Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve methods to solve the lnear operator equaton Au = f, (1) posed on a fnte dmensonal Hlbert space V equpped wth an nner product (, ). Here A : V V s a symmetrc postve defnte (SPD) operator, f V s gven, and we are lookng for u V such that (1) holds. The X-Z dentty for the multplcatve subspace correcton method for solvng (1) s ntroduced and proved n Xu and Zkatanov [2002]. Alternatve proves can be found n Cho et al. [2008] and Vasslevsk [2008]. In ths paper we derve the X-Z dentty from the auxlary space method Nepomnyaschkh [1992], Xu [1996]. A basc lnear teratve method for solvng (1) can be wrtten n the followng form: startng from an ntal guess u 0, for k = 0, 1, 2, u k+1 = u k + B(f Au k ). (2) Here the non-sngular operator B A 1 wll be called terator. Let e k = u u k. The error equaton of the basc teratve method (2) s e k+1 = (I BA)e k = (I BA) k e 0. Thus the teratve method (2) converges f and only f the spectral radus of the error operator I BA s less than one,.e., ρ(i BA) < 1. Gven an terator B, we defne the mappng Φ B v = v +B(f Av) and ntroduce ts symmetrzaton Φ B = Φ B t Φ B. By defnton, we have the formula for the error operator I BA = (I B t A)(I BA), and thus The author s supported n part by NSF Grant DMS-0811272, and n part by NIH Grant P50GM76516 and R01GM75309. Ths work s also partally supported by the Bejng Internatonal Center for Mathematcal Research.
2 Long Chen B = B t (B t + B 1 A)B. (3) Snce B s symmetrc, I BA s symmetrc wth respect to the A-nner product (u, v) A := (Au, v). Indeed, let ( ) be the adjont n the A-nner product (, ) A. It s easy to show I BA = (I BA) (I BA). (4) Consequently, I BA s postve sem-defnte and thus λ max (BA) 1. We get I BA A = max{ 1 λ mn (BA), 1 λ max (BA) } = 1 λ mn (BA). (5) From (5), we see that I BA s a contracton f and only f B s SPD whch s also equvalent to B t + B 1 A beng SPD n vew of (3). The convergence of the scheme Φ B and ts symmetrzaton Φ B s connected by the followng nequalty: ρ(i BA) 2 I BA 2 A = I BA A = ρ(i BA), (6) and the equalty holds f B = B t. Hence we shall focus on the analyss of the symmetrc scheme n the rest of ths paper. The terator B, when t s SPD, can be used as a precondtoner n the Precondtoned Conjugate Gradent (PCG) method, whch admts the followng estmate: ( ) k u u k A κ(ba) 1 u u 0 2 (k 1), A κ(ba) + 1 ( κ(ba) = λ ) max(ba). λ mn (BA) A good precondtoner should have the propertes that the acton of B s easy to compute and that the condton number κ(ba) s sgnfcantly smaller than κ(a). We shall also dscuss constructon of multlevel precondtoners n ths paper. 2 Auxlary Space Method In ths secton, we present a varaton of the fcttous space method Nepomnyaschkh [1992] and the auxlary space method Xu [1996]. Let Ṽ and V be two Hlbert spaces and let Π : Ṽ V be a surjectve map. Denoted by Π t : V Ṽ the adajont of Π wth respect to the default nner products (Π t u, ṽ) := (u, Πṽ) for all u V, ṽ Ṽ. Here, to save notaton, we use (, ) for nner products n both V and Ṽ. Snce Π s surjectve, ts transpose Π t s njectve. Theorem 1. Let Ṽ and V be two Hlbert spaces and let Π : Ṽ V be a surjectve map. Let B : Ṽ Ṽ be a symmetrc and postve defnte operator. Then B := Π BΠ t : V V s also symmetrc and postve defnte. Furthermore (B 1 v, v) = nf Πṽ=v ( B 1 ṽ, ṽ). (7)
