For the paralel solution of boundary value problems BVPs for partial differential equatio s thre main directions can b distinguished: approaches based on "parallelism across the problem"
Trang 1PARAMETRIC EXTRAPOLATION AS A PARALLEL METHOD
DANG QUANG A
Abstract In recent y ars we have developed a parallel method for mathematical physics problems It is th metho ofparametric extra olation In this paper we give an overview of our results concern ing this metho for constructin parallel algor hms for some problems of mathematical physics
Torn tlit Trong n h iing n arn gan day ch ung toi d a ph at trie'n mot ph u'o'ng phap son song gili mot so bai torin bien cii a v at Iy- toano Do la ph tro'rig phap ngoai suy theo tham so Bai b o nay Ii tc!ng qua c ac k t quti nghien ctru cii a ch n toi lien quan de ph u'o'ng ph ap nay de' xfiy du'ng c ac th ufit toan song song giai mot so bai to.in bien cho ph trong trln elliptic cap hai va cap bon o·rnu'c vi phfm cling nh iro ·rmrc roi r ac
1 INTRODUCTION
Now, c ping with large-scale problems of physics, mechanics, oceanology, meteorolo y,
hydrol-o y, one has to use parallel c mputing systems in ord r to reduce computation time For this reason i should construct p ralell methods and algorithms for the problems to be realized on the paralel systems For the paralel solution of boundary value problems (BVPs) for partial differential equatio s thre main directions can b distinguished: approaches based on "parallelism across the problem", "parallelism acros method" and on "parallelism across steps" Among the directions, the second approach of method-parallelism receive much attention Here it is worth to mention
th domain d c mposition methods a d the parallel splitting u methods In recent years we hav developed an another parallel method for mathematical p ysics problems It is the method of para-metric extra olation In this paper we give an o erview of our results concerning this method for constructing parallel algorithms for some problems of mathmatical p ysics
2.1 From the method of parametric correction of difference schemes
The idea of the method is orginate fom th method of parametric correction of difference schemes proposed by Belotserkovski and his colleagues [3] in 1984 Their goal the was to solve the conflict between the stability and hig order approximation of difference sch mes for hyperb lic problems a d to increase the effectiveness of iterative processes for second order elliptic problems
In order to do this for each BVP the constructe a manifold of diferenc schemes depending on two or more parameters instea of one as it was usu lly d n before Due to this manifold of difference schemes th y could get n w pro er es of the difference scheme which is a appropiat e linear combination of basic difference sch mes Speakin roughly, the idea of th method of parametic correction of difference sch mes is that a "good" difference scheme may be obtained in the result of combining "bad" o es by the suitable selection of parameters The realization of this method leds
to th concept of th gen ralized difference scheme as a combination of the basic difference schemes with some weights, whic was discussed in [4] and applied for studying discontinuous solutions of the wave equ tion in [5] The results of computation in the latter p p r allows to con lude that the consideration of a family of difference schemes constructed by special way not only opens a possibility
• This work is s u p or t ed by t h e Nat i o al Basic Research Program in Natural Sciences. TH\J VI EN
TRU~~ 'f N
VA eN Quae GIA
Trang 2to increase the effectiveness of difference scheme but also reaches more adequacy of discrete model
to the phen mena studied Indeed, it is proposed to construct the discrete model of continuous media from several discrete models, each of those isnot adequated to the continuous model But the difference between the discrete models is organized so that they may be controlled The family of
these models due to their constructive chara ter may be made ratio al and when being considered as
a new model can possess new properties which each separate mo el does not have For this reason the method of parametric correctio of difference schemes is considered as a new principle in the construction of discrete models in mechanics ofcontinuo s media
The method of parametric correction of difference schemes were used by o rselves in [7] for
constructing generalized difference schemes quasimonotone and having high order of accuracy for some equations and systems It is in the latter paper, the conflict between the stability and high
order approximation solved not completely in [3]was solved fully But the problem, in which we are interested most, is the constructio of efficient iterative meth ds for solving BVPs for elliptic
equations on both differental and difference levels
2.2 To the parameter extrapolation method
Below we present the idea of the parameter extrap latio method for a general equation
Let A be a linear symmetric, positive definite o erator in a Hilbert space H. Consider the equation
This equation may be solved by known iterative methods with the rate of convergence depend-ing on the ratio M [ i , Here M and m are maximal and minimal eigenvalues of the operator A,
respectively In the case, where H is of infinite dimension and 0isthe limit point of the spectrum
of the operator, in general o e has not obtained or obtained very bad results of the convergence rate
of the methods In order to overcome this difficulty, and also to increase the convergence rate ofthe iterative processes, we propose instead (1) to solve some perturbed problems
where P is alinear symmetric, positive definite operator suitably selected for every specified operator
A. Then, we extrapolate by the parameter e the soluto s of (2), i.e., take the combination
N+ l
ir LIkUe / k
k = l
(3)
Ik = k !(N +1- k)!
