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11
Secrets of SupercomputingSecrets of SupercomputingThe Conservation LawsThe Conservation Laws
Supercomputing Challenge KickoffSupercomputing Challenge KickoffOctober 21-23, 2007October 21-23, 2007
I. Background to SupercomputingI. Background to Supercomputing
II. Get Wet! With the Shallow Water EquationsII. Get Wet! With the Shallow Water EquationsBob Robey - Los Alamos National LaboratoryBob Robey - Los Alamos National Laboratory
Randy Roberts – Los Alamos National LaboratoryRandy Roberts – Los Alamos National Laboratory
Cleve Moler -- MathworksCleve Moler -- Mathworks
LA-UR-07-6793Approved for public release;distribution is unlimited
22
IntroductionsIntroductions• Bob Robey -- Los Alamos National Lab, X division
– [email protected], 665-9052 or home: [email protected], 662-2018
– 3D Hydrocodes and parallel numerical software– Helped found UNM and Maui High Performance Computing
Centers and Supercomputing Tutorials
• Randy Roberts -- Los Alamos National Lab, D Division– Java, C++, Numerical and Agent Based Modeling– [email protected]
• Cleve Moler– Matlab Founder– Former UNM CS Dept Chair– SIAM President– Author of “Numerical Computing with Matlab” and “Experiments with
Matlab”
33
Conservation LawsConservation Laws• Formulated as a conserved quantity
– mass– momentum– energy
• Good reference is Leveque’s book and his freely available software package CLAWPACK (Fortran/MPI) and a 2D shallow water version Tsunamiclaw
Conserved variable
Change
Leveque, Randall, Numerical Methods for Conservation Laws
Leveque, Randall, Finite Volume Methods for Hyperbolic Problems
CLAWPACK http://www.amath.washington.edu/~claw/
Tsunamiclaw http://www.math.utah.edu/~george/tsunamiclaw.html
44
I. Intro to SupercomputingI. Intro to Supercomputing
• Classical Definition of Supercomputing– Harnessing lots of processors to do lots of small
calculations
• There are many other definitions which usually include any computing beyond the norm– Includes new techniques in modeling, visualization,
and higher level languages.
• Question for thought: With greater CPU resources is it better to save programmer work or to make the computer do bigger problems?
55
II. Calculus QuickstartII. Calculus Quickstart
Decoding the Language of Decoding the Language of WizardsWizards
66
Calculus Quickstart GoalsCalculus Quickstart Goals
• Calculus is a language of mathematical wizards. It is convenient shorthand, but not easy to understand until you learn the secrets to the code.
• Our goal is for you to be able to READ calculus and TALK calculus.
• Goal is not to ANALYTICALLY SOLVE calculus using traditional methods. In supercomputing we generally solve problems by brute force.
77
Calculus TerminologyCalculus Terminology
• Two branches of Calculus – Integral Calculus– Derivative Calculus
• P = f(x, y, t)– Population is a function of x, y, and t
• ∫f(x)dx – definite integral, area under the curve, or summation
• dP/dx – derivative, instantaneous rate of change, or slope of a function
• ∂P/∂x – partial derivative implying that P is a function of more than one variable
88
Matrix NotationMatrix Notation
0
xtd
c
b
a
The first set of terms are state variables at time t and usually called U. The second set of terms are the flux variables in space x and usually referred to as F.
UF
This is just a system of equations
a + c = 0
b + d = 0
99
Parallel Algorithms• Data Parallel -– most
common with MPI• Master/Worker – one
process hands out the work to the other processes – great load balance, good with threads
• Pipeline – bucket brigade
Implementation Patterns• Message Passing• Threads• Shared Memory• Distributed Arrays, Global
Arrays
Patterns for Parallel ProgrammingPatterns for Parallel Programming
Patterns for Parallel Programming, Mattson, Sanders, and Massingill, 2005
1010
Writing a ProgramWriting a ProgramData Parallel ModelData Parallel Model
P(400) – distributed
Ptot -- replicated
Proc 1
P(1-100)
Ptot
Proc 2
P(101-200)
Ptot
Proc 3
P(201–300)
Ptot
Proc 4
P(301-400)
Ptot
Serial operations are done on every processor so that replicated data is the same on every processor.
