VNU Joumal of Science, Mathematics - Physics 23 (2007) 201-209 Fully parallel methods for a class of linear partial differential-algebraic equations Vu Tien Dung* D epartm ent o f M athem atics, M echanics , ỉrựòrmatics, College o f Science, VNU 334 Nguyên Trai, Thanh Xuan, Hanoi, Vietnam Received 30 N ovem ber 2007; received in revised fo rm 12 December 2007 A b s t r a c t T h is note deals vvith tvvo fư lly parallel methods fo r solvin g linear partial d iíĩe re n tia lalgebraic equations (P D A E s) o f the form : Aut + B A u = /( x , t) (1) where A is a singular, symmetric and nonnegative matrix, while B is a symmetric positive define matrix The stability and convergence of proposed methods are discussed Some numerical experim ents on high-perform ance computers are also reported Keywords: DiíTerential-algebraic equation (DAE), partial diíĩerential-algebraic equation (PDAE), nonnegative penciỉ o f m atrices, parallel method In tro d u c ỉio n Recentỉy there has bcen a growing interest in the analysis and numerical solution o f PDAEs because o f their importance in various applications, such as plasm a physics, magneto hydro dynam ics, electrical, mechanical and chem ical engineering, etc A lthough the numerical solution for differential-algebraic equations (DAEs) and (PDA Es) has been studied intensively [1 ,2 ], until novv we have not found any results on parallel methods for PDAEs This problem will be studied here for a special case The paper is organized as follows Section deals with some properties o f the so called nonnegative pencils of matrices In Section we describe tw o parallel m ethods for solving linear PDAEs, w hose coefficients found a nonnegative pencil o f matrices T he solvability and convergence o f these methods are studied Finally in section some numerical examples are discussed Properties of nonnegativc pencils of matrices In what follows we will consider a pencil o f matrices {^4, B }, w here A G R nxn is a singular, symmetric and nonnegative m atrix vvith rank (j4) = r < Tí and B € R nXn is a sym m etric positive dìne matrix Such a pencil will be called shortly a nonnegative pencil We begin w ith the íollovving property o f nonnegative pencils • Tel.: 084-48686532 E-mail: duzngvt@gmail.com 201 202 Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209 P ro p osition Any nonnegative pencil { A , B } can be reduced to the Kronecker-Weierstrass form { d ia g (/r , On- r), d ia g (D , /n - r ) } with a symmetric andpositive defìne matrix D e R r x r Here Ir and On-r stand fo r the identity and zero matrices o f appropriate dimensions respectively Proof T he sym m etric and nonnegative matrix A can be diagonalized by an orthogonal matrix u , such as ƯT A U = d i a g ( A i , A r , 0, 0), vvhere Ai > Ả2 > > Ar > are positive eigenvalues o f A Defm e two m atrices s := d i a g ^ A i ) ^ , 1, 1) and Ô := S U T DUS Clearly, Ỗ is also symmetric and positive deíine Morever, S U T A s = d ia g (/r , O n_ r) Now let It is easy to veriíy that Bị and its Schur complement defm ed by B\ — B ìB ị 1B$ are also symmetric and positive define Putting p = (lr B 2B Ị l \ P:= u V, ^ J; B - Ị B ì - B 2B ^ B z \ { Bi) and q = ( b ị ' b /„ _ r ) we get Ế = P Ỗ Q and P d i a g ( / r , On-r)Q = d ia g ự r , O n-r) From the last relations it follows P ~ l Ỗ Q ~ l = Ổ and p ~ x d ia g (/r , O n_r )Q _ = d ia g (/r , On-r) Finally, letting p = d ia g (/r , B ị X) we find P d i a g ( / r , On-r) = d ia g (/r , „ _ r ) and P ỗ - d ia g (D , In-r), where D D\ B$ and D t = D > Thus, there hold decompositions M A N = d ia g ( /r , On-r), M B N — d ia g (5 , In- r) with nonsingular matrices M := P P S U T and N := Ư S Q ~ l , vvhich was to be proved In \vhat follows, vve need two Toeplitz tridiagonal m atrices p and Q o f dimension k X k, vvhere as a rule k is much greater than TI ( p = - \ _1 ị - -1 - * ; Q = - *• -1 *• -1 - ) \ ^ *• -! -1 / Clearly, if D R rx r is a symmetric and positive dìne m atrix, then the Kronecker product p D and be a positive parameter P ro p osition Let the pencil {A, B ) be nonnegative and ỉet M and N be two nonsingular matrices, such as M A N = d i ă g ( I r ) O n - r ) , M B N = diag(£>, / n- r ) , where D r — D > Then One can explicitily deỳìne two nonsingular mrices K and H, tranforming the pencil {Ik ® A, jf?p B} to the corresponding Kronecker-Weierstrass form {diag(//c, 0*;(n_ r )), diag ( Ặ ĩ > , ( „ - r ) ) } , (3) with symmetric and positive defìne matrix D Proof A ccording to Proposition 1, the nonnegative pencil {Jfc A2 > > > Then, the matrix s := p I n- k is diagonalized, (U ® I n - r ) S ( U ® / „ - r ) = A In -r = d ia g (A i/n_ r , A fc/„ _ fc) Let M := Ik ® _ M ; N := I k N-,A := Ik d ia g (/r , On-dir)', B := Ặ p diag(L>, / „ - r ) Clearly, M ự k ® = i4 and M ( & P ® B ) N = B Now we define tw o special matrices £ € R rx n and E e R (n - r )xr as / 0 0 \ d i a g ( r , r ® Ịn-À:), h J \ A \ M \H 0 0 ((Ê \ ằô 0 r-4 & = 0 := ; E2 = • » and put ị0 ^ ® /* )T (£ 0 0 / đ /fc)T ) Further, let D := p ® D \ J X := := diag(/A:r ,ỉ7 /n_ r ) ; J := diag(/fcr , / l - ® / n-r ) Finally, set /c := N A i h h - We vviil show that /c and H transíòrm the pencil {Ik A , jp P ® B } to the canonical form (3) Indeed, a sim ple calculation shows that £ ii £ = diag(/*;r , ơfc(n_fc)) and t i B ị ĩ = d i a g ( ^ Z ? ,5 ) ^ Futher , KựkA)H = A ị2h h = d ia g (/fc r,0 * (n -r))- Similarly, ® B )H = M x B h h h = diag (^jjD , h(n-k))- Thus the proposition is complete Fully p a llel m ethods fo r lin e a r PD A Es In this section vve study the numerical solution o f the following initial boundary value problems (IBVPs) for linear PDAEs: A u t + B A u = f {x, t), E u ( x , 0) = uo(x), ÍỈ, í £ (0 ,1 ), X u ( x t t) = 0, d ,where A u := Gn, X € dữ, = {^ (^ Ìi •••>Zd);0 < Xi < 1;» = l , d } , (5) (6 ) (7) »= *• A, B, E are given n x n matrices and the pencil {v4, B} is nonnegative Further, u, f are vector íunctions, u, / : Q X [0,1] —♦ R n and the gi ven íìinction / ( X , t) is assumed to be sufficiently smooth We propose tw o parallel methods for solving the IBVP (5)-(7) where the parallelism vvill be períorm ed across both the problem and the method According to Proposition 1, there exist nonsingular matrices M, N such as M A N = d ia g ( /r , „ _ r ) and M D N = d ia g (jD ,/n_ r ), where as above, r=rank (i4) and D = D T > We will partition N ~ xu , M f and UQ into tw o parts, N ~ l u := (vT , w T)T \ M f := (F ị , F Ị ) T t uo := (v q , w Ị ) t , vvhere VũyV, F\ € K r and w0)w, F2 G R n~r From (5) we get M A N § ị ( N ~ lu) + M B N A ( N ~ ì u) = M Ị , or equivalently, Vt + D A v = F \ , and A w = F2 Further, as in D A E ’s case, the initical condition ( ) cannot be given arbitrarily It m ust satisíy some so-calleđ hidden constraints Indeed, suppose that the matrix E N is partitioned accorđingly to the 204 Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209 ìu{x, 0) such that E N = ịẸ} partition