be an approximate solution of (1) For the error ofthe approximate solution we have the estimate
I U C - u * 11 :::;CeN+1.
This re ult is obtained with the help of the expansion
N
* '\' k N+ l
k= l
where u" is the solutio of the original equato (1), v» are elements of H independent of e , We I S
uniformly bounded in e, N is an integer depending on A.
The mentio ed above fact is pro ed in [1 ]
Thus, the direct solving of (1) is replaced by solving N perturbed problems (2) with the param-eters elk, (k = 1, ,N). These problems may be solved simultaneously on processors of parallel computers The advantage of this method is that known iterative meth ds applied directly to (1)
Trang 3are slowly converged, even may be, are deverged, while known iterative metho s applied to (2) wil
converge fastly with the rate of geometric progression
Comment 1 (Tikhonov regularization) The equation (2) in some sense is the Tikhonov reg
u-lariz~d equation for (1) (for Tikhonov regularization see e.g [28]) Here we extrapolate is solution
depending on the regularization parameter for obtaining the solution of the original equatio (1)
Comment 2 (Richardson extrapolation) In the proposed method, the extrapolation is perfomed
by a small parameter introduced into the original equation in order to make some perturbation Differently fom this, the well-known Richardson extrapolatio (see, e.g [23]) is by the stepsize of
discretization of differential problem Due to this extrapolatio the order of accura y of diference
scheme is increased It is possible to be realized with the help of the asymptotic error expansions to
finite difference schemes
In the following sections we shall summarize results of using the method of parametric extrapo-lation for some problems on differential and difference levels It should be noticed that for differential
problems, in order to apply this method, the most tmportan t s t e p is the r du c ti o of the pro bl ems und e consideration to an equation unih a symmetri c , p os itiv e defi ni a nd c om plete ly co n tinuous
i n a Hilbert space Therefore, in Sections 2 and S we only ske tch h w t o r educe o r igi nal BVPs to
c orr es ponding operator equations tn Hilbert space
3 THE DIRICHLET PROBLEM FOR SECOND ORDER ELLIPTIC EQUATION
WITH DISCONTINUOUS COEFFICIENTS Let 0 be a bounded domain in the m-dimensional Eucledean space R ' " with Lipshiz boundary
S Denote by 0+ a proper subdomain of 0 with boundary r ,LI the outward normal to r
Consider the boundary value problem (BVP)
t J = l
a ( x ) ~ 6 >0, [u] = 0, [:~] r = 0, u ls = <1>,
(4)
where [u]r is the jump of u through r : [u] = u + - u- , u ± ( x l = u( x ) , x E o ,
conormal derivatives of u ±
By the introduction of a boundary operator K , defned as follows
K: g - , w ]r,
where 9 is a boundary functio defined o r ,w is the solution of the problem
~~ : l r = g, [ ~~ = t0, the problem (4) is reduced to the operator equation
Kg = F,
(5)
(6)
here F is afunction depending linearly on f and <1>.There was proved that Ki s a linear, symmetric,
positive definite and completely continuous in the space L2 ( r ). Instead of solving (6), we consider perturbed equation
where I is the identity operator
Trang 4This equation is lead from the perturbed problem
LUe = f(x) , x EOW, u el8 = <1> ,
E: +' U e r = 0, - = 0,
The simple iterative method applied to (7) is converged with the rate of geometric progression,
while the iterative method of Osmolovskij & Rivkind [24] for the original problem (4) only is con
-vereged with the rate O(l / N Q ) , where N is the number of iterations, a is a number depending o
the smooth ess of the solution It is interesting that the realization of the iterative method for (6)
and for (7) leads to the successive solutio of a sequence of BVP in each of the subdomains, where the Neumann condition on the interface is step by step made more precise
(8 )