This may seem like a waste of work, but it is easier than synchronizing data values.
Sections of distributed data are “owned” by each processor. This is where the parallel speedups occur.
Often ghost cells around each processor’s data is a way to handle communication.
1111
2007-2008 Sample2007-2008 SampleSupercomputing ProjectSupercomputing Project
• Evaluation Criteria – Expo (Report slightly different). Use these to evaluate the following project.– 15% Problem Statement– 25% Mathematical/Algorithmic Model– 25% Computational Model– 15% Results and Conclusions– 10% Code– 10% Display Evaluate Us!!
1212
Get Wet!Get Wet! With the Shallow Water Equations With the Shallow Water Equations
• The shallow water model for wave motion is important for water flow, seashore waves, and flooding
• Goal of this project is to model the wave motion in the shallow water tank
• With slight modifications this model can be applied to:– ocean or lake currents– weather– glacial movement
1313
The water experiences 5 splashes which generate surface gravity waves that propagate away from the splash locations and reflect off of the bathtub walls. Wikipedia commons, Author Dan Copsey
Go to shallow water movie. http://en.wikipedia.org/wiki/Image:Shallow_water_waves.gif
Output from a shallow water equation Output from a shallow water equation model of water in a bathtub.model of water in a bathtub.
1414
Mathematical EquationsMathematical EquationsMathematical ModelMathematical Model
0)( xt huhConservation of Mass
0)()( 2212 xt ghhuhu
Conservation of Momentum
Shallow Water Equations
Notes: mass equals height because width, depth and density are all constant
h -> heightu -> velocityg -> gravity
References: Leveque, Randall, Finite Volume Methods for Hyperbolic Problems, p. 254
Note: Force term, Pressure P=½gh2
1515
Shallow Water EquationsShallow Water EquationsMatrix NotationMatrix Notation
.02
212
xtghhu
uh
hu
h
0ghspeedwave The maximum time step is calculated so as to keep a wave from completely crossing a cell.
1616
Numerical ModelNumerical Model
• Lax-Wendroff two-step, a predictor-corrector method– Predictor step estimates the values at the zone
boundaries at half a time step advanced in time– Corrector step fluxes the variables using the predictor
step values• Mathematical Notes for next slide:
– U is a state variable such as mass or height.– F is a flux term – the velocity times the state variable
at the interface– superscripts are time– subscripts are space
1717
The Lax-Wendroff MethodThe Lax-Wendroff Method
)(2
)(5.0 112
1
2
1ni
ni
ni
ni
n
iFF
x
tUUU
)( 2
1
2
1
2
1
2
1
1
n
i
n
i
n
i
n
i FFxt
UU
Half Step
Whole Step
Explanation graphic courtesy of Jon Robey and Dov Shlacter, 2006-2007 Supercomputing Challenge
1818
Explanation of Lax-Wendroff ModelExplanation of Lax-Wendroff Model
Physical model
Original
Half-step
Full step
Ghost cell
ti
t+1i
t+.5i+.5
Data assumed to beat the center of cell.
Space index
Explanation graphic courtesy of Jon Robey and Dov Shlacter, 2006-2007 Supercomputing Challenge. See appendix for 2D index explanation.
1919
Extension to 2DExtension to 2D
• The extension of the shallow water equations to 2D is shown in the following slides.– First slide shows the matrix form of the 2D
shallow water equations– Second slide shows the 2D form of the Lax-
Wendroff numerical method
2020
2D Shallow Water Equations2D Shallow Water Equations
.02
212
2212
yxt ghhv
huv
vh
huv
ghhu
uh
hv
hu
h
Note the addition of fluxes in the y direction and a flux cross term in the momentum equation. The U, F, and G are shorthand for the numerical equations on the next slide. The U terms are the state variables. F and G are the flux terms in x and y.
