Overlapping additive and multiplicative Schwarz iterations for H -matrices

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1 Lnear Algebra and ts Applcatons 393 (2004) Overlappng addtve and multplcatve Schwarz teratons for H -matrces Rafael Bru a,1, Francsco Pedroche a, Danel B. Szyld b,,2 a Departament de Matemàtca Aplcada, Unverstat Poltècnca de Valènca, Camí de Vera s/n, Valènca, Span b Department of Mathematcs, Temple Unversty, 1805 N. Broad Street, Phladelpha, PA , USA Receved 7 June 2003; accepted 28 October 2003 Submtted by M. Tsatsomeros Abstract In recent years, an algebrac framework was ntroduced for the analyss of convergence of Schwarz methods for the soluton of lnear systems of the form Ax = b. Wthn ths framework, addtve and multplcatve Schwarz were shown to converge when the coeffcent matrx A s a nonsngular M-matrx, or a symmetrc postve defnte matrx. In ths paper, many of these results are extended to the case of A beng an H -matrx. The case of nexact local solves s also consdered. In addton, the two-level scheme s studed,.e., when a coarse grd correcton s used n conjuncton wth the addtve or the multplcatve Schwarz teratons Elsever Inc. All rghts reserved. AMS classfcaton: 65F10; 65F35; 65M55 Keywords: Lnear systems; Schwarz teratve methods; Block methods; Overlap; H -matrces Correspondng author. E-mal addresses: rbru@mat.upv.es (R. Bru), pedroche@mat.upv.es (F. Pedroche), szyld@math. temple.edu (D.B. Szyld). 1 Supported by Spansh DGI grant BFM Supported n part by the Natonal Scence Foundaton under grant DMS /$ - see front matter 2003 Elsever Inc. All rghts reserved. do: /j.laa

2 92 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Introducton We consder lnear systems of equatons of the form Ax = b, (1) where A R n n s an H -matrx (and thus nonsngular; see, e.g., [5] and the references theren) and x and b are vectors of V = R n ; we revew some defntons later n ths secton. We study the soluton of (1) by Schwarz teratons wth p overlappng blocks. These are teratve methods orgnally developed for operators A arsng from dscretzatons of partal dfferental equatons (p.d.e.). In these cases, Schwarz teratons correspond to doman decomposton methods; see, e.g., [12,13]. Schwarz methods are most often used as precondtoners, but n some nstances they are used as statonary teratve methods of the form x k+1 = Tx k + c, k = 0, 1,..., (2) where x 0 s an ntal guess, c s an approprate vector, and T s the teraton matrx; see, e.g., [3,8], and the references gven theren for such use. In our context, the convergence of the teraton (2), whch holds for any ntal vector x 0 f and only f ρ = ρ(t) < 1(ρ(T) denotng the spectral radus; see, e.g., [2]), ndcates that the spectrum of the precondtoned matrx I T (= B 1 A for some nonsngular matrx B),.e., ts set of egenvalues, s contaned n a ball centered at one wth radus ρ. In the rest of ths secton we revew the addtve and multplcatve teratons, each correspondng to a dfferent matrx T n (2). Our exposton s based on the algebrac formulaton presented n [6] and [1]. In these references, convergence of the Schwarz teratons was studed when the matrx A n (1) s ether a nonsngular M-matrx or symmetrc postve defnte. In ths paper, n Sectons 2 and 3 we study the convergence of these teratons when A s an H -matrx. The analyss for the case of nexact local solves s also ncluded. In Secton 4 we extend the convergence results to the two-level teratons,.e., when a coarse grd correcton s used. A nonsngular matrx A havng all nonpostve off-dagonal entres s called an M-matrx f the nverse s (entry-wse) nonnegatve,.e., A 1 O; see, e.g., [2,11]. For any matrx A = (a j ) R n n, ts comparson matrx A =(α j ) s defned by α = a, α j = a j, /= j. A matrx A s sad to be an H -matrx f A s an M-matrx. In partcular, A s an H -matrx f and only f t s generalzed dagonally domnant,.e., a u > a j u j, = 1,...,n (3) /=j for some postve vector u = (u 1,...,u n ) T. H -matrces were ntroduced n [11] as generalzaton of M-matrces. They appear n many applcatons, e.g., when dscretzng certan nonlnear parabolc operators usng hgh order fnte elements and suffcently small tme steps [4]. The characterzaton (3) also ndcates how general these matrces are; see further [15].

