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					 Chapters 5-8 of
SOURCEBOOK

  Spring Semester 2005
      Geoffrey Fox
      Community
   Grids Laboratory
   Indiana University
     505 N Morton
        Suite 224
    Bloomington IN
    gcf@indiana.edu
   Computational Fluid Dynamics
       (CFD) in Chapter 5 I
• This chapter provides a thorough formulation of CFD with a
  general discussion of the importance of non-linear terms and
  most importantly viscosity.
• Difficult features like shockwaves and turbulence can be traced
  to the small coefficient of the highest order derivatives.
• Incompressible flow is approached using the spectral element
  method, which combines the features of finite elements (copes
  with complex geometries) and highly accurate approximations
  within each element.
• These problems need fast solvers for elliptic equations and
  there is a detailed discussion of data and matrix structure and
  the use of iterative conjugate gradient methods.
• This is compared with direct solvers using the static
  condensation method for calculating the solution (stiffness)
  matrix.
Computational Fluid Dynamics
   (CFD) in Chapter 5 II
• The generally important problem of adaptive
  meshes is described using the successive
  refinement quad/oct-tree (in two/three
  dimensions) method.
• Compressible flow methods are reviewed and
  the key problem of coping with the rapid change
  in field variables at shockwaves is identified.
• One uses a lower order approximation near a
  shock but preserves the most powerful high
  order spectral methods in the areas where the
  flow is smooth.
• Parallel computing (using space filling curves for
  decomposition) and adaptive meshes are
  covered.
Hair Combing problem
       Environment and Energy in
             Chapter 6   I
• This article describes three distinct problem areas – each
  illustrating important general approaches.
• Subsurface flow in porous media is needed in both oil
  reservoir simulations and environmental pollution studies.
   – The nearly hyperbolic or parabolic flow equations are characterized by
     multiple constituents and by very heterogeneous media with possible
     abrupt discontinuities in the physical domain.
   – This motivates the use of domain decomposition methods where the
     full region is divided into blocks which can use different solution
     methods if necessary.
   – The blocks must be iteratively reconciled at their boundaries (mortar
     spaces).
   – The IPARS code described has been successfully integrated into two
     powerful problem solving environment: NetSolve described in chapter
     14 and DISCOVER (aimed especially at interactive steering) from
     Rutgers university.
         Environment and Energy in
               Chapter 6  II
• The discussion of the shallow water problem uses a method
  involving implicit (in the vertical direction) and explicit (in the
  horizontal plane) time-marching methods.
• It is instructive to see that good parallel performance is
  obtained by only decomposing in the horizontal directions and
  keeping the hard to parallelize implicit algorithm sequentially
  implemented.
• The irregular mesh was tackled using space filling curves as
  also described in chapter 5.
• Finally important code coupling (meta-problem in chapter 4
  notation) issues are discussed for oil spill simulations where
  water and chemical transport need to be modeled in a linked
  fashion
• . ADR (Active Data Repository) technology from Maryland is
  used to link the computations between the water and
  chemical simulations. Sophisticated filtering is needed to
  match the output and input needs of the two subsystems.
Molecular Quantum Chemistry in
         Chapter 7 I
• This article surveys in detail two capabilities of the
  NWChem package from Pacific Northwest Laboratory. It
  surveys other aspects of computational chemistry.
• This field makes extensive use of particle dynamics
  algorithms and some use of partial differential equation
  solvers.
• However characteristic of computational chemistry is the
  importance of matrix-based methods and these are the
  focus of this chapter. The matrix is the Hamiltonian
  (energy) and is typically symmetric positive definite.
• In a quantum approach, the eigensystems of this matrix
  are the equilibrium states of the molecule being studied.
  This type of problem is characteristic of quantum
  theoretical methods in physics and chemistry; particle
  dynamics is used in classical non-quantum regimes.
   Molecular Quantum Chemistry in
            Chapter 7 II
• NWChem uses a software approach – the Global Array (GA) toolkit,
  whose programming model lies in between those of HPF and
  message passing and has been highly successful.
• GA exposes locality to the programmer but has a shared memory
  programming model for accessing data stored in remote processors.
• Interestingly in many cases calculating the matrix elements
  dominates (over solving for eigenfunctions) and this is a pleasing
  parallel task.
• This task requires very careful blocking and staging of the
  components used to calculate the integrals forming the matrix
  elements.
• In some approaches, parallel matrix multiplication is important in
  generating the matrices.
• The matrices typically are taken as full and very powerful parallel
  eigensolvers were developed for this problem.
• This area of science clearly shows the benefit of linear algebra
  libraries (see chapter 20) and general performance enhancements
  like blocking.
               General Relativity
• This field evolves in time complex partial differential equations
  which have some similarities with the simpler Maxwell
  equations used in electromagnetics (Sec. 8.6).