Comment 3 (domain decomposition methods) The proposed above method is applicable for
the problem where the domain °consists of two sub domains which except for the interface have
their pro er b undary Thus, the approximate solution is constructed by the extrap lation of the
solutions found by a domain decompositio meth d It should be notce that at present domain
decomposi o methods attract great attention from many researchers (see c.f [ 2, 2 1 , 22, 26, 29 ]
due to the needs to solve BVPs in geometrically complicated domains Besides the way of making
the Neumann boundary values more precise on the interface as in our work [ 1 ] , many other authors
proposed to do so with the Dirichlet boundary values or alternatively exchange the Neumann and
Dirichlet boundary values
Comment 4 (boundary element methods) After red cing the original and the associated
perurbed problems to boundary operator equatio s we don't intend to solve them by n merical
meth ds, for example, boundary element methods, but only use them as means for studing the
convergence of iterative process for BVPs
4 BVPs FOR BIHARMONIC, BIHARMONIC TYPE AND
TRIHARMONIC EQUATIONS
4.1 Solving BVPs for the fourth order differential equatio by the reductio of them to BVPs for
the second order equations with the aim to use alot of efficient algorithms for the latter ones attracts attento from many researchers Namely, for the biharmonic equato t ::.2 u = f wih the Dirichlet
boundary condition, there isintensively develo ed the iterative metho , which leads the problem to two problems for the Poisson equatio at each iteratio (see e.g [ 2 ,25 ] But unforunately, in these
works the convergence rate of the iterative process either was not obtained [ 2 ] or isvery low, namely,
is of order 0(1/ N) , where N is the number of iterations [ 2 ] In order to elab rate faster algorithms
for the biharmonic equation, in [8] irst time we ap lied the parameter extrap lation technique to this
equatio For reducing the Dirichlet problem for the biharmo ic equatio to a b undary o erator
equatio we defned the boundary operator via Green functions as was done in [6] The result of
computation implemented in [9]confirmed the advantage of the parametric extap latio technique
4.2 The technique for reducing BVP for biharmonic equatio to b undary operator equation in the
mentioned above papers is improved in our further works when being applied to a mixed BVP for
the biharmonic equation [ 16 ] and for BVPs for biharmonic type equation [ 13 - 15 ] Below we briefly
demonstrate this technique for the Dirichlet problem
aU I
u l r = 11, 0 , av I'= U v·
Here °is a b unded domain in R m, t :: is the Laplace operator, a 2: 0, b 2 : 0
4.2.1 Suppose that a > °and
Trang 5We introduce boundary operator B by the formula
a U I
where V a is afunction defined on I', U solves the problems
Here L1, L are the factors in the factorization of L , whos formulae are given III [131 Then the
problem (9) is reduced to the o erator equation
with B = B * >0 and completely continuous in L (f) , linearly expressin through U o, u v, f Rather than (11) we solve the perturbed equatio
This equation is obtained from the perturbed problems
L U h == !:: :: 2 n- a!:: :: u + b U n =f( x ) , x E0,
u "l r =Uo, 0(~!:::: U 6 - u ) I + a U " 1 = UV '
where J - L = ~ (a + J a2 - 4b)
It should be emphasized that the simple iteratation meth d for the equation (1 2 ) is convergent
with the rate of geometric progression and is realized by solving a s quence of BVPs for second order
equatio s, while the iterative method for the biharmonic type equatio (9) in [1 is not proved to be
convergent
4.2.2 Now co sider the case, where and the condition (10) is not satisfied For brief we set U o =
U v = O
We introduce a mixed domain-boundary operator B , defined by the formula B : w - > Bw ,
where
D + bu
u isthe function found from the problems
!:::: u = v, x E0, u = o.