3 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) As we have mentoned, n ths paper we study the soluton of (1) when A s an H - matrx, usng Schwarz teratve methods. These are block teratve methods n whch the blocks are overlappng,.e., some varables are common to more than one block. When there s no overlap, addtve Schwarz reduces to block Jacob, whle multplcatve Schwarz reduces to block Gauss Sedel. To formally descrbe the overlappng blocks, let V be subspaces of V = R n of dmenson n, = 1,...,p, (p > 1) such that the sum of these subspaces span the whole space. These subspaces are not parwse dsjont; on the contrary, ther ntersecton s precsely the overlap, and thus p n >n. The restrcton and prolongaton operators map vectors from V to V and vceversa. The restrcton operators used here are of the form R =[I O], = 1,...,p, (4) where I s the dentty on R n,and s a permutaton matrx on R n. The prolongaton operator consdered here s R T. We now defne the followng matrces E = R T R, = 1,...,p. (5) Note that the dagonal matrces E gven by (5) have nonzero dagonal elements (wth value one) only n the columns whch have a nonzero element the matrx R. We denote by q the measure of overlap,.e., the maxmum over all possble rows, of the number of matrces E wth a nonzero n the row. Thus E qi; (6) and usually q p. The restrcton of the matrx A to the subspace V s A = R AR T, (7) whch s a symmetrc permutaton of an n n prncpal submatrx of A, = 1,..., p. These are precsely the p overlappng blocks; see [6] for more detals. We are ready to descrbe the Schwarz teratons for the soluton of (1). The damped addtve Schwarz teraton has the followng form x k+1 = x k + θ R T A 1 R (b Ax k ), (8) where 0 <θ 1sthedampng factor. In practcal mplementatons of the addtve Schwarz teratons (8), for each teraton, the resdual vector r k = b Ax k s restrcted to each subspace V usng the operator R. Then, the local problem A e = r k = R r k (9)

4 94 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) s solved, the obtaned local errors, e k = e, are prolongated, summng them (and dampng ts value) and fnally that correcton s added to x k yeldng the new approxmaton vector x k+1. It follows that the process (8) s (2) wth the teraton matrx T θ = I θ R T A 1 R A = I θ P, (10) where P = R TA 1 R A s a projecton onto V. In the case of A symmetrc and postve defnte, ths projecton s orthogonal wth respect to the A-nner product; see, e.g., [13]. The teraton matrx for the multplcatve Schwarz teratons s T µ = (I P p )(I P p 1 ) (I P 1 ) = (I P ). (11) In contrast to the addtve Schwarz teraton (8), here, the correcton n each subspace s followed by another correcton, untl all correctons have been made. When the local problem (9) s not solved exactly, but only approxmately, ts soluton can be consdered the exact soluton of another approxmate local problem, namely à e = r k. The matrx à could be, for example, an ncomplete factorzaton of A ; see, e.g., [13]. In ths case, the teraton matrx for the damped addtve Schwarz teraton wth nexact local solves s T θ = I θ R T à 1 R A = I θ P, (12) where P = R Tà 1 R A. Smlarly, the teraton matrx for the multplcatve Schwarz teratons wth nexact local solves s T µ = (I P p )(I P p 1 ) (I P 1 ) = (I P ). (13) We proceed n the next sectons to study the convergence of the teratons (10) (13). 2. Convergence of addtve Schwarz teratons We begn by establshng a dfferent algebrac representaton of the teraton matrx (10). Gven a matrx A = (a j ), we defne the matrx A =( a j ). It follows that A O and that AB A B for any two matrces A and B of compatble sze. Let A =[O I ] A T [O I ] T, (14)