o f the vector N where E \ , E ị are square matrices T X T and ( n — r ) X ( n - r ) , respectively F o r the sake o f s im p lic ity , w e assume that Ẽ = 0, E ị — a n d E\ is nonsingular T h e n co n d itio n (6 ) c a n b e revvritten as E iv(x,0 ) = ^ o (x ) and E$v(x, ) = tơ o (z )- From the last re la tion s, it it cle a r th a t the value w (x, ) w ill not p articipa te o f d im e n s io n in fu rth e r co m p u ta tio n s Besides, the in itia l co n d itio n l í o ( i ) = (U(f (:r), tL>(f(x))7 satisfies a hidden co n stra in t E E ị lvo(x) = wQ(x) (8) T h u s, IB V P (5 )-(7 ) is s p lit in to an IB V P fo r the p arab o lic equation Vị + D & v = F \ , (9) v(x, 0) = E ì «o(x), X í) v(x, t) = , x e dQ, t € (0 , ), (10) (11) and a B V P fo r the e llip tic equation A w(x,t) = 0, w = F2 (1 ) X dũt t e (0 ,1 ), (1 ) A p a lle l fra c tio n a l step (PFS) m ethod, proposed in [3 ] and developed in [4 ], w ill be e xp lo ite d fo r s o lv in g the IB V P ( ) - ( l l ) F o r by c h o o sin g a m esh size h > this purpose, w e firs t d iscre tize in the sp atia l va ria b le X= ( li, .,Xd) and a pproxim ate the p ro b le m in the discrete d om ain Q /, by u sing the second o rd e r centered d iíĩe re n c e ío rm u la It leads to the O D E (1 ) Vh{0) = vOh (15) T h a n k s to th e s y m m e tr y a n d p o sitiv e d e f in ite n e s s o f D , in m a n y ca.ses, th e m a trix H and p o s itiv e d e íin e F o r exam ple, in 1D case (d = 1) and using the H = ^ ĩL D, m atrices w here p and Q dìned b y (2 ) w e get ( Q - I - I Q H is s y m m c tric = ẢP ộộ D \ - I L = for D case -I Q -I \ (d = -I Q / 2) F u rth e r, suppose H can be split into the sum of symmetric, painvise commutative and p o s itiv e semidefinite matrices Hk, d H = Y J Hk\ H Ị = H k > 0; HỵHị = H i H M = l, d ; k= i We d is c re tiz e the tim e in te rv a l [0 ,1 ] w ith step T > and a p p ly the PFS m e tho d [4 ]; (1 ) Vu Tien Dung / VNU Jo u rn a l o f Science, M athem atics - Physics 23 (2007) 201-209 205 A lg o r it h m PFS Step In itia liz e v ° : = Voh Stcp For g ive n m > and V™ (an a p p ro xim a tio n o f Vh(mr)) fin d vm+1’k b y s o lv in g (in p a lle l) systcms o f lin e a r equations ( /+ y f W w here Ffh := +1'‘ = ( / - y £ j= (17) H i) v m + T ~ F íkh /,( ( /; + / ) r ) Step C om pute ^ ' m +1’* + (1 - É - (1 ) d fc=l N o tc that the lin c a r systems (1 ) can be solved by any p arallel ite tiv e m ethods [5 ,6 ,7 ,8 ,9 ] N ow vve tu m to the B V P (12 )-( 13) F o r its s o lu tio n vve im p le m e n t the p a lle l s p littin g up (P S U ) m ethod, c proposed b y T L u , p N e itta a n m a k i, and X Tai [3 ] D is c re tiz in g the B V P ( ] )-( 13) One obta in s a large-scale system o f lin e a r equations Lw - g, (1 ) vvhere L is a sym m etric positive de fin e m atrix o f d im e nsion p X p, vvhere p = p(h) dep en d s on the d iscre tiza tio n param cter h A ssu m c tha t L can be decom poscd in to the sum m o f s y m m e tric and positive defmc matrices, vvhich commute with each other 771 L ^ L/ị — L ị ^ 0; LfịL/j = L j L ị ) z, J = 1?771 (20) i= i T he PSU m ethod consists o f the fo llo w in g steps: A lg o r it h m PSU Step Choose an in itia liz a tio n Step S upposing WJ up is k n o w n , w e co m p ute the ĩra c tio n a l stcp values L iiẮ p + t t z= f — L k w \i ^ = , ,771 (2 ) k=2 ,k& Step For chosen parameters set m w j + _ ^Ịj_ + -|- (1 — £Jj-)uíJ (22) »=1 j system (2 ) can be solved b y p a lle l processors