• Key difficulties are the boundary conditions which are
  outgoing waves at infinity and the difficult and unique multiple
  black hole surface conditions internally.
• Finite difference and adaptive meshes are the usual
  approach.
  Lattice Quantum Chromodynamics
   (QCD) and Monte Carlo Methods I
• Monte Carlo Methods are central to the numerical approaches
  to many fields (especially in physics and chemistry) and by their
  nature can take substantial computing resources.
• Note that the error in the computation only decreases like the
  square root of computer time used compared to the power
  convergence of most differential equation and particle dynamics
  based methods.
• One finds Monte Carlo methods when problems are posed as
  integral equations and the often-high dimension integrals are
  solved by Monte Carlo methods using a randomly distributed
  set of integration points.
• Quantum Chromodynamics (QCD) simulations described in this
  subsection are a classic example of large-scale Monte Carlo
  simulations which perform excellently on most parallel
  machines due to modest communication costs and regular
  structure leading to good node performance.
    From Numerical Integration Lecture:
                 Errors
• For an integral with N points
• Monte Carlo has error 1/N0.5
• Iterated Trapezoidal has error 1/N2
• Iterated Simpson has error 1/N4
• Iterated Gaussian is error 1/N2m for our a basic
  integration scheme with m points
• But in d dimensions, for all but Monte Carlo
  must set up a Grid of N1/d points on a side; that
  hardly works above N=3
    – Monte Carlo error still 1/N0.5
    – Simpson error becomes 1/N4/d etc.
       Monte Carlo Convergence
• In homework for N=10,000,000 one finds errors
  in π of around 10-6 using Simpson’s rule
• This is a combination of rounding error (when
  computer does floating point arithmetic, it is
  inevitably approximate) and error from formula
  which is proportional to N-4
• For Monte Carlo, error will be about 1.0/N0.5
• So an error of 10-6 requires N=1012 or
• N=1000,000,000,000 (100,000 more than
  Simpson’s rule)
• One doesn’t use Monte Carlo to get such
  precise results!
  Lattice Quantum Chromodynamics
  (QCD) and Monte Carlo Methods II
• This application is straightforward to parallelize and very
  suitable for HPF as the basic data structure is an array.
  However the work described here uses a portable MPI code.
• Section 8.9 describes some new Monte Carlo algorithms but
  QCD advances typically come from new physics insights
  allowing more efficient numerical formulations.
• This field has generated many special purpose facilities as the
  lack of significant I/O and CPU intense nature of QCD allows
  optimized node designs. The work at Columbia and Tsukuba
  universities is well known.
• There are other important irregular geometry Monte Carlo
  problems and they see many of the same issues such as
  adaptive load balancing seen in irregular finite element
  problems.
               Ocean Modeling
• This describes the issues encountered in optimizing a
  whole earth ocean simulation including realistic
  geography and proper ocean atmosphere boundaries.
• Conjugate gradient solvers and MPI message passing
  with Fortran 90 are used for the parallel implicit solver
  for the vertically averaged flow.
Tsunami Simulations
                 • These are still
                   very
                   preliminary; an
                   area where
                   much more
                   work could be
                   done
   Multidisciplinary Simulations
• Oceans naturally couple to atmosphere
  and atmosphere couples to environment
  including
  – Deforestration
  – Emissions from using gasoline (fossil fuels)
  – Conversely atmosphere makes lakes acid etc.
• These are not trivial as very different
  timescales
         Earthquake Simulations
• Earthquake simulations are a relatively young field and it is not
  known how far they can go in forecasting large earthquakes.
• The field has an increasing amount of real-time sensor data,
  which needs data assimilation techniques and automatic
  differentiation tools such as those of chapter 24.
• Study of earthquake faults can use finite element techniques or
  with some approximation, Green’s function approaches, which
  can use fast multipole methods.
• Analysis of observational and simulation data need data mining
  methods as described in subsection 8.7 and 8.8.
• The principal component and hidden Markov classification
  algorithms currently used in the earthquake field illustrate the
  diversity in data mining methods when compared to the
  decision tree methods of section 8.7.
 World-Wide Forecast Hotspot Map for Likely Locations of
              World-Wide Forecast M > 5, 1965-2000
        World-Wide Earthquakes, Hotspot Map
  Great Earthquakes M  7.0 For the Decade 2000-2010
 Green Circles = Large Earthquakes M  7 from Jan 1, 2000 – Dec 1, 2004
    Blue Circles: Large Earthquakes M  7 from Jan 2004 - Present
Green Circles = Large Earthquakesfrom December 1, 1, 2000 – Dec 1, 2004




             Dec. 26 M ~ 9.0
             Northern Sumatra



                          Dec. 23 M ~ 8.1
                          Macquarie Island
          Cosmological Structure
             Formation (CSF)
• CSF is an example of a coupled particle field problem.
• Here the universe is viewed as a set of particles which
  generate a gravitational field obeying Poisson’s equation.