Dervng the X-Z Identty from Auxlary Space Method 3 Proof. We adapt the proof gven by Xu and Zkatanov [2002] (Lemma 2.4). It s obvous that B s symmetrc and postve sem-defnte. Snce B s SPD and Π t s njectve, (Bv, v) = ( BΠ t v, Π t v) = 0 mples Π t v = 0 and consequently v = 0. Therefore B s postve defnte. Let ṽ = BΠ t B 1 v. Then Πṽ = v by the defnton of B. For any w Ṽ ( B 1 ṽ, w) = (Π t B 1 v, w) = (B 1 v, Π w). In partcular ( B 1 ṽ, ṽ ) = (B 1 v, Πṽ ) = (B 1 v, v). For any ṽ Ṽ, denoted by v = Πṽ, we wrte ṽ = ṽ + w wth Π w = 0. Then nf Πṽ=v ( B 1 ṽ, ṽ) = nf ( B 1 (ṽ + w), ṽ + w) Π w=0 = (B 1 v, v) + nf Π w=0 = (B 1 v, v) + nf Π w=0 ( B 1 w, w) = (B 1 v, v). ( 2( B 1 ṽ, w) + ( B 1 w, w) The symmetrc postve defnte operator B may be used as a precondtoner for solvng Au = f usng PCG. To estmate the condton number κ(ba), we only need to compare B 1 and A. Lemma 1. For two SPD operators A and B, f c 0 (Av, v) (B 1 v, v) c 1 (Av, v) for all v V, then κ(ba) c 1 /c 0. Proof. Note that BA s symmetrc wth respect to A. Therefore λ 1 mn (BA) = λ max((ba) 1 ((BA) 1 u, u) A (B 1 u, u) ) = sup = sup u V\{0} (u, u) A u V\{0} (Au, u). Therefore (B 1 v, v) c 1 (Av, v) mples λ mn (BA) c 1 1. Smlarly (B 1 v, v) c 0 (Av, v) mples λ max (BA) c 1 0. The estmate of κ(ba) then follows. By Lemma 1 and Theorem 1, we have the followng result. Corollary 1. Let B : Ṽ Ṽ be SPD and B = Π BΠ t. If c 0 (Av, v) then κ(ba) c 1 /c 0. nf ( B 1 ṽ, ṽ) c 1 (Av, v) for all v V, (8) Πṽ=v Remark 1. In lterature, e.g. the fcttous space lemma of Nepomnyaschkh [1992], the condton (8) s usually decomposed as the followng two condtons. 1. For any v V, there exsts a ṽ Ṽ, such that Πṽ = v and ṽ 2 B 1 c 1 v 2 A. 2. For any ṽ Ṽ, Πṽ 2 A c 1 0 ṽ 2 B 1. )
4 Long Chen 3 Auxlary Spaces of Product Type Let V V, = 0,..., J, be subspaces of V. If V = =0 V, then {V } J =0 s called a space decomposton of V. Then for any u V, there exsts a decomposton u = =0 u. Snce =0 V s not necessarly a drect sum, decompostons of u are n general not unque. We ntroduce the ncluson operator I : V V, the projecton operator Q : V V n the (, ) nner product, the projecton operator P : V V n the (, ) A nner product, and A = A V. It can be easly verfed Q A = A P and Q = I t. Gven a space decomposton V = =0 V, we construct an auxlary space of product type Ṽ = V 0 V 1... V J, wth the nner product (ũ, ṽ) := =0 (u, v ). We defne Π : Ṽ V as Πũ = =0 u. In operator form Π = (I 0, I 1,, I J ). Snce V = =0 V, the operator Π s surjectve. Let R : V V be nonsngular operators, often known as smoothers, approxmatng A 1. Defne a dagonal matrx of operators R = dag(r 0, R 1,, R J ) : Ṽ Ṽ whch s non-sngular. An addtve precondtoner s defned as B a = Π RΠ t = I R I t = =0 I R Q. (9) Applyng Theorem 1, we obtan the followng dentty for precondtoner B a. Theorem 2. If R s SPD on V for = 0,..., J, then B a defned by (9) s SPD on V. Furthermore (Ba 1 v, v) = nf (R 1 J =0 v=v v, v ). (10) To defne a multplcatve precondtoner, we ntroduce the operator à = Πt AΠ. By drect computaton, the entry ã j = Q AI j = A P I j. In partcular ã = A. The symmetrc operator à may be sngular wth nontrval kernel ker(π), but the dagonal of à s always non-sngular. Wrte à = D + L + D Ũ where = dag(a 0, A 1,, A j ), L and Ũ are lower and upper trangular matrx of operators, and L t = Ũ. Note that the operator R 1 + L s nvertble. We defne B m = ( R 1 + L) 1 and ts symmetrzaton as =0 =0 B m = B m t + B m B mã t B m = B m( t t B m + The symmetrzed multplcatve precondtoner s defned as B 1 m Ã) B m. (11) B m := Π B m Π t. (12) We defne the dagonal matrx of operators R = dag(r 0, R 1,, R J ), where, for each R, = 0,, J, ts symmetrzaton s R = R t (R t + R 1 A )R.
