected and what
    stresses are created that could affect the
    aneurysm. A precise understanding of this dynamic
    will be of great benefit to brain surgeons when
    they have to make a decision about whether or not
    to operate A traditional Singular Value
    Decomposition algorithm running on a conventional
    computer does not preserve the symmetry of the
    molecule, making it difficult to isolate and
    study a proteins characteristics. The team
    developed more accurate algorithms that they
    parallelized to run quickly on the
    supercomputer

8
Example of HPC for economic competitiveness
  • CRAY SUPERCOMPUTERS PLAY KEY ROLE IN DESIGNING
    BOEING 787 DREAMLINER 800,000 Simulation Hours
    Helped Create Design For Highly Successful
    Commercial Aircraft SEATTLE, WA, July 5, 2007 --
    Global supercomputer leader Cray Inc. (Nasdaq GM
    CRAY) reported today that 800,000 processor hours
    of computing time on Cray supercomputers went
    into the design of the highly successful Boeing
    787 Dreamliner. Supercomputer-based modeling
    and simulation is far more efficient,
    cost-effective and practical than physical
    proto-typing for testing large numbers of design
    variables. While physical prototyping is still
    important for final design validation, Boeing
    engineers were able to build the 787 Dreamliner
    after physically testing only 11 wing designs,
    versus 77 wing designs for the earlier Boeing 767
    aircraft. The Boeing 787 Dreamliner is 20
    lighter and produces 20 fewer emissions than
    similarly sized airplanes, while providing 10
    better per-seat costs per mile, according to
    Boeing.

9
Step 2 mathematical analysis
  • Analysis gives range of analytical techniques of
    posing or formally solving boundary value
    problems
  • What are mathematical properties of equations
    are equations parabolic, elliptic, hyperbolic?
    what are appropriate BCs? are initial conditions
    continuous, differentiable? should we allow for
    weak solutions? what are continuity properties
    of operators?
  • Can perturbation theory be used to find
    first-order solutions? Are there
    transformations of independent, dependent
    variables that simplify problem? are there
    analytical, limiting solutions? are there
    symmetry properties to take advantage of?
  • What are possible methods of solution
    eigenfunction expansions, Greens function,
    method of characteristics, transform methods,
    divergence theorem, Stokes theorem? are there
    special functions that form an expansion set with
    desired continuity, orthogonality, completeness
    properties?
  • Is the solution derivable from a variational
    principle? are there constraints in the problem
    Lagrange multipliers?
  • Many of these techniques apply not only as
    traditional mathematical solutions but also as
    numerical solutions on computers, only now linear
    spaces have finite dimensionality

10
Resolving coordinate-induced singularities
  • For problems involving cylindrical or
    spherical domains, introduce artificial
    singularities at origin by cylindrical or
    spherical coordinates ? must eliminate singular
    solutions, made worse by increasing resolution.
    For expansions
    , determine minimum continuity conditions on
    function that allows only analytic
    solutions about the origin. Do this separately
    for scalar quantities versus vector
    components . This
    can be done independent of any PDE.
  • For example, for simple scalar case, for
    ,
  • Derivatives of the latter functional form
    eventually will diverge at the origin, unless we
    have . For , we
    have
  • . Thus
  • Thus the analytic solution is of the form
  • Furthermore, in finite element
    implementations shape functions that intersect
    the origin can be made explicit functions of .
    Shape functions whose finite elements do not
    reach the origin can be functions of .

11
Use of Fourier Expansion
  • For theoretical purposes, decomposition of
    solution in terms of Fourier wavelengths and
    frequencies will always be important for
    judging physical scales of interest
  • Specific applicability for a) periodic boundary
    conditions, b) smooth, long wavelength solutions,
    c) when basis functions needs to be small as
    possible, and d) when analysis or computation
    aided by orthogonal basis
  • Currently used in the following HPC applications
  • Turbulence modeling
  • Long range molecular forces in molecular dynamics
    and materials science
  • Application that have geometries (cylindrical,
    spherical) with periodic BCs
  • Some weather/climate codes for latitude-longitude
  • But, Fourier expansion not for every problem
  • Solutions with steep gradients exhibit Gibbs
    phenomena, where Fourier solution has strong,
    spurious oscillations around discontinuities
  • Complicated boundaries difficult with global
    expansion functions
  • some problems have coordinate singularities in
    Fourier expansions

