Solving Linear Equations using Parallel Algorithms
| |
Overview
• Introduction
• Parallelization of Methods
– Direct (Gaussian Elimination + Back
Propagation)
– Iterative (Jacobi, Gauss-Seidel, Conjugate
Gradient)
• Applications |
Why important?
• Encounter linear equations when solving ODEs/PDEs
numerically .
• Many domains
– Structural Analysis (civil engineering)
– Heat Conduction & Fluid Dynamics (mechanical
engineering)
– Power Grid Analysis (electrical engineering)
– Regression Analysis (statistics)
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Linear vs . Nonlinear Equations
• Linear system of equations 
• Non- linear system
|
Linear vs. Nonlinear Equations
• Navier-Stokes Equation for Compressible
Flow, a non-linear PDE
• With some assumptions for periodicity in
 , it is reduced to a linear PDE
 |
System of Equations
• General system with 4 unknowns
• Can obtain a system with finite differences
|
| System as a Matrix Equation
• Equation system as Matrix equation Ax=b
• Solve x =A-1b.
For dense systems we do
not solve by computing the inverse of A
directly, and for sparse systems we never
compute the inverse. |
Overview
• Introduction
• Methods and Parallelization
– Direct (Gaussian Elimination + Back
Propagation)
– Iterative (Jacobi, Gauss-Seidel, Conjugate
Gradient)
• Applications
|
Back Propagation
|
Back Propagation
How to parallelize?
• New
 values
must be computed
sequentially
• Updating the
equations can be
done in parallel |
 |
|
Back Propagation
• Row-oriented parallel algorithm
– Each process gets a row of the
coefficient matrix and its
corresponding  value
– Computation O(n2/p)
– Communication O(n log p )
|
 |
• Column-oriented parallel
algorithm
– Each process stores the
coefficients of a single and
entire b vector, and fires when
appropriate
– Computation O(n2) (no
computational concurrency)
– Communication O(n2) |
 |
|
Back Propagation
Which approach is better?
|
Gaussian Elimination
• How to get a dense matrix into the upper
triangular form for back-propagation?
|
Gaussian Elimination
• Robust implementation
includes row pivoting to
overcome roundoff errors
when dividing by small
coefficients
• Serial Pseudocode
– Choose pivot row O(n2)
– Perform elimination O (n3)
– Back substitution O (n2)
• Overall O(n3)
|
 |
|
Gaussian Elimination
• Parallelization – options for row
& column oriented approaches |
 |
• Tournament style evaluation to
determine pivot using
MPI_Allreduce with
MPI_MAXLOC |
• Elimination done in parallel
after divisor is broadcast |
• Use already developed parallel
back propagation algorithm |
| |
|
Gaussian Elimination
• Bad Decomposition Approach
– Row & Column Oriented give same time complexity ,
computation O(n3/p) and communication O(n2p log p)
– Poor Scalability, M = Cp(log p)2
– Poor because computation and communication are
done separately
• Good Decomposition Approach
– Pipelined row oriented algorithm, still computation
O(n3/p)
– Pivoting done column-wise, results sent in a ring
O(np)
– Overlaps communication and computation by forming
a ring, M = C and therefore scalable
|
Pipelined Gaussian Elimination
 |
• Divide rows to processes
in an interleaved manner
• Each process gets the
current row used for
elimination and the pivot
column from the task
master at that iteration
• Elimination is performed,
and then the next process
in line becomes the task
master
|
|
Pipelined Gaussian Elimination
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= 4/2 = 2
= 6/2 = 3
= -12/2 = -6
= 9/1 = 9