CONJUGATE GRADIENT METHOD

Monica GarikaChandana Guduru

METHODS TO SOLVE LINEAR SYSTEMSDirect methods Gaussian elimination method LU method for factorization Simplex method of linear programmingIterative method Jacobi method Gauss-Seidel method Multi-grid method Conjugate gradient method

Conjugate Gradient methodThe CG is an algorithm for the numerical solution of particular system of linear equations

Ax=b. Where A is symmetric i.e., A = AT and positive

definite i.e., xT * A * x > 0 for all nonzero vectors

If A is symmetric and positive definite then the function

Q(x) = ½ x`Ax – x`b + c

Conjugate gradient methodConjugate gradient method builds up a

solution x*€ Rn

in at most n steps in the absence of

round off errors.Considering round off errors more than n

steps may be needed to get a good approximation of exact solution x*

For sparse matrices a good approximation of exact solution can be achieved in less than n steps in also with round off errors.

In oil reservoir simulation, The number of linear equations corresponds to the number of grids of a reservoirThe unknown vector x is the oil pressure of

reservoirEach element of the vector x is the oil

pressure of a specific grid of the reservoir

Practical Example

Linear System

Square matrixUnknown vector

(what we want to find)Known vector

Matrix Multiplication

Positive Definite Matrix

> 0[ x1 x2 … xn ]

ProcedureFinding the initial guess for solution 𝑥0Generates successive approximations to 𝑥0Generates residualsSearching directions

x0 = 0, r0 = b, p0 = r0

for k = 1, 2, 3, . . . αk = (rT

k-1rk-1) / (pTk-1Apk-1) step length

xk = xk-1 + αk pk-1 approximate solution

rk = rk-1 – αk Apk-1 residual

βk = (rTk rk) / (rT

k-1rk-1) improvement

pk = rk + βk pk-1 search direction

Conjugate gradient iteration

Iteration of conjugate gradient method is of the form

x(t) = x(t-1) + s(t)d(t) where, x(t) is function of old value of vector x s(t) is scalar step size x d(t) is direction vector Before first iteration, values of x(0), d(0) and g(0) must be set

Steps to find conjugate gradient methodEvery iteration t calculates x(t) in four steps : Step 1: Compute the gradient g(t) = Ax(t-1) – b Step 2: Compute direction vector d(t) = - g(t) + [g(t)` g(t) / g(t-1)` g(t-1)] d(t-1) Step 3: Compute step size s(t) = [- d(t)` g(t)]/d(t)’ A d(t)] Step 4: Compute new approximation of x x(t) = x(t-1) + s(t)d(t)

Sequential Algorithm1) 𝑥0 = 02) 𝑟0 := 𝑏 − 𝐴𝑥03) 𝑝0 := 𝑟04) 𝑘 := 05) 𝐾𝑚𝑎𝑥 := maximum number of iterations to be done6) if 𝑘 < 𝑘𝑚𝑎𝑥 then perform 8 to 167) if 𝑘 = 𝑘𝑚𝑎𝑥 then exit8) calculate 𝑣 = 𝐴𝑝k

9) αk : = rkT rk/pT

k v10) 𝑥k+1:= 𝑥k + ak pk

11) 𝑟k+1 := 𝑟k − ak v12) if 𝑟k+1 is sufficiently small then go to 16end if13) 𝛽k :=( r T

k+1 r k+1)/(rTk 𝑟k)

14) 𝑝k+1 := 𝑟k+1 +βk pk

15) 𝑘 := 𝑘 + 116) 𝑟𝑒𝑠𝑢𝑙𝑡 = 𝑥k+1

Complexity analysisTo Identify Data DependenciesTo identify eventual communicationsRequires large number of operationsAs number of equations increases complexity

also increases .

Why To Parallelize?Parallelizing conjugate gradient method is a

way to increase its performanceSaves memory because processors only store

the portions of the rows of a matrix A that contain non-zero elements.

It executes faster because of dividing a matrix into portions

How to parallelize?For example , choose a row-wise block

striped decomposition of A and replicate all vectors

Multiplication of A and vector may be performed without any communication

But all-gather communication is needed to replicate the result vector

Overall time complexity of parallel algorithm is

Θ(n^2 * w / p + nlogp)

Row wise Block Striped Decomposition of a Symmetrically Banded Matrix

( a ) (b )