12
DNS code and Fourier Transforms
  • E.g. 3-D turbulence. Direct Numerical Simulation
    (DNS) code seeks to understand distribution of
    energy in density/velocity fluctuations versus
    wavelengths for the incompressible Navier-Stokes
    equations, for a specified source of turbulence.
    Understanding turbulence needed for efficient
    design of airplanes, vehicles, combustion
    systems, etc.
  • Code written in Fortran 90 with MPI
  • Time evolution Runge Kutta 2nd order
  • Spatial derivative calculation pseudospectral
    method
  • Typically, FFTs are done in all 3 dimensions.
  • Parallel 3D FFT so-called transpose strategy, as
    opposed to direct strategy. That is, make sure
    all data in direction of 1D transform resides in
    one processors memory. Parallelize over
    orthogonal dimension(s).
  • Data decomposition N3 grid points over P
    processors
  • Originally 1D (slab) decomposition divide one
    side of the cube over P, assign N/P planes to
    each processor. Limitation P lt N
  • Currently 2D (pencil) decomposition divide side
    of the cube (N2) over P, assign N2/P pencils
    (columns) to each processor.
  • DNS code needs PFLOPS sustained performance to
    achieve turbulence scaling on grid in
    40 hours

13
Issues of Fourier expansions for climate models
  • Pick geometry, Boundary Conditions (BC),
    coordinate system
  • In climate/weather studies, use of
    latitude-longitude (and height) coordinates leads
    to efficient Fourier expansions in longitude,
    Legendre transforms in latitude. But there are
    issues with this approach 1) artificial
    singularities at poles ? must eliminate singular
    solutions at poles ? do complicated Fourier
    filtering ? but poor load balancing on
    computers 2) 1D data decomposition in either
    longitude or latitude dimensions leads to limited
    scalability on computers ? cannot use more
    processors to reduce wall time. Furthermore, when
    switching between Fourier phase or Legendre phase
    data must be re-distributed across processors via
    global transposes, which require very high
    bandwidth networks. For these reasons the
    climate/weather community is moving away from
    Fourier/spectral methods to finite element
    approaches

14
More Detailed Models with High Resolution
Sea surface temperature (degreesC) on the last
day of year 4 from the 1/10 degree, 42 level POP
spinup simulation on Jaguar.  The result of this
spinup run will be used as the ocean initial
condition for a fully coupled climate run.
(Mat Maltrud, LANL)
(courtesy M. Gunzberger)
15
New methods and scaling
  • Scaling to 100,000 processors using cubed sphere
  • Finite Volume with GFDL (Lin, Kerr, Putman)
  • Spectral Elements with NCAR (Taylor, Nair)
  • Cloud resolving icoshedral dynamical core being
    developed by Randall at CSU under SciDAC2

16
Example of perturbation theory, Hamiltonian
dynamics, etc. in Magnetic Fusion
  • Plasma physics is regime of high temperature,
    ionized gases. At T gt 100M nuclear fusion can
    occur. Energy of fusion reactions released as
    high energy neutrons, photons, or alpha
    particles, depending on type of reaction. Energy
    can be captured in a reactor to make electricity
    -- ultimate energy source
  • After equilibrium and gross stability are
    ensured, need to reduce microturbulence-
  • induced thermal transport to walls so
    applied heating can raise temperature
    Microturbulence requires particle description
    rather than continuous fluid model
  • Collisionless plasma described by Vlasov equation
    for -- 6D phase space
  • Electromagnetic field given by Maxwells
    equations, where plasma particles are sources
    of charge density and current density. Highly
    coupled, nonlinear system
  • Solving Vlasov equation equivalent to solving
    equations of motion for millions of particles
    subject to applied and self-consistent forces
    Particle-In-Cell (PIC)
  • Ion temperature gradient instability drives
    microturbulence, but Tokamak fusion
  • devices have a small parameter of
    , is major
    radius of torus. Perturbation analysis for small
    is used to simplify Hamiltonian