T h e o re n i The PFS-PSƯ melhod (17)-(18),(2I)-(22)Ịor solving the IBVP (5)-(7) with a nonnegative pertcil {.4 , B ) and the consistent initial condiiions (6), (8), is convergent N ote that fo r d iíĩe re n t Proof T h an ks to the n o n n e g a tiv ity o f the p en cil { A , B } , w e can s p lit the IB V P (5 )-(7 ) in to the IB V P ( ) - ( ] I ) and the B V P (1 )-(1 ) Theorem s 4.11 and 5.2 [4 ] ensure that the PFS m ethod in the s y m m e tric and co m m u ta tive case is stable p ro vide d r convergent w ith global e rro r (h -f < 2{d m a x | | i/ f c | | } _ \ í), we can revvrite the last system of equations as (/m -1 ® B)U = F, + ^ (P ® (23) w here the m a trix p is d eterm ined by (2 ) B y P ro p o sitio n w e can fìn d n o n sin g u la r m atrices /4, and p u ttin g to the K ro ne cke r-W eie rstra ss fo rm (3 ) M u ltip ly in g b o th H ~ l u = ( V T , W T)T ; K F = ( F 1r , F2r )T, V, F\ vvherc € R(M~ì'>r R ( W - )(n - r ), w e com e to the system " + (24) w = w here as in P ro p o sitio n 2, D Ẽ2, (25) is a sym m e tric and p o s itiv c d e íìn itc m a trix N o tc that the boundary con- d itio n (7 ) has been in clu d cd in E quation (23) N o w let H~l (u0 ( x i ) , uỊịxM-ị))1 = w here Vo € R ( A /- )r and ÍVo R ( M _ )(n _ r ) T h e n , the in itia l c o n d itio n (6 ), w ith y(0) = (VoT , E = I v0 W Ị)'1 , becomes (2 ) M o re o ver, th c in itia l c o n d itio n (6 ) in u st sa tisfy a h id de n co n stra in t Wữ = Í2 (0 ) For s o lv in g the ỈV P (2 -2 ) in p a lle l, the PFS m ethod describcd above fo r the p ro ble m (1 )(1 ) can be a p p lie d We sh a ll not g ive the le n g th y details Numerical experiment C o n s id c r the b ou n da ry - value problem (5 )-(7 ) w ith the fo llo w in g data: ị\ n = ;d = 2M = \0 T he fu n c tio n u f( x , = 103( í x i ( l - 0\ / ;B = - 0) \ -0.5 l\ ;E = I (27) 1/ í) is chosen such that the exact s o lu tio n o f the B V P (5 )-(7 ) is x \)x ị{ \ - X2), tx \{ - Xl)X2(l - X2),ÍX1X2(1 - X i)(l - X 2))1 U s in g n o n s in g u la r m atrices /1 Aí = Ị \0 -1 \ 7V = / / 0\ 0 ì) (28) 207 Vu Tien Dung / VNU Journaỉ o f Science, Mathematics - Physìcs 23 (2007) 201-209 w e can s p lit the IB V P (5 )-(7 ) in to an IB V P fo r the p a b o lic equation Vt - W{vXìXì + VX2X2) = v(x, 0) = v(x, t) = F i ( x i ,X2,t) X íì XG ỡ íì (29) and t€ c and (0 , 1) and a B V P fo r the e llip tic equation ^ 12 x2 ) -^2 ( ^ ì %2ĩ tu(x, í) = T h e PFS m ethod and PSU m ethod [4,3] are im p le m e n te d in MPI and executed on a L in u x C lu s te r 1350 w ith e ig h t c o m p u tin g nodes o f 51.2G FIops Each node contains tw o In te l X eo n dual core G H z , 2G B Ram T h e ĩo llo v v in g table show s the dependence o f the e rro r o f the a pp ro xim a te s o lu tio n s on the num ber N = ị w h ile the tio p N Residuaỉ Residuaỉ r ti* rem ains constant 16 24 30 40 50 60 0.5 0 0 0 00 0 0 0.000014 0 0 0 0 0.2 0.0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 In vvhat fo llo \v s , w c stud y the re la tio n bet\veen the tota l (C P U ) tim e spent on the períbrm ance o f a program , the spccdup and the e ữ ìc ie n c y o f th is períbrm ance T h e specdup o f the p eríbrm ance is d efined as s = Ts/Tp, w h e re Ts (Tp) is serial e xe cu tio n tim e (p a lle l c xccu tio n tim e ), respectively T he eíT iciency o f the p crform a nce is d cte rm in e d as E = s /p , vvhcre p is the n um be r o f processors T h e resnlt o f an c x p c rim c n t w ith PFS m ethod