5 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) where I s the (n n ) (n n ) dentty matrx, and let [ ] M = T A O O H, (15) where H s some (n n ) (n n ) nonsngular matrx such that H A = H A. (16) In fact, ths condton gves us a lot of freedom n choosng H. In [1,6], dfferent choces were H = A or H = D = dag A. These choces clearly satsfy our condton (16). It follows then from the form of the matrces (5) and (15) that E M 1 = R T A 1 R. (17) Usng ths equalty, the teraton matrx T θ can be expressed as T θ = I θ E M 1 A. (18) Our proof of convergence conssts of showng that f θ 1/q, thenρ(t θ )<1; cf. [6]. Our strategy s to show that T θ T θ, for the matrx T θ = I θ R T A 1 R A, (19) whch we show n the next subsecton to be nonnegatve and to have spectral radus less than one Propertes of T θ Let us consder the followng lnear system assocated wth the orgnal problem (1), A x = b, (20) and apply the addtve Schwarz teratons wth the same p overlappng blocks that we consdered n Secton 1. Then, gven an ntal approxmaton x 0 for the soluton of (1), the damped addtve Schwarz teraton appled to (20) reads, for k = 0, 1,..., x k+1 = x k + θ R T A 1 R (b A x k ), (21) where the R are gven by (4), and n a way smlar to (7), A = R A R T, and the teraton matrx for ths scheme s precsely (19).

6 96 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) It s not hard to see that A = A, and snce any prncpal submatrx of an M-matrx s also an M-matrx [2,11], we have the followng useful result. Lemma 1. If A R n n s an H -matrx, then any prncpal submatrx of A, and any symmetrc permutaton of t s an H -matrx. In partcular the matrx A gven by (7) s an H -matrx. We note that the coeffcent matrx of (20) s an M-matrx, and therefore, we can use the results n [6]. In partcular, we have that B = R T A 1 R = E M 1 s nonnegatve and nonsngular, where E s gven by (5). We also have that the teraton matrx (19) can be wrtten as T θ = I θ E M 1 A (22) and that f θ 1/q, T θ O, and the damped addtve Schwarz teraton (21) converges to the soluton of (20) for any ntal vector x 0. Therefore we have ρ( T θ )<1. (23) 2.2. Convergence for H -matrces Before proceedng wth the convergence analyss of (21) we prove an mportant result concernng the matrces M defned n (15). A splttng A = M N s called regular f M 1 O and N O [14];tscalledH-compatble f A = M N ; see [5]. Theorem 1. Let A R n n be an H -matrx and let the matrces M be of the form (15), satsfyng (16). Then, A= M N,= 1,...,p, are H -compatble splttngs. Proof. Frst, from the defnton of A notce that TA = T A snce the dagonal of the matrx TA s a permutaton of the dagonal of A. Let A be wrtten by blocks as [ ] A = T A U V A, where U and V are matrces of approprate sze. Then, we have that A = T [ ] A U V A (24)

7 and M = T R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) [ ] A O O H. From these expressons we may wrte, usng (16) [ ] [ ] M A = T A O O H T A U V A [ ] = T O U V H A. Consder now the splttngs A = M N, = 1,...,p. Then, [ ] [ ] N = M A = T A O O H T A U V A [ ] = T O U V H A. Hence, we have that N = M A and the proof s complete. We are ready now to prove the followng convergence result. Theorem 2. Let A R n n be an H -matrx. Let the matrces R be of the form (4). Then, f θ 1/q, the damped addtve Schwarz teraton (8) converges to the soluton of (1) for any ntal vector x 0. Proof. We frst show that T θ T θ. (25) From the expressons (22) and (18) we have [ ] T θ T θ = θ E A M 1 A M 1 and applyng Theorem 1, we have [ ] T θ T θ = θ E M 1 N M 1 N. (26) Let us recall that f A s an H -matrx, then A 1 A 1 [11]. Then, the rght hand sde of (26) s nonnegatve, snce M 1 N M 1 N M 1 N M 1 N (27) and thus T θ T θ. (28)