Dependency Graph in CG

Algorithm of a Parallel CG on each Computing Worker (cw)1) Receive 𝑐𝑤𝑟0,𝑐𝑤 𝐴,𝑐𝑤 𝑥02) 𝑐𝑤𝑝0 =𝑐𝑤 𝑟03) 𝑘 := 04) 𝐾𝑚𝑎𝑥 := maximum number of iterations to be done5) if 𝑘 < 𝑘𝑚𝑎𝑥 then perform 8 to 166) if 𝑘 = 𝑘𝑚𝑎𝑥 then exit7) 𝑣 =𝑐𝑤 𝐴𝑐𝑤𝑝𝑘8) 𝑐𝑤𝑁𝛼𝑘9) 𝑐𝑤𝐷𝛼𝑘10) Send 𝑁𝛼𝑘,𝐷𝛼𝑘11) Receive 𝛼𝑘12) 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘𝑝𝑘13) Compute Partial Result of 𝑟𝑘+1: 𝑟𝑘+1 =𝑟𝑘 − 𝛼𝑘𝑣14) Send 𝑥𝑘+1, 𝑟𝑘+115) Receive 𝑠𝑖𝑔𝑛𝑎𝑙16) if 𝑠𝑖𝑔𝑛𝑎𝑙 = 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑟𝑒𝑎𝑐ℎ𝑒𝑑 go to 2317) 𝑐𝑤𝑁𝛽𝑘18) 𝑐𝑤𝐷𝛽𝑘19) Send 𝑐𝑤𝑁𝛽𝑘 , 𝑐𝑤𝐷𝛽𝑘20) Receive 𝛽𝑘21) 𝑐𝑤𝑝𝑘+1 =𝑐𝑤 𝑟𝑘+1 + 𝛽𝑘 𝑐𝑤𝑝𝑘22) 𝑘 := 𝑘 + 123) Result reached

Speedup of Parallel CG on Grid Versus Sequential CG on Intel

Communication and Waiting Time of the Parallel CG on Grid

We consider the difference between f at the solution x and any other vector p:

1) 21) 21 1) ( )2 2

1 1simplify) ( )2 21 1replace ) ( )2 2

1 12 2

t

t

t t

t t

t t t t

t t

i f

ii f

ii i f f

f f

f f

t

t

t t

t t

p p Ap b p c

x x Ax b x c

x p x Ax b x c p Ap b p c

x p x Ax b x p Ap b p

b x p x Ax x Ax p Ap x Ap

x Ax p A

12

positive-definitess) ( ) 0

t

t

f f

p x Ap

x p A x p

x p

Parallel Computation Design– Iterations of the conjugate gradient method can be executed only in sequence, so the most advisable approach is to parallelize the computations, that are carried out at each iteration The most time-consuming computations are the multiplication of

matrix A by the vectors x and d– Additional operations, that have the lower computational complexity order, are different vector processing procedures (inner product, addition and subtraction, multiplying by a scalar).While implementing the parallel conjugate gradient method, it can be used parallel algorithms of matrix-vector multiplication,

Parallel Computation Design

“Pure” Conjugate Gradient Method (Quadratic Case)

0 - Starting at any x0 define d0 = -g0 = b - Q x0 , where gk is the column vector of gradients of the objective function at point f(xk)

1 - Using dk , calculate the new point xk+1 = xk + ak dk , where

2 - Calculate the new conjugate gradient direction dk+1 , according to: dk+1 = - gk+1 + bk dk where

T a k = - gk dk

dkTQdk

b k = gk+1TQd kdkTQd k

ADVANTAGES Advantages:1) Gradient is always nonzero and linearly

independent of all previous direction vectors.2) Simple formula to determine the new

direction. Only slightly more complicated than steepest descent.

3) Process makes good progress because it is based on gradients.

ADVANTAGESAttractive are the simple formulae for

updating the direction vector.Method is slightly more complicated than

steepest descent, but converges faster.Conjugate gradient method is an Indirect SolverUsed to solve large systemsRequires small amount of memoryIt doesn’t require numerical linear algebra

ConclusionConjugate gradient method is a linear solver

tool which is used in a wide range of engineering and sciences applications.

However, conjugate gradient has a complexity drawback due to the high number of arithmetic operations involved in matrix vector and vector-vector multiplications.

Our implementation reveals that despite the communication cost involved in a parallel CG, a performance improvement compared to a sequential algorithm is still possible.

ReferencesParallel and distributed computing systems by

Dimitri P. Bertsekas, John N.TsitsiklisParallel programming for multicore and cluster

systems by Thomas Rauber,Gudula RungerScientific computing . An introduction with parallel

computing by Gene Golub and James M.OrtegaParallel computing in C with Openmp and mpi by

Michael J.QuinnJonathon Richard Shewchuk, ”An Introduction to

the Conjugate Gradient Method Without the Agonizing Pain”, School of Computer Science, Carnegie Mellon University, Edition 1 1/4