fo r (2 ) is reported in the fo llo w in g table Table S peed up and E íĩiciency on C luster 1350 w ith N = I2 Processors Toltal times(minutes) Speedup Ẹffìciency 10 252 126 62 43 37 32 8 7.8 1 97 0.85 U s in g processors o f C lu s te r 1350 and a p p ly in g the PSU m ethods to the B V P (3 ) w e observe that the to ta l tim e increascs tog cth er vvith the g ro w th o f the num be r N = ị Table N Toltal times(seconds) 24 30 40 120 180 300 F o r b e tte r convergcnce, w e use o th e r m ethods, such as the p a lle l Jacobi m ethod [5 ], the p a lle l SOR R e d /B la c k [6 ,7 ,8 ,9 ] m ethod T h e p a lle l Jacobi m ethod and P arallel S O R R e d /B la c k m ethod are im p le m e n te d in c and M P I and executed on node o f A I X C lu s te r 1600 o f c o m p u tin g nodes, vvhose tota l c o m p u tin g p o w e r is 40 G F lo ps Each node co nta in s C P U Pow er4 b it R S IC 1.7G Hz B e lo w are some results fo r p a lle l Jacobi m ethod and p a lle l SO R R e d /B la c k m ethod Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209 208 T able S peed up and Eflficiency on node o f C luster 1600, N = 240 Parallel Jacobi method Processors Toltal times(seconds) 937 484 232 191 Speedup 1.94 4.0 4.9 Effìciency 0.97 0.81 155 6.05 0.75 A lth o u g h t the p a lle l Jacobi m ethod converges íaster than the PSU m ethods, it is re ly used as a p a lle l so lv e r fo r e lip tic problem s T able S peed up and E ffìciency on C lu ster 1600 w ith N = 1200 Parallel Red - Black SOR method Processors Toltal times(seconds) 275 154 64 54 83 Speedup 1.79 3.31 4.3 5.1 Effìciency 0.9 0.83 0.72 0.64 T h e n um b e r o f ite tio n s needed fo r convergence and the to ta l tim e fo r the serial co m p u ta tio n o f Red - B la c k S O R and Jacobi m ethod are g iven in the fo llo w in g tables T able N u m b er o f Iterations o f sequential Red - B lack SO R and Jacobi m ethod N SOR Jacobi 60 284 10599 120 565 39680 180 836 86119 240 1101 149311 300 1351 T able Total tim es o f R ed - B lack SO R m ethod and Jacobi m ethod N 60 SOR(seconds) Jacobi (seconds) 120 45 180 200 240 12 720 300 19 T h e Red - B la c k S O R m ethod is c le a rly the íastest o n c in term s o f serial tim c and the num be r o f ite tion s T a ble 1,3,4 show th a t w hen the n um be r o f processors increases, the speedup increases T h e actual speedup is sm a lle r than the ideal speedup because the co m m u n ic a tio n cost is r đ a tiv e ly h ig h e r w h e n im p le m e n te d and executeđ on a L in u x C lu s te r 1350 and A I X C lu s te r 1600 From T a ble 1,3,4 it is cle a r th a t the m ore processors are used, the c o m m u n ic a tio n cost increases, and the e íĩic ie n c y decreases Acknovvlcdgements T h e a u th o r thanks Prof Dr Pham K y A n h fo r su gg e sting the considered to p ic and fo r h e lfu l discussions P a rtia lly supported b y the V N U ’ s K e y P ro je ct Q G T 05.10 Reĩerences [1] [2] w Lucht, K Strchmcl, c Eichler-Licbcnow, Lincar Partial DiíTcrcntial Algcbraic Equation, Report No 18 (1997)430 w Marszalck, z Trzaska, A Boundary-valuc Problem for Lincar PDAEs, IntJ.Appl.MQth.Comput.Sci., Vol 12, No (2002) 487 [3] T Lu, p NeittaaniTiaki, Tai, A Parallel Splitting-up Method for Partial DiíTerential Equations and Its Apptications to Navier-Stockcs Equations, Applied Mathematics Letters Vol 4, No (1992) 25 x.c Vu Tien Dung / VNU Journal o f Science, Mathematics - 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