It was proved that B = B * > 0 and B is bounded in the space L 2( f ) x L2(0)and has expansion
B = Eo + h,where E o = B( ~ > 0 is completely continuo s, h is a projector on L ( 0) , namely,
Then the BVP isreduced to the o erator equatio
here for brevity we omit the concrete expre sion of F Ifapply any iterative method immediately, for
example, the two-layer iterative scheme to (13) then we c n not say anythin ab ut its convergence
Hence, instead of (13) we consider the equation
(B +O JdW 60 = F , ,0< 0 < 1
where II is a projector on L2(f) ,i.e I1 w = ( v~» )
(14)
Trang 6We have B +5I , 2:B i , +5I 2:5 I Consequently, two-layer iterative scheme for (14) will be convergent
The perturbed problem (14) islead from the original problem, where the boundary condition ~~I = 0
4.3 The technique for reducing BVP to boundary operator equation in order to apply the metho of
a l
au l
Bv u , = - av r'
(15)
where Wo is a function defined on ran u solves the problems
Then the problem (15) is reduced to the operator equation
with B = B * > 0 and completely continu us in L (I'} , F linearly expressing through j The
(aU 6 2 ) I
and the realization of the iterative method for it leads to the solutio of three Dirichlet problems for the Poisson equation
of parametric correction of difference schemes [3 - 5, 7], where the difference o erator approximating
a differential equation is made perturbed, is that in our works [13, 14-16, 1 ] we consider a family
of BVPs with one perturbed boundary condition Hence, after the reduction of them to boundary operator equation we obtained a family of boundary o erator equatio s depending o a parameter
and the extrapolation is performed by this parameter
5 ACC ELERATING THE C ONVERGENCE RATE OF ITERATIVE METHODS
FOR SOLV I NG G RID EQUATIONS AND DEGENERATE S Y S TEM
OF ALG EBRAI C EQUATIONS
5.1 The design of fast algorithms for large-scale systems of linear algebraic equatio s is a very actual
problem attracting great attentio from both mathematicians and engineers These large systems usually arise in the result of discretization of BVPs for two- or three-dimensional elliptic equations
on thin grids There are a lot of works concerning this problem (see e.g mo ographs [2 , 27] an
references therein) In [10, 11, 17] we proposed to use the meth d of parametric extrap latio for
Trang 7Consider the operator equation
Tk +
(17)
i1B~A~i2B, i2~i1>0.
c 1R < A < c2R, C2 ~ C 1>O
specifically as follows:
triangles method to (2)
R2R3 +R1R3 +YhR1R2R 3, where h is the grid step for discretization of differential problems, and
[ 17 ]
extrapolation by the regularization parameter a (see [19]) We have obtained the following estimate
I WE - u *11 ak + 1
Trang 8where u* is the n rmal soluto , UC is the extrapolated solution by k:+1solutions of (18), A rn in is
the smallest eigenvalue of A From this estimate we establish that if applying the simple iterative
meth d [22,27]to (18) then for achievin the normal soluto of [La] with the given accuracy e by
usin the parametric extrap latio wereduce the computational amount G times in comparison with
using only one shifted equato (18),
G= [ k +l)(k: 2)ck /(k+ ) ]
In the case if the system (la) is inconsistent then the solutio of (18) d e not approximate the
normal soluto Nevertheless, we proved that extrap lating k +1solutio s of (18)with parameters
a (j = 1, , k +1) we get an approximatio ofthe n rmal solution wih the estimate
Il U e - u ' ll a k
- "- -: , ; < , -
"-Il u 'll - A ~n i n
Remark. In the case when the matrix A is n t symmetric the Tikhonov regularization leads to the