8 98 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Now, from the expressons (22) and (18), we also have [ ] T θ + T θ = 2I θ E M 1 A +M 1 A, usng agan Theorem 1 and smplfyng, we have [ ] [ ] T θ + T θ = 2 I θ E + θ E M 1 N +M 1 N. (29) Now, from (6), f θ 1/q,wehave I θqi θ E, and then the frst term of the rght hand sde of (29) s nonnegatve. Moreover, from (27), the second term of the rght hand sde of (29) s also nonnegatve. Therefore, we have that f θ 1/q, then T θ T θ. (30) Combnng (28) and (30) we have the desred result (25). To conclude the proof, we recall that f A, B R n n and A B then ρ(a) ρ(b); see, e.g., [10, 2.4.9]. Applyng ths to (25) we have, usng (23), that f θ 1/q, then ρ(t θ ) ρ( T θ ) ρ( T θ )< Inexact local solves Gven the matrces Ã, = 1,...,p, representng the nexact local solves, for the convergence analyss one consders the matrces ] [à M = T O O H, (31) cf. (15). We assume as before that H satsfes (16). As n (17) and (18), we have now E M 1 = R Tà 1 R, = 1,...,p,and T θ = I θ p E M 1 A. The condtons we mpose on the local solves to guarantee convergence of the addtve Schwarz methods are the followng: à 1 O and (32) à A = à A, = 1,...,p. (33) We note that condton (32) s satsfed automatcally f à s an H -matrx. Ths occurs, e.g., f à s an ncomplete factorzaton of A [9,14]. Condton (33) s equvalent to havng the splttng A = à (à A ) be H -compatble. Note also that under the condtons (32) (33), snce we have that ( à A ) O, we conclude that A = à ( à A ) s a regular splttng. These condtons also provde us wth the counterpart to Theorem 1. Its proof s analogous, and therefore t s omtted.

9 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Theorem 3. Let A be an H -matrx and let the matrces M be defned as n (31). Assume further that the nexact solves are such that the condton (33) holds. Then, the splttngs A = M Ñ,= 1,...,p,are H -compatble splttngs. The followng counterpart to the result (23) s obtaned by applyng [6, Theorem 3.5] to the M-matrx A. Theorem 4. Let A be an H -matrx and let the nexact solves be such that (32) holds and that à A O, = 1,...,p, whch hold f one mposes the condton (33) as well. Then, f θ 1/q, the damped addtve Schwarz teraton wth nexact local solves, defned by (2) wth the teraton matrx T θ = I θ R T à 1 R A converges to the soluton of A x = b for any ntal vector x 0,.e., we have that ρ( T θ )<1. We are now ready to show the convergence of the damped addtve Schwarz teraton matrx wth nexact local solves for H -matrces. Theorem 5. Let A be an H -matrx and let the nexact solves be such that the condtons (32) (33) hold. Then, f θ 1/q, the damped addtve Schwarz teraton wth nexact local solves, defned by (2) wth the teraton matrx (12) converges to the soluton of (1) for any ntal vector x 0. Proof. The proof s analogous to that of Theorem 2. Usng the same technques one shows that T θ T θ, and usng Theorem 4, we conclude that ρ( T θ ) ρ( T θ ) < Multplcatve Schwarz teratons We frst observe that usng (17), the teraton matrx for multplcatve Schwarz T µ gven n (11) can be wrtten as T µ = (I R T A 1 R A) = (I E M 1 A). (34) As was the case for addtve Schwarz, we consder the soluton of the lnear system (20) by multplcatve Schwarz teratons wth the same blocks as for the system (1). Then, the new teraton matrx for the multplcatve Schwarz teraton s now T µ = (I R T A 1 R A ).

10 100 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Usng the results of [1] appled to the system (20) (whose coeffcent matrx s an M-matrx), we have that T µ = (I E M 1 A ) O. and that ρ( T µ )<1. Furthermore, usng Theorem 1, we have that T µ = (I E + E M 1 N ). (35) We are ready to prove the convergence of multplcatve Schwarz teratons for H -matrces. Theorem 6. Let A R n n be an H -matrx. Let the matrces R be of the form (4). Then, the multplcatve Schwarz teraton (2) wth teraton matrx (11) converges to the soluton of (1) for any ntal vector x 0. Proof. As n the proof of Theorem 2, to prove that ρ(t µ )<1 we show that T µ T µ. To that end, we bound I E + E M 1 N I E + E M 1 N = I E + E M 1 N I E + E M 1 N, (36) where the last nequalty follows from E M 1 N E M 1 N E M 1 N =E M 1 N. From (34), usng that A = M N,wehave T µ = (I E M 1 A) I E M 1 A = I E + E M 1 N, and usng 36 and (35) we conclude that T µ (I E + E M 1 N ) = T µ. (37) The convergence of the multplcatve Schwarz teraton s proved.

