totally parallel multilevel algorithms for sparse elliptic systems

Research

TOTALLY

PARALLEL FOR SPARSE

Institute

for Advanced Computer Science NASA Ames Research Center

MULTILEVEL ALGORITHMS ELLIPTIC SYSTEMS

Paul O. Frederickson

March, 1989

A-! Research I.nsdtute for Advanced Computer Science NASA Ames Research Center

RIACS Technical Report 89.10

NASA Cooperative

Agreement

Number NCC 2-387

(NASA-CR-ldSd37) TOTALLY PARALLEL MULTILEVEL ALGORITHMS FOR SOARSE ELLIPTIC SYSTEMS (Research Inst. for Advanced Computer Science) 7 p CSCL

N92-10333

09B G3/62

Unclas 0043023

f

TOTALLY

PARALLEL FOR SPARSE

MULTILEVEL ALGORITHMS ELLIPTIC SYSTEMS

Paul O. Frederickson

Research Institute for Advanced Computer NASA Ames Research Center

Science

RIACS Technical Report 89.10 March, 1989

The fastest known algorithms for the solution of a large elliptic boundary value problem on a massively parallel hypercuhe all require O(log(n)) floating point operations and O(Iog(n)) distance-1 communications, ff we define massively parallel to mean a number of processors proportional to the size n of the problem. The algorithm TPMA (for Totally Parallel Multilevel Algorithm) that we describe below has, as special cases, four of these'fasialgorkhms. These four algorithms are PSMG (Parallel Superconvergent Multigrid) of Frederickson and McBryan, Robust Muldgrid of Hackbusch, the FFT based Spec|ral Algorithm, and Paral/eI Cyctic Reduction. The algorithm TPMA, when described recursively, has four steps: (I)

Project to a collection of interlaced, coarser problems at the next lower level

(2)

Apply TPMA, recursively, to each of these lower level problem, Solving directly at the lowest level

(3)

Interpolate these approximate solutions to the finer grid, and a verage them to form an approximate soludon on this grid.

(4)

Refine this approximate solution with a defect-correction inverse.

step, using a local approximate

Choice of the projection operator P, the interpolation operator Q, and the smoother S determines the class of problems on which TPMA is most effective. There are special cases in which _e first three steps produce an exact solution, and the smoother is not needed (e.g. constant coefficient oeprators).

Key Words: Multilevel algorithm; Multigrid; Cyclic reduction; Spec_'al method

Work reported herein was supported in part by Cooperative Agreement NCC2-387 between the National Aeronautics and Space Administration (NASA) and the Universities Space Research Association (USRA).

TOTALLY

PARALLEL FOR SPARSE

MULTILEVEL ALGORITHMS EI,LIPTIC SYSTEMS

Paul

Research

Institute NASA

O. Frederic_on

for'Advanced Ames Research

1: Introduction. The fa_tc_t elliptic

known

boundary

alg(,rithms

f,,t the s,,lution

of an

value problem

.4. =.,

(:.:)

Computer Cenmr

rithms are PSMG (Parallel Superconvetgent Multigrid) of Fredcrickson and McBryan, Rol)ust Multigrid of Hackbusch,

the FFT

Cyclic

Reduction.

the

parallel

hypercul)e,

hypercube

with

to the size

n of the problem,

in structure. ceed

a number

point

stages

by which

of processors apparent

(or levels)

operations

we nlean

executed

a

proportional

are all very closely

It is imme(iiately

in O(log(n))

floating

p

related

that

all pro-

consisting

of

O(n)

in parallel

and

O(p)

parallel communications with a nearest neighb{,r. On closer examination one observes that in the k th level of any of these

algorithms

A*u * = v'

consists

being

solved,

in a number

problems.

It is this

in the following The

and that

(1.2) this problem

of independent observation

actually

and interleaved that

we wish

sub-

to clarify

sections.

algorithm

TPMA

(for

Totally

Parallel

Multi-

level Alg,,rithm) that we describe i)el,w has, as special cases, four of these- fast algorithms. These four algo-

Algorithm,

of the projection

operator

Qk

S I' in the alg,,rith,u

them

the characteristics

and

TPM

It is useful,

when

A determines

between

to understand

having

the same color.

ample,

the graph in Fig.

graph

2.

al-

sul)grids These are

triangulations,

graph

can be col-

Fewer

colors

may suffice.

