Notes on convergence of an algebraic multigrid method

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1 Appled Mthemtcs Letters 20 (2007) Notes on convergence of n lgebrc multgrd method Zhohu Hung,, Peln Sh b Stte Key Lbortory of Spce Wether, Center for Spce Scence nd Appled Reserch, Chnese Acdemy of Scences, Bejng , People s Republc of Chn b College of Scence, Tyun Unversty of Technology, Tyun , People s Republc of Chn Receved 5 Februry 2006; receved n revsed form 11 My 2006; ccepted 30 My 2006 Abstrct The convergence theory for lgebrc multgrd (AMG) lgorthms proposed n Chng nd Hung [Q.S. Chng, Z.H. Hung, Effcent lgebrc multgrd lgorthms nd ther convergence, SIAM J. Sc. Comput. 24 (2002) ] s further dscussed nd smller nd elegnt upper bound s obtned. On the bss of element-free AMGe [V.E. Henson, P.S. Vsslevsk, Elementfree AMGe: Generl lgorthms for computng nterpolton weghts n AMG, SIAM J. Sc. Comput. 23(2) (2001) ] we rewrte the nterpolton opertor for the clsscl AMG (camg), present unform expresson nd then, by ntroducng sprse pproxmte nverse n the Frobenus norm, gve generl convergence theorem whch s suted for not only camg but lso AMG for fnte elements nd element-free AMGe. c 2006 Elsever Ltd. All rghts reserved. Keywords: Algebrc multgrd; Interpolton opertor; Convergence; Mtrx nlyss; Sprse pproxmte nverse 1. Introducton Algebrc multgrd (AMG) hs recently developed rpdly snce Brndt et l. frst ntroduced the clsscl AMG (camg) n the 1980s [1 4] due to ts symptotclly optml convergence nd the ncresng pplcton need. AMG s key pont s tht the lgebrclly smooth error components fter relxton must be removed by determnng proper corsenng process nd desgnng the pproprte prolongton opertors. For camg, the smooth error s chrcterzed by smll r bsed on the defect equton Ae = r [5]; the element-bsed AMGe [6] ndρamge [7] usng spectrl decomposton employed smlr crteron to produce the corse grds nd trnsfer opertors. Element-free AMGe [8] gve generl mtrx expresson for computng AMG nterpolton weghts nd descrbed the results of camg nd AMGe by proposng n extenson opertor. A new method of AMG corse grd selecton bsed on the element stffness mtrx nd the technque of locl relxton for the smooth process re gven n [9]. Here we would lke to present generl frmework for constructng the nterpoltng opertor whch s obtned by solvng the defect equton pproxmtely. Correspondng uthor. E-ml ddresses: hzh@cssr.c.cn (Z. Hung), pelnsh@yhoo.com.cn (P. Sh) /$ - see front mtter c 2006 Elsever Ltd. All rghts reserved. do: /j.ml

2 336 Z. Hung, P. Sh / Appled Mthemtcs Letters 20 (2007) The generl nterpolton opertor Frst, we strt from the defect equton + k m em k + j m em j 0, F m (2.1) m em k C m j D m where m s the corse grd level ndex whch wll be omtted whle not cusng ny confuson, C m = C m S m, C m denotes the corse grd set, F m the fne grd ponts, S m the set of ll strong connecton ponts of the pont, ndd m s the complementry set of C m n the neghborhood. Let A hve the block form ( ) AFF A A = FC, (2.2) A CF A CC nd A FF, A CC re squre mtrces. Evdently, e m j ( j Dm ) must be pproxmted by e m, ek m(k Cm ) or ther (lner) combnton n order to obtn the nterpolton opertor. When we let e m j pe m + q k C m g m jk em k (p, q R), g m jk = m jk / k C m m jk, j Dm. Here, e j pe + q k C g jk e k s equvlent to ddng p j to nd q j g jk to k,.e., we replce the j-th row of A FF by the zero row nd then plce 1 n the column j nd p n the column, nd the block A FC s modfed to  FC by zerong the j-th row; we modfy A FF by zerong out the off-dgonl entres, nd replcng the dgonl entry jj wth â jj = q k C jk, keepng the off-dgonl entres. Then we hve ) )  FF = ( + p j D j,  FC = ( k + q j D j g jk AFC + B FC, F, k C. Tht s to sy, the nterpolton opertor Im+1 m cn be obtned by pproxmtely solvng the equton A FF e F + A FC e C = 0, nd hs the opertor form k e F =  1 FF (A FC + B FC )e C = (Im+1 m ) FCe C I FC e C. (2.4) As for the camg, the nterpolton weghts re m k + q m wk m = k D m j gm jk m + p, k C m. (2.5) j D m m j It s esy to see tht the formul (2.5) s generl nd t my nclude Ruge nd Stübe, Chng Wong Fu [10] nd Chng Hung [11] forms, etc., even ny nterpolton opertor whch s constructed bsed on mtrx elements. As we choose p = 1 for the wek connecton ponts nd q = 1 for the strong connecton ponts n D, we wll get Ruge nd Stüben nterpolton formule. Furthermore, by ntroducng two geometrc ssumpton, nd p = 0, ±1 for ll ponts n D, q = 1/2, 1, 2forD s, q = 1, 2forDw, we cn deduce esly Chng Wong Fu nterpolton formule, nd n ddtonl Jcob or Guss Sedel terton wll get Chng Hung nterpolton formule, Guss Sedel-type AMG nterpolton opertor [12]. Eq. (2.5) s more flexble nd resonble becuse t wll scrfce the ccurcy tht the set D m s emprclly dvded nto strong nd wek, n prtculr, when the sze of the mtrx elements s lmost the sme order. Fnlly, we hope to emphsze tht the set D m s cut prtly to sve the computng work nd memory n the prctcl computton, s lso shown n the followng theoretcl nlyss. 3. Convergence nlyss Let (x, y) E (or (x, y) for smplcty) nd be the Euclden nner product nd the ssocted norm respectvely. If the mtrx A s symmetrc postve defnte (.e. A > 0), we lso use the followng three nner products: (2.3)

3 Z. Hung, P. Sh / Appled Mthemtcs Letters 20 (2007) (u,v) 0 = (Du,v), (u,v) 1 = (Au,v), (u,v) 2 = (D 1 Au,v), long wth ther ssocted norms ( = 0, 1, 2), nwhchd = dg(a). Denote by Fr the Frobenus norm of the mtrx. To prove the convergence of the AMG method, by the theory of AMG [3,8,10 20], we need only to demonstrte the nterpolton opertor stsfyng mn em I m e m+1 m+1 em β em 2 1, where β s ndependent of e m nd m. Frst, we gve the lemm: (3.1) Lemm 3.1. Let A > 0, B > 0, then we hve (Ae, e) E c(be, e) E wth the constnt c, f nd only f ρ(b 1 A) c, where ρ s the spectrl rdus of the mtrx. Proof. Becuse A > 0, B > 0, the followng equlty holds. ρ(b 1 A) = ρ(b 1/2 AB