11 3.1. Inexact local solves R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) In order to analyze the convergence of the multplcatve Schwarz teraton wth nexact local solves, we frst consder the soluton of the auxlary system (20), and apply [6, Theorem 4.5] to t. We thus obtan the followng result. Theorem 7. Let A be an H -matrx. Assume that à A O, = 1,...,p, whch hold f one mposes condton (33). Then the multplcatve Schwarz teraton matrx wth nexact local solves, defned by (2) wth the teraton matrx T µ = (I E M 1 A ) converges to the soluton of A x = b for any choce of the ntal vector x 0,.e., we have that ρ( T µ )<1. We proceed as n Theorem 6, usng Theorem 3 one can prove that T µ T µ mplyng that ρ( T µ ) ρ( T µ )<1. We have then the followng convergence result. Theorem 8. Let A be an H -matrx and let the nexact solves be such that condton (33) holds. Then, the multplcatve Schwarz teraton wth nexact local solves, defned by (2) wth the teraton matrx (13) converges to the soluton of (1) for any ntal vector x Coarse grd correctons for H -matrces In ths secton, we study the convergence of the addtve and multplcatve Schwarz teratons when a coarse grd correcton s appled. We follow the structure used n [1,6]. The coarse grd s represented algebracally by an addtonal subspace V 0 of dmenson n 0, wth p n 0 <n. For ths subspace, we defne a restrcton operator R 0 as before wth (4) ( = 0), A 0 as n (7), and E 0 as n (5), mplyng O E 0 I. (38) We also defne the correspondng matrx M 0 as n (15), and thus, by Theorem 1, the splttng A = M 0 N 0 s also H -compatble. In other words, Theorem 1 holds for = 0, 1,...,p, and ths s how we use t throughout ths secton. If the coarse grd equaton A 0 e 0 = r 0, s solved approxmately, we defned the matrx à 0 so that the approxmaton solves exactly à 0 e 0 = r 0. We assume that the condtons (32) (33) hold (for = 0). One can then defne a matrx M 0 as n (31). The analyss of the two-level methods presented n the sequel can be appled to the

12 102 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) case of nexact local solves, and/or nexact correctons usng the same technques. We omt the detals Multplcatve Schwarz wth multplcatve correcton Followng [1], let the teraton matrx of the multplcatve Schwarz teraton wth a multplcatve coarse grd correcton be H µ = (I G 0 A)T µ, (39) where G 0 = R0 TA 1 0 R 0. It follows that (17) holds [1],.e., E 0 M 1 0 A = G 0A. (40) Let us consder the lnear system (20). Applyng the results of [1] to the M-matrx A, we have that the teraton matrx Ĥ µ = (I Ĝ 0 A ) T µ (41) s nonnegatve and has spectral radus less than one, where Ĝ 0 = R0 T A 1 0 R 0 = R0 T A 0 1 R 0 = E 0 M 0 1. By Theorem 1, we know that A = M 0 N 0 and thus we can rewrte the matrx (41) as Ĥ µ = (I E 0 + E 0 M 0 1 N 0 ) T µ. Theorem 9. Let A R n n be an H -matrx and consder the soluton of (1) by multplcatve Schwarz teratons wth multplcatve correcton,.e., the teraton (2) wth T = H µ as n (39). Then, the teratons converge to the soluton, for any ntal vector x 0. Proof. We can rewrte the teraton matrx (39) usng (40) and Theorem 1 as H µ = (I E 0 + E 0 M 1 0 N 0)T µ. (42) As n the proof of Theorem 6, we have I E 0 + E 0 M 1 0 N 0 I E 0 + E 0 M 0 1 N 0. (43) From (42) and (43), we have that H µ = (I E 0 + E 0 M 1 0 N 0)T µ I E 0 + E 0 M 1 0 N 0 T µ (I E 0 + E 0 M 0 1 N 0 ) T µ. We use now (37), whch says that T µ T µ, then we have that (I E 0 + E 0 M 0 1 N 0 ) T µ (I E 0 + E 0 M 0 1 N 0 ) T µ = Ĥ µ. Thus, H µ Ĥ µ and therefore ρ(h µ ) ρ(ĥ µ )<1.