Dervng the X-Z Identty from Auxlary Space Method 5 Substtutng B 1 m = R 1 + L, and à = D + L + Ũ nto (11), we have B m = ( R t + L t ) 1 ( R t + R 1 D)( R 1 1 + L) = ( R t + L t ) 1 R t R R 1 ( R 1 + L) 1. (13) It s obvous that B m s symmetrc. To be postve defnte, from (13), t suffces to assume R,.e. each R, s symmetrc and postve defnte whch s equvalent to the operator I R A s a contracton and so s I R A. (A) I R A A < 1 for each = 0,, J. Theorem 3. Suppose (A) holds. Then B m defned by (12) s SPD, and (B 1 m v, v) = v 2 A + nf =0 v=v =0 In partcular, for R = A 1, we have Proof. Let (B 1 m v, v) = v 2 A + R(A t P nf =0 v=v =0 j= P v j R 1 j=+1 v ) 2. (14) R 1 v j 2 A. (15) M = R t + R 1 D = R t R R 1, U = D + Ũ R 1, L = U t. then R 1 + L = M + L and à = M + L + U. We then compute B 1 m = ( R 1 + L)( R t + R 1 D) 1 ( R t + L t ) = (M + L )M 1 (M + U ), = à + L M 1 U [ = à + Rt ( D + Ũ R R 1[ 1 )]t Rt ( D + Ũ R ] 1 ). For any ṽ V, denoted by v = Πṽ, we have (Ãṽ, ṽ) = (Πt AΠṽ, ṽ) = (AΠṽ, Πṽ) = v 2 A. Usng component-wse formula of Rt ( D + Ũ R 1 )ṽ, e.g. (( D + Ũ)ṽ) = j= ãjv j = j= A P v j, we get (M 1 U ṽ, U ṽ) = The dentty (14) then follows. R(A t P =0 j= v j R 1 v ) 2. R 1
6 Long Chen If we further ntroduce the operator T = R A P : V V, then T = R ta P, T := T + T T T = R A P, and (R 1 w, w ) = (A T 1 w, w ) = (T 1 w, w ) A. Here T 1 := (T V ) 1 : V V s well defned due to the assumpton (A). We then recovery the orgnal formulaton n Xu and Zkatanov [2002] (B 1 m v, v) = v 2 A + nf =0 v=v =0 (T 1 T w, T w ) A, wth w = j= v j T 1 v. Wth these notaton and w = k> v k, we can also use (13) to recovery the formula n Cho et al. [2008] (B 1 m v, v) = nf =0 v=v (T 1 (v + T w ), v + T w ) A. =0 4 Method of Subspace Correcton In ths secton, we vew the method of subspace correcton Xu [1992] as an auxlary space method and provde denttes for the convergence analyss. Let V = =0 V be a space decomposton of V. For a gven resdual r V, we let r = Q r be the restrcton of the resdual to the subspace V and solve the resdual equaton n the subspaces A e = r approxmately by ê = R r. Subspace correctons ê are assembled together to gve a correcton n the space V. There are two basc ways to assemble subspace correctons. Parallel Subspace Correcton (PSC) Ths method s to do the correcton on each subspace n parallel. In operator form, t reads u k+1 = u k + B a (f Au k ), (16) where B a = I R I t = =0 I R Q. (17) Thus PSC s also called addtve methods. Note that the formula (17) and (9) are dentcal and thus dentty (10) s useful to estmate κ(b a A). Successve Subspace Correcton (SSC) Ths method s to do the correcton n a successve way. In operator form, t reads v 0 = u k, v +1 = v + R Q (f Av ), = 0,..., N, u k+1 = v J+1. (18) =0