U F G
2121
The Lax-Wendroff MethodThe Lax-Wendroff Method
)(2
)(5.0
)(2
)(5.0
,1,,1,
,,1,,1
2
1
2
1,
2
1
,2
1
nji
nji
nji
nji
n
i
nji
nji
nji
nji
n
i
GGy
tUUU
FFx
tUUU
j
j
)()( 2
1
2
12
1
2
12
1
,2
12
1
,2
1 ,,,1
,
n
ji
n
ji
n
i
n
i
nji
nji GG
y
tFF
x
tUU
jj
Half Step
Whole Step
2222
2D Shallow Water Equations2D Shallow Water EquationsTransformed for ProgrammingTransformed for Programming
.0
/
/
/
/2
212
2212
yxt gHHV
HUV
V
HUV
gHHU
U
V
U
H
Letting H = h, U = hu and V = hv so that our main variables are the state variables in the first column gives the following set of equations.
H is height (same as mass for constant width, depth and density) U is x momentum (x velocity times mass)V is y momentum (y velocity times mass)
2323
Sample ProgramsSample Programs
• The numerical method was extracted from the McCurdy team’s model (team 62) from last year and reprogrammed from serial Fortran to C/MPI using the programming style from one of the Los Alamos team’s project (team 51) with permission from both teams.
• Additional versions of the program were made in Java/Threads and Matlab
2424
Programming ToolsProgramming ToolsThree optionsThree options
1. Matlab– Computation and graphics integrated into Matlab desktop
2. Java/Threads– Eclipse or Netbeans workbench– Graphics via Java 2D and Java Free Chart
3. C/MPI– Eclipse workbench -- An open-source Programmers
Workbench http://www.eclipse.org.– PTP (parallel tools plug-in) – adds MPI support to Eclipse
(developed partly at LANL)– OpenMPI – a MPI implementation (developed partly at LANL)– MPE -- graphics calls that come with MPICH. Graphics calls
are done in parallel from each processor!
2525
Initial Conditions and Boundary Initial Conditions and Boundary ConditionsConditions
• Initial conditions– velocity (u and v) are 0 throughout the mesh– height is 2 with a ramp to the height of 10 at the right
hand boundary starting at the mid-point in the x dimension
• Boundary conditions are reflective, slip– hbound=hinterior; uxbound=0; vxbound=vinterior
– hbound=hinterior; uybound=uinterior; vybound=0
– If using ghost cells, force zero velocity at the boundary by setting Uxghost= -Uinterior
2626
Results/ConclusionsResults/Conclusions
• The Lax-Wendroff model accurately models the experimental wave tank– matches wave speed across the tank
• Some of the oscillations in the simulation are an artifact of the numerical model– OK as long as initial wave is not too steep– numerical damping technique could be added
but is beyond the scope of this effort
2727
AcknowledgementsAcknowledgements
Work used by permission:• Awash: Modeling Wave Movement in a Ripple Tank,
Team 62, McCurdy High School, 2006-2007 Supercomputing Challenges
• A Lot of Hot Air: Modeling Compressible Fluid Dynamics, Team 51, Los Alamos High School, 2006-2007 Supercomputing Challenge
We all have bugs and thanks to those who found mine• Randy Roberts and Jon Robey for finding and fixing a
bug in the second pass• Randy Leveque for finding a missing square in the
gravity forcing term
2828
Lab ExercisesLab Exercises
• TsunamiClaw
• Matlab • Experimental demonstration
• Java Serial• Java Parallel
• C/MPI
Java Wave StructureJava Wave Structure
• Wave class does most of the work– main(String[] args) calls start()– start() creates a WaveProblemSetup– start() calls methods to do initialization and
boundary conditions– start() calls methods to iterate and update the
display
Java Wave Structure (continued)Java Wave Structure (continued)
• WaveProblemSetup stores the new and old arrays
• swaps the new and old arrays when asked to by Wave
Java Wave Program FlowJava Wave Program Flow
• Create arrays for new, old, and temporary data
• Initialize data
• Set boundary data to represent correct boundary conditions
• Iterate for the given number of iterations
Java Wave Iteration FlowJava Wave Iteration Flow
• Update physics into new arrays from data in old arrays
• Set boundary data to represent correct boundary conditions with updated arrays
• Update display
• Swap new arrays with old arrays
Java ThreadsJava Threads
• How do you take advantage of new Multi-Core processors?
