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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)
 

 

 

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
 
Solve = 4/2 = 2

Plug =2 into other
equations and simplify
Serial Pseudocode, O(n2)

Solve = 6/2 = 3

 

Solve = -12/2 = -6

 

Solve = 9/1 = 9
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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