solution of the system
Using the extrapolatio by the parameter a we obtained the same result as (19) for both consistent
and inconsistent systems (la)
6 CONCLUDING REMARK
The major work in the realization of the method of parametric extrapolation is the parallel
soluton of the perturbed problem with some various values of the parameter, each on a processor
The computatio of the extrapolated solution as a combination of the perturbed solutions isonly the
last simple work Thus, the degree of parallelization of the method is very high
Acknowledgement. We wish to thank an anonymous referee for his valuable comments and
sug-gestions which improved the paper
REFERENCES
[ Abramov A A and Ulijan va V.I On a method
sin ularly small parameter, J of C omp o Math
(Russian)
[2] Agoshkov V I Lebedev B.I., Poincare-Steklov operators and domain decomposition methods
in variational problems, In book: Computing Proces s es and System s , Issue 2, Nauka, Moscow,
1 85,173-227 (Russian)
[3] Belots rko skij O M Panarin A.I Tshennikov V.V., Method of parameter correction of
dif-ference schemes, J of Co mp o Math and Math Phy s. 24 (1) (1984) 65-74 (Russian)
[4] Belots rko skij O.M., Panarin A I Tshennikov V V, Generalized difference schemes and the
method of parameter correction of difference schemes, In book: Cybernetic s and Computing
T ech ni c , Isue 1, Moscow, 1986,99-117 (Russian)
[5] Belotserk vskij O.M Panarin A.I Tshennikov V V., Discontinuous solutions and generalized
difference schemes, In book: C ybern e ti c a nd Computing Technic s, Issue 2, Moscow, 1 86, 95
-104 (Russian)
[6] Dang Quang A, On an iterative method for a boundary value problem for a fourth order
diferential equation, Math Phys and Nonlin Mech. 44 (10) (1988) 54-59 (Russian).
[7] Dan Quang A, Constructio of quasimonotone generalized difference schemes having high order
of ac uracy for some equations, J of C omp o Sci and Cyber 6 (2) (1990) 15-20 (Vietnamese)
[8] Dang Quan A, Applicatio of extrap lation to constructing effective method for solving the
Dirichlet problem for biharmonic equatio , In s titute o f C omputer Sci e nce , Pre print No.5, 1990
for solving biharmonic type equation with
and Math Phy s 32 (4) (1992) 567-575
Trang 9[9)Dan Quan A,Numerical method for solving the Dirichlet problem for fourth order differential
equation, Proc of the Sr d National Conferenc on Gas and Fluid Mecha ics, Han i 1991, 195
-199 (Vietnamese)
and Cyber. 11 (2) (1995) 1-6
coeffi-cients, Journal of Comput and Applied Math. 51 (2) (1994) 193-203
-tio , Vietnam Journal of Math. 22 (1&2 (1994) 114-120
[14) Dan Quan A, Iterative method for solving the s cond boundary value problem for biharmonic
type equation, J of Compo Sci and Cyber. 14 (4) (1998) 66-7
[15) Dang Quang A, Mixed boundary-domain o erator in ap roximate solution of biharmonic type
September 17-2 , 1 97, Sci and Tech Publ Ho s , Han i, 1999,47-55
Sci and Cyber. 13 (4) (1997) 33-45
Vietnam Journal of Math. (ac epted)
Decompostion Methods, SIAM, Philadelphia, 1988
[2 ) Marchuk G.1 Methods of Numerical Mathematics, Nauka, Mos ow, 1989 (Russian)
(1981) 33-38 (Russian)
method for elliptc partal differential equations, Journal of Comput and Applied Math 8
[28) Tikho o A.N., Goncharsky A.V., Stepan v V.V., and Yagola A.G., Numerical Methods for the Solution of Ill-Posed Problem s, Dordrecht - Kluwer Acad Publ, 1995
El-ement Metho s, Proc of the Fifth Japan-China Symposium on Boundary Element Methods,
Received December 14, 1999
R evise d J an u ary SO, 2001
In s titute of Information Technology