Domain Decomposed Parallel Heat Distribution Problem in Two Dimensions

1
Domain Decomposed Parallel Heat
DistributionProblem in Two Dimensions

• Yana Kortsarts
• Jeff Rufinus
• Widener University
• Computer Science Department

2
Introduction

• 2004 Office of Science in the Department of
  Energy issued a twenty-year strategic plan with
  seven highest priorities ranging from fusion
  energy to genomics.
• To achieve the necessary levels of algorithmic
  and computational capabilities it is essential to
  educate students in computation and computational
  techniques.
• Parallel Computing topic is one of the attractive
  topics in computation science field

3
Introductory Parallel Computer Course

• Computer Science Department, Widener University,
  CS and CIS majors
• Series of two courses Introduction to Parallel
  Computing I and II
• Resources computer cluster of six nodes, each
  node has two 2.4 GHz processors and 1 GB of
  memory, nodes are connected by Gigabit Ethernet
  switch.

4
Course Curriculum

• Matrix Manipulation
• Numerical Simulation Concepts
• Direct applications in science and engineering
• Introduction to MPI libraries and their
  applications
• Concepts of parallelism
• Finite difference method for the 2-D heat
  equation using parallel algorithm

5
2-D Heat Distribution Problem

• The problem to determine the temperature
  u(x,y,t) in an isotropic two-dimensional
  rectangular plate
• The model

6
Finite Difference Method

• The finite difference method begins with the
  discretization of space and time such that there
  is an integer number of points in space and an
  integer number of times at which we calculate the
  temperature

(xi , yj)
tk
tk1
?t
?y
?x
7
We will use the following notation
We will use the finite difference approximations
for the derivatives
Expressing ui,j,k1 from this equation yields
8
Finite Difference Method Explicit Scheme
ui,j1,k
ui,j,k1
ui,j,k
ui-1,j,k
ui1,j,k
k ? k 1
ui,j-1,k
9
Single Processor Implementation

• double u_oldn1n1, u_newn1n1
• Initialize u_old with initial values and boundary
  conditions
• while (still time points to compute)
• for (i 1 i lt n i)
• for (j 1 j lt n j)
• compute u_newi, j using formula (1)
• //end of for
• // end of for
• u_old ? u_new
• // end of while

10
Parallel ImplementationDomain Decomposition

• Dividing computation and data into pieces
• Domain could be decomposed in three ways
• Column-wise adjacent groups of columns (A)
• Row-wise adjacent groups of rows (B)
• Block-wise adjacent groups of two dimensional
  blocks (C)

11
Domain Decomposition and Partition

• Example column-wise domain decomposition method,
  200 points
• to be calculated simultaneously, 4 processors
• MPI_Send and MPI_Recv

Processor 4 x149x199
Processor 1 x0x49
Processor 2 x50x99
Processor3 x100x149
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Load Imbalance

• When dividing the data into processes we have to
  pay attention to the number of loads being
  processed by each processor
• Uneven load distribution may cause some processes
  to finish earlier than others
• Load imbalance is one source of overhead
• Good task mapping is needed
• All tasks should be mapped onto processes as
  evenly as possible so that all tasks complete in
  the shortest amount of time and the idle time is
  minimized

13
Communication

• Communication time depends on the latency and the
  speed of communication network these two
  factors are much slower than CPUs communication
  time
• There is a catch of using too many communications

14
Running Time and Speedup

• The running time of one time iteration of the
  sequential algorithm is ?(MN), where M and N are
  numbers of grid points in each direction
• The running time of one time iteration of the
  parallel algorithm is computational time
  communication time
• ??(MN/p) B
• where p is the number of processors and B is
  the total send-receive communication time that is
  required for one time iteration
• The speedup is always defined as

15
Results
The temperature distribution on the
two-dimensional plate at a much later time
16
Results

• Two cases were considered
• M x N 1000
• M x N 500,000.
• Next slide shows the speed-up versus the number
  of processors for two different inputs 500,000
  (the top chart) and 1000 (the bottom chart).
• The dashed line indicates the speed-up equals to
  one, which is the sequential version of the
  algorithm.
• The higher the speed-up (at a specific number of
  processors) means the better the performance of
  the parallel algorithm.
• Most of the results come from the column-wise
  domain decomposition method.

17
Results
18
Results

• For the case of input 1000, the sequential
  version (with p 1) is faster than
  the parallel version (p 2). The parallel
  version is slower because of the latency and
  speed of the communication network which does not
  exist in the sequential version.
• The top chart shows the speedup versus the number
  of processors for total input 500,000. In this
  case, as we increase the number of processors,
  the speedup also increases, reaching the speedup
  of 4.13 at p 10.
• For a large number of inputs the communication
  time begins to catch up with the CPUs
  computation time, resulting in a better
  performance of the parallel algorithm.

19
Speedup comparisons for column-wise and
block-wise decomposition methods for number of
processors equals to 4 and 9

Total number of inputs Speedup Column-wise Decomposition P 4 Speedup Column-wise Decomposition P 9 Speedup Block-wise Decomposition P 4 Speedup Block-wise Decomposition P 9
1,000 0.093 0.05 0.065 0.04
500,000 2.12 3.82 2.19 3.51
20
Results

• Overall, the speedups between the two methods are
  not very different.
• For number of inputs 1,000 the column-wise
  decomposition produces better speed-ups than the
  block-wise decomposition.
• For number of inputs 500,000, we have a mixed
  result. The column-wise method performs better
  for 9 processors while the block-wise method
  performs (slightly) better for 4 processors.
• The results given in the table do not give a
  conclusive idea of which decomposition method is
  better, unless the number of inputs and the
  number of processors could be extended beyond the
  ones used in here.

21
Summary

• Numerical simulation of two-dimensional heat
  distribution has been used as an example that can
  be used to teach parallel computing concepts in
  an introductory course.
• With this simple example we introduce the core
  concepts of parallelism
• Domain decomposition and partitioning
• Load balancing and mapping
• Communication
• Speedup
• We show the benchmarking results of the parallel
  version of two-dimensional heat distribution
  problem with different number of processors.

22
References

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