13 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Addtve Schwarz wth multplcatve correcton The multplcatve correcton (I G 0 A) appled to the addtve Schwarz teratons gves rse to the followng teraton matrx [1] H θ = (I G 0 A)T θ. (44) As t can be apprecated, the structure of ths teraton matrx s the same as that of (39). Usng the same logc as n the prevous subsecton, and usng the fact that T θ T θ, one can compare (44) wth Ĥ θ = (I Ĝ 0 A ) T θ, the teraton matrx of addtve Schwarz teratons for the system (20), for whch ρ(ĥ θ )<1fθ 1/q; see [1]. Thus, we have the counterpart to Theorem 9. Theorem 10. Let A R n n be an H -matrx and consder the soluton of (1) by addtve Schwarz teratons wth multplcatve correcton,.e., the teraton (2) wth T = H θ as n (44). Then, f θ 1/q, the teratons converge to the soluton for any ntal vector x 0. We remark that unlke the stuaton n the M-matrx case [1], the coarse grd correcton not always leads to a decrease of the spectral radus of the teraton matrx. The followng smple example llustrates ths stuaton. Consder the followng H - matrx 14/15 1/15 7/90 1/20 A = 1/5 4/5 7/45 3/20 1/30 4/15 29/30 3/10, 1/10 1/45 1/10 89/90 and let [ ] R 1 =, R = , wth the coarse grd correcton gven by [ ] R 0 = The spectral rad of the teraton operators are ρ(t µ ) , ρ(h µ ) , ρ(t θ ) , ρ(h θ ) for the value θ = 1/3. On the other hand, for the case of A symmetrc postve defnte, one can say the followng. It holds that I G 0 A s an orthogonal projecton onto V 0 usng the A- nner product, and thus I G 0 A A = 1. Therefore H µ A = (I G 0 A)T µ A T µ A and H θ A = (I G 0 A)T θ A T θ A.

14 104 R. Bru et al. / Lnear Algebra and ts Applcatons 393 (2004) Addtve Schwarz wth addtve coarse grd correcton Havng a coarse grd correcton addtvely conssts of havng an extra term n (10) correspondng to the correcton n the subspace V 0. The correspondng teraton matrx s thus T θ = I θ =0 R T A 1 R A. (45) In [6], ths teraton was shown to be convergent for systems where the coeffcent s an M-matrx, and θ 1/(q + 1). Thus, f we consder the soluton of system (20) by the addtve Schwarz wth addtve coarse grd correcton, we have that the matrx T θ = I θ =0 R T A 1 R A s nonnegatve and has spectral radus less than one, f θ 1/(q + 1). Usng Theorem 1 (for = 0, 1,...,p) we can rewrte t as T θ = I θ (E E M 1 N ) O. =0 Theorem 11. Let A R n n be an H -matrx. Then, f θ 1/(q + 1), the damped addtve Schwarz teraton wth addtve correcton defned by the teraton matrx T θ of (45) converges to the soluton of (1) for any ntal vector x 0. Proof. The proof s analogous to that of Theorem 2. On one sde we have that T θ T θ = θ =0 On the other, we wrte [ T θ + T θ = 2 I θ ( M 1 N M 1 N ) O. ] E + θ =0 =0 E [ M 1 N +M 1 N ], whch s nonnegatve snce from (6) and (38), for θ 1/(q + 1), I θ(q + 1)I θ E. =0 Then, we conclude that T θ T θ and thus ρ( T θ ) ρ( T θ )<1.

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