• Run parts of the problem on different cores at the same time!
Java Threads (continued)Java Threads (continued)
• WaveThreaded program– partitions the problem into domains using
SubWaveProblemSetup objects– runs calculations on each domain in separate
threads using WaveWorker objects– adds complexity with synchronization of
thread's access to data
3535
C/MPI Program DiagramC/MPI Program Diagram
Update Boundary CellsMPI CommunicationExternal Boundaries
First Pass x half step y half step
Second Pass
Swap new/oldGraphics Output
Conservation Check
Calculate RuntimeClose Display, MPI & exit
Allocate memorySet Initial Conditions
Initial Display
Repeat
3636
MPI Quick StartMPI Quick Start• #include <mpi.h>• MPI_Init(&argc, &argv)
• MPI_Comm_size(Comm, &nprocs) // get number of processors• MPI_Comm_rank(Comm, &myrank) // get processor rank 0 to nproc-1
• // Broadcast from source processor to all processors• MPI_Bcast(buffer, count, MPI_type, source, Comm)
• // Used to update ghost cells• MPI_ISend(buffer, count, MPI_type, dest, tag, Comm, req)• MPI_IRecv(buffer, count, MPI_type, source, tag, Comm, req+1)• MPI_Waitall(num, req, status)
• // Used for sum, max, and min such as total mass or minimum timestep• MPI_Allreduce(&num_local, &num_global, count, MPI_type, MPI_op, Comm)
• MPI_Finalize()
• Web pages for MPI and MPE at Argonne National Lab (ANL) -- http://www-unix.mcs.anl.gov/mpi/www/
3737
SetupSetup
• The software is already setup on the computers
• For setup on home computers, there are two parts. First download the files from the Supercomputing Challenge website for the lab in C/MPI if you haven’t already done that.
• Untar the lab files with “tar –xzvf Wave_Lab.tgz”
3838
Setting up SoftwareSetting up SoftwareInstructions in the README fileInstructions in the README file
Setting up System Software• Need Java, OpenMPI and
MPE package from MPICH
• Download and install according to instructions in openmpi_setup.sh
• Can install in user’s directory with some modifications
Setting up User’s workspace
• Download eclipse software including eclipse, PTP and PLDT
• Install according to instructions in eclipse_setup.sh
• Import wave source files and setup eclipse according to instructions in eclipse_setup.sh
3939
Lab ExercisesLab Exercises
• Try modifying the sample program (Java and/or C versions)– Change initial distribution. How sharp can it be before it goes
unstable?– Change number of cells– Change graphics output– Try running 1, 2, or 4 processes and time the runs. Note that you
can run 4 processes even if you are on a one processor system.– Switch to PTP debug or Java debug perspective and try
stepping through the program• Comparing to data is critical
– Are there other unrealistic behaviors of the model?– Design an experiment to isolate variable effects. This can greatly
improve your model.
4040
Appendix A.Appendix A.Calculus and SupercomputingCalculus and Supercomputing
• Calculus and Supercomputing are intertwined. Why?
• Here is a simple problem – Add up the volume of earth above sea-level for an island 500 ft high by half a mile wide and twenty miles long.
• Typical science homework problem using simple algebra. Can be done by hand. Not appropriate for supercomputing. Not enough complexity.