17
Example of choosing proper coordinates
Gyrokinetic Toroidal Code (GTC)
  • Tokamaks are main candidates for fusion research
    today
  • Plasma contained in toroidal (donut) shaped
    device. Long way around is toroidal direction,
    short way around is poloidal, and minor radial
    direction.
  • obvious independent spatial coordinates are
  • Better to transform from to poloidal flux,

18
Efficiency of Global Field-aligned Mesh
  • Transform from toroidal coordinates to canonical
    magnetic coordinates

  • Use perturbation analysis to find canonical
    coordinates with .
    Following particle motion with magnetic
    coordinates in Hamiltonian straightens out
    and changes charge deposition step of PIC from 3D
    to 2D process. This allows for coarse
    grid and saves factor 100 in CPU time. This is
    huge and easily justifies a team spending a year
    doing analysis to get this right

R
19
Domain Decomposition
  • Domain decomposition
  • each MPI process holds a toroidal section
  • each particle is assigned to a processor
    according to its position
  • Initial memory allocation is done locally on each
    processor to maximize efficiency
  • Communication between domains is done with MPI
    calls (runs on most parallel computers)

20
Step 2 Example of taking advantage of symmetry
  • Find eigenfunctions, eigenvalues of operator
    in 1D geometry with periodic boundary
    conditions at , but which also has
    periodicity length
    . The eigenvalue problem is
  • ,
    . The eigenfunctions are of the form
  • ,
    where ,
    .
  • That is, rather than find eigenfunctions
    over interval , find reduced
    eigenfunctions over interval , for
    each value of . This is an example of the
    Floquet-Bloch theorem.

21
Step3 Analysis in computer solution
  • Many new, critical problems of interest will
    involve computer solution. Embrace this reality,
    as analysis can greatly aid implementation of
    computer solutions
  • Do contributions from new sciences enter as
    source terms to old models or are new PDEs
    introduced? What is coupling at interfaces what
    continuity conditions, conservation laws, and
    boundary conditions are appropriate?
  • What are asymptotic solutions, symmetric
    solutions, or other first-order solutions? They
    can be crucial in understanding properties of
    general solution space. Special solutions often
    can be used as sanity checks, to improve choice
    of basis functions, reduce computation, or
    improve convergence.
  • Applied math needed to steer among myriad of
    possible numerical algorithms, based on
    mathematical properties of final system of
    equations methods for different types of PDEs
    finite differencing versus finite elements
    global eigenfunction expansions versus basis
    functions with compact support direct versus
    iterative solvers explicit versus implicit
    iterative methods convergence and stability of
    iterative methods what kind of iterative solver
    multi-grid, conjugate gradient,
    pre-conditioners?
  • How is problem distributed among processors? are
    scalable algorithms chosen for inter-processor
    communication? is code written to allow full
    expression of parallelism to CPUs, including
    vectorization?

22
How do university math departments view analysis?
Is there an applied math major?
  • In my opinion, applied math majors should take
  • Required 2 semesters calculus-based introductory
    physics
  • 1 semester undergraduate mechanics
  • 1 semester undergraduate atomic/quantum physics
  • 1 semester undergraduate numerical
    methods/analysis
  • 1 semester programming for scientific
    applications, including Fortran/C/C/Matlab
  • 1 semester undergraduate electromagnetism
  • 22 25 credits outside math courses
  • Each of the physics courses has graduate
    counterparts
  • Or, rather than physics could focus on
    engineering fields, chemistry, biology, medicine,
    etc
  • Math I found most useful calculus, vector
    analysis, complex variables, linear vector
    spaces, ODEs, PDEs, probability, calculus of
    variations, linear algebra

23
Conclusions
  • Analysis and applied math are crucial components
    of training professionals needed help mankind to
    understand the environment, achieve fusion
    energy, enable drugs by design, develop new
    materials, etc. These also keep the US on top
    economically and in terms of national security
  • Todays problems are more targeted to solving
    specific problems that are directly tied to
    industrial, national, or international need
  • Many of todays problems are multi-disciplinary
    and all require sound mathematical foundation and
    understanding
  • Todays big problems are solved on
    supercomputers, and use of these machines
    requires solid understanding of algorithms and
    computer system architecture
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