• The field then determines the force needed to evolve each
  particle in time. This structure is also seen in Plasma physics
  where electrons create an electromagnetic field.
• It is hard to generate compatible particle and field
  decompositions. CSF exhibits large ranges in distance and
  temporal scale characteristic of the attractive gravitational
  forces.
• Poisson’s equation is solved by fast Fourier transforms and
  deeply adaptive meshes are generated.
• The article describes both MPI and CMFortran (HPF like)
  implementations.
• Further it made use of object oriented techniques (chapter
  13) with kernels in F77. Some approaches to this problem
  class use fast multipole methods.
        Cosmological Structure
           Formation (CSF)
• There is a lot of structure in universe
 Computational Electromagnetics (CEM)
• This overview summarizes several different approaches to
  electromagnetic simulations and notes the growing importance
  of coupling electromagnetics with other disciplines such as
  aerodynamics and chemical physics.
• Parallel computing has been successfully applied to the three
  major approaches to CEM.
• Asymptotic methods use ray tracing as seen in visualization.
  Frequency domain methods use moment (spectral) expansions
  that were the earliest uses of large parallel full matrix solvers 10
  to 15 years ago; these now have switched to the fast multipole
  approach.
• Finally time-domain methods use finite volume (element)
  methods with an unstructured mesh. As in general relativity,
  special attention is needed to get accurate wave solutions at
  infinity in the time-domain approach.
                 Data mining
• Data mining is a broad field with many different
  applications and algorithms (see also sections 8.4 and
  8.8).
• This article describes important algorithms used for
  example in discovering associations between items
  that were likely to be purchased by the same
  customer; these associations could occur either in
  time or because the purchases tended to be in the
  same shopping basket.
• Other data-mining problems discussed include the
  classification problem tackled by decision trees.
• These tree based approaches are parallelized
  effectively (as they are based on huge transaction
  databases) with load balance being a difficult issue.
  Signal and Image Processing
• This samples some of the issues from this field,
  which currently makes surprisingly little use of
  parallel computing even though good parallel
  algorithms often exist.
• The field has preferred the convenient
  programming model and interactive feedback of
  systems like MATLAB and Khoros.
• These are problem solving environments as
  described in chapter 14 of SOURCEBOOK.
          Monte Carlo Methods and
            Financial Modeling I
• Subsection 8.2 introduces Monte Carlo methods and this
  subsection describes some very important developments in the
  generation of “random” numbers.
• Quasirandom numbers (QRN’s) are more uniformly distributed
  than the standard truly random numbers and for certain
  integrals lead to more rapid convergence.
• In particular these methods have been applied to financial
  modeling where one needs to calculate one or more functions
  (stock prices, their derivatives or other financial instruments) at
  some future time by integrating over the possible future values
  of the underlying variables.
• These future values are given by models based on the past
  behavior of the stock.
          Monte Carlo Methods and
            Financial Modeling II
• This can be captured in some cases by the volatility or
  standard deviation of the stock.
• The simplest model is perhaps the Black-Scholes equation,
  which can be derived from a Gaussian stock distribution,
  combined with an underlying "no-arbitrage" assumption. This
  asserts that the stock market is always in equilibrium
  instantaneously and there is no opportunity to make money
  by exploiting mismatches between buy and sell prices.
• In a physics language, the different players in the stock
  market form a heat bath, which keeps the market in adiabatic
  equilibrium.
• There is a straightforward (to parallelize and implement)
  binomial method for predicting the probability distributions of
  financial instruments. However Monte Carlo methods and
  QRN’s are the most powerful approach.
Quasi Real-time Data analysis of
  Photon Source Experiments
• This subsection describes a successful
  application of computational grids to accelerate
  the data analysis of an accelerator experiment. It
  is an example that can be generalized to other
  cases.
• The accelerator (here a photon source at
  Argonne) data is passed in real-time to a
  supercomputer where the analysis is performed.
  Multiple visualization and control stations are
  also connected to the Grid.
    Forces Modeling and Simulation
• This subsection describes event driven simulations which as
  discussed in chapter 4 are very common in military
  applications.
• A distributed object approach called HLA (see chapter 13) is
  being used for modern problems of this class.
• Some run in “real-time” with synchronization provided by wall
  clock and humans and machines in the loop.
• Other cases are run in “virtual time” in a more traditional
  standalone fashion.
• This article describes integration of these military standards
  with Object
• Web ideas such as CORBA and .NET from Microsoft. One
  application simulated the interaction of vehicles with a million
  mines on a distributed Grid of computers.
   – This work also parallelized the minefield simulator using threads (chapter
     10).
        Event Driven Simulations
• This is a graph based model where independent objects
  issue events that travel as messages to other objects
• Hard to parallelize as no guarantee that event will not arrive
  from past in simulation time
• Often run in “real-time”


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posted:11/2/2011
language:English
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