4141
Add ComplexityAdd Complexity
• The island profile is a jagged mountainous terrain cut by deep canyons. How do we add up the volume?
• Calculus – language of complexity– Addition – summing numbers– Multiplication – summing numbers with a constant
magnitude– Integration – summing numbers with an irregular
magnitude
4242
Divide and ConquerDivide and Conquer
• In discrete form
• Divide the island into small pieces and sum up the volume of each piece.
• Approaches the solution as the size of the intervals grows smaller for a jagged profile.
∑ -- Summation symbol
∆ -- delta symbol or x2-x1
4343
Divide and ConquerDivide and Conquer
• In Continuous Form – Integration
• Think of the integral symbols as describing a shape that is continuously varying
• The accuracy of the solution can be improved by summing over smaller increments
• Lots of arithmetic operations – now you have a “computing” problem. Add more work and you have a “supercomputing” problem.
4444
Derivative CalculusDerivative CalculusDescribing ChangeDescribing Change
• Derivatives describe the change in a variable (numerator or top variable) relative to another variable (denominator or bottom). These three derivatives describe the change in population versus time, x-direction and y-direction.
y
Pandx
P
t
P
,
4545
Appendix B. Appendix B. Computational MethodsComputational Methods
1.1. Eulerian and LagrangianEulerian and Lagrangian
2.2. Explicit and ImplicitExplicit and Implicit
4646
Two Main Approaches to Divide up Two Main Approaches to Divide up ProblemProblem
• Eulerian – divide up by spatial coordinates– Track populations in a location– Observer frame of reference
• Lagrangian – divide up by objects– Object frame of reference– Easier to track attributes of population since they
travel with the objects– Agent based modeling of Star Logo uses this
approach– Can tangle mesh in 2 and 3 dimensions
4747
EulerianEulerian
Eulerian – The area stays fixed and has a Population per area. We observe the change in population across the boundaries of the area.
Lagrangian – The population stays constant. The population moves with velocity vx and vy and we move with them. The size of the area will change if the four vertexes of the rectangle move at different velocities. Changes in area will result in different densities.
Eulerian
Population moves out of cell
Lagrangian
Population moves and so does region
4848
Explicit versus ImplicitExplicit versus Implicit
• Explicit – In mathematical shorthand, Un+1= f(Un). This means that the next timestep values can be expressed entirely on the previous timestep values.
• Implicit – Un+1=f(Un+1,Un). Next timestep values must be solved iteratively. Often uses a matrix or iterative solver.
• We will stick with explicit methods here. You need more math to attempt implicit methods.
4949
Appendix CAppendix C
Index Explanation for 2D Lax Index Explanation for 2D Lax WendroffWendroff
5050
ProgrammingProgramming
• Most difficult part of programming this method is to keep track of indices – half step grid indices cannot be represented by ½ in the code so they have to be offset one way or the other.
• Errors are very difficult to find so it is important to be very methodical in the coding.
• Next two slides show the different sizes of the staggered half-step grid and the relationships between the indices in the calculation (courtesy Jon Robey).
5151
y yy
y
y
y y
y y
yy y
x
x
x x x x
x
x x
x x
x
0
1
2
3
4
j
0 1 2 3 4
i
0,0 -- 1,0 | 1,1
j,i -- j+1,i | j+1,i+1
X step grid Main grid
0,0 -- 0,1 | 1,1
j,i -- j,i+1 | j+1,i+1
Y step grid Main grid
1st Pass
5252
y yy
y
y
y y
y y
yy y
x
x
x x x x
x
x x
x x
x
1,1
1,1
-- 0,0 | 1,0
-- 0,0 | 0,1
j,i -- j-1,i-1 | j,i-1
0
1
2
3
4
j
0 1 2 3 4
i
j,i -- j-1,i-1 | j-1,i
Main grid X step grid
Main grid Y step grid
2nd Pass