Dervng the X-Z Identty from Auxlary Space Method 7 For each subspace problem, we have the operator form v +1 = v +R (f Av ), but t s not easy to wrte out the terator for the space V. We defne B m such that the error operator I B m A = (I R J Q J A)(I R J 1 Q J 1 A)...(I R 0 Q 0 A). Therefore SSC s also called multplcatve method. We now derve a formulaton of B m from the auxlary space method. In the sequel, we consder the SSC appled to the space decomposton V = J =0 V wth smoothers R, = 0,, J. Recall that Ṽ = V 0 V 1... V J and à = Π t AΠ. Let f = Π t f. Followng Grebel and Oswald [1995], we vew SSC for solvng Au = f as a Gauss-Sedel type method for solvng Ãũ = f. Lemma 2. Let à = D + L + Ũ and B = ( R 1 + L) 1. Then SSC for Au = f wth smothers R s equvalent to the Gauss-Sedel type method for solvng Ãũ = f: ũ k+1 = ũ k + B( f Ãũk ). (19) Proof. By multplyng R 1 + L to (19) and rearrangng the term, we have Multplyng R, we obtan R 1 ũ k+1 = R 1 ũ k + f Lũ k+1 ( D + Ũ)ũk. ũ k+1 = ũ k + R( f Lũ k+1 ( D + Ũ)ũk ), and ts component-wse formula, for = 0,, J Let u k+1 1 = u k + R (f j=0 ã j u k+1 j 1 = u k + R Q (f A j=0 1 v 1 = u k+1 j + j=0 u k+1 j u k j. j= ã j u k j ) j= A u k j ). Notng that v v 1 = u k+1 u k, we then get, for = 1,, J + 1 v = v 1 + R Q (f Av 1 ), whch s exactly the correcton on V ; see (18). Lemma 3. For SSC, we have j= B m = Π B m Π t and B m = Π B m Π t.
8 Long Chen Proof. Let u k = Πũ k. Applyng Π to (19) and notng that we then get f = Π t f, and Ãũ k = Π t Au k, u k+1 = u k + Π BΠ t (f Au k ). Therefore B m = Π B m Π t. The formulae for B m s obtaned smlarly. Combnng Lemma 3, (5), (6), and Theorem 3, we obtan the X-Z dentty. Theorem 4 (X-Z dentty). Suppose assumpton (A) holds. Then where (I R J Q J A)(I R J 1 Q J 1 A)...(I R 0 Q 0 A) 2 A = 1 1 1 + c 0, (20) c 0 = sup nf v A =1 =0 v=v =0 In partcular, for R = A 1, where R(A t P j= v j R 1 v ) 2. R 1 (I P J )(I P J 1 ) (I P 0 ) 2 A = 1 1 1 + c 0, (21) c 0 = sup nf v A =1 =0 v=v =0 P j=+1 v j 2 A. References D. Cho, J. Xu, and L. Zkatanov. New estmates for the rate of convergence of the method of subspace correctons. Numer. Math. Theor. Meth. Appl., 1:44 56, 2008. M. Grebel and P. Oswald. On the abstract theory of addtve and multplcatve Schwarz methods. Numer. Math., 70:163 180, 1995. S. V. Nepomnyaschkh. Decomposton and fcttous domans methods for ellptc boundary value problems. In Ffth Internatonal Symposum on Doman Decomposton Methods for Partal Dfferental Equatons (Norfolk, VA, 1991), pages 62 72. SIAM, Phladelpha, PA, 1992. P. S. Vasslevsk. Multlevel Block Factorzaton Precondtoners: Matrx-based Analyss and Algorthms for Solvng Fnte Element Equatons. Sprnger Verlag, 2008. J. Xu. Iteratve methods by space decomposton and subspace correcton. SIAM Rev., 34:581 613, 1992. J. Xu. The auxlary space method and optmal multgrd precondtonng technques for unstructured meshes. Computng, 56:215 235, 1996. J. Xu and L. Zkatanov. The method of alternatng projectons and the method of subspace correctons n Hlbert space. J. Amer. Math. Soc., 15:573 597, 2002.