1 corresponding

nodes For ex-

to an equilat-

Observe

that

each subgrid

corresponds

,J" the: square o1"the original graph. to the statement that nodes connected

requires

connect

over

only

distance the

hypercube,

two

by connecting

by the factor

points

two apart x/_.

colors. nodes

The

five point two subgrids

of like color which

in the original

grid

when

for communication

computing distance

bounded

by Cooperative Agreement (NASA) aml the: Univt:t_ith_

by

Laplathat are a

grid are larger

Use of this grid offers some

triangular

to a This is by an

that are separated

advantage

on a binary along

axis remains a power of two as the algorithm through the levels, llyl)ercube communication is therefore

(USIL¢)

the

distance two apart we fi)rm three interlaced subeach larger by a factor v_ and rotated, as shown

grapll grids,

are formed

was suI)Imrted in part Sl,a(:l.' Adndnlstrati(m

new

eral triangulation of the plane requires only three colors. If we connect the nodes of the same color that are a

clan

herein and

of

these.

two edges of the original graph. Similarly, the graph of the familiar

reported Acrtntautics

What

ored using at most four colors with no two adjacent

edge in the subgrid

Work tit,]tai

which

(1.1).

2: Graph Coh)rings and S,hgrids. The nodes of an arbitrary planar

in Fig.

1

op-

these known ones is yet

attempting

with

P,

Which algothings, among

in the case of plane

we begin

Parallel

smoothing

gorithm TPMA, to see clearly the intertwined at each of the multiple levels of the algorithm.

subgraph equivalent

Fig.

and

operator

the

of the problem

algorithms fill the space to be determined.

more easily visualized

the problem

Spectral

Choice

these particular algorithms is tel)resented. rithm one wishes to use depends on many

_and hence

is, in effect,

based

interpolation

erat,,r on a massivc.ly

Scienc_

each grid descends distance

by four. NCC2-387 bctween the NaSl)m:t_ llt..st:arch Association

t. -'.

!

ins equation at level k - 1. The easiest example of a projection operator, P_ = I, is used in some versions of

i ..--':--.i. -". !

the algorithm L.-I

, "'-.:.-"

I

r-..,..-:...l. I .'_" 9 II "

,

:_'..J,.-/,

, -

_ll.

.,_ I , "--_l_/

.

I.',

-,"

".¢'[_

"".'*.'"

I

,_ -..!q

.'_

._,. : ._ , "-. , .-" I "

,

,

i/

,'-.

.,_ I : __l

-

-

I-.

I

/,-

I ,.-"

o.,_

,

I " -

"

.I..

,

/,-

I

: _.t/','-.,

.-

I

jecting

In this

TI'MA.

equation

case

k - 1 is particularly easy. S,pp,_e that we haw been struct an approximate solution equation

A_-ta

interpolation solution

the first step

(3.1) to the interleaved

_-_

at level

able, s_mehow, to conu *-t in X _-_ to the

=

¢_-_

_

to map it into an approximate

operator

t=_' =

in pro-

subgrids

Q_=_-_

.

'l'hen

of equation

we will use the

(3.1).

The effect of _ is to combine the approximate solutions from all of the interleaved subgrids of a given grid I "

,

, "__ I_/

"-. , .-" I " .

-

, _.

I

Fig. 2 When cretized

using TPMA

to solve an elliptic

onto an unstructured

triangular

to generate interlaced s,,bgrids eolori,g, i,t which the number

problem

dis-

grid we are able

using an imperfect of a, ijaeent n,-lcs

threeof" the

same color is minimized, or an imperfect two-col.ring. This construction of interlaced subgrids is repeated recursively: each of these sul_gri,ls is a triangulation in its Finally, Laplacian, factor

we observe although

The

four

that

not

of TPMA

the CM2

in particular.

is col,ruble

subgrids,

each

to a very

high

on highly

parallel

grid.

Except

at

interleaved

subgrids

of a grid

at a yet higher

level.

In

many cases Q_ is best described as an averaging operator, while in other cases it will simply be the identity. The er)nvergenee the, ry is partleular;x easy to state when the two operators _ and I '_ are adjolnt each other. This is the situation, for example, arc constructed

\ /

by the

nsing

cases

the

or dual to when they

Rahigh-Ritz-Calerkin

pc,co-

A k-t

at level

k -

1 is

A _-' =

P_A_Q

.(3.2)

_,

and via the commutative

diagram

,,_,k

.._.,AS

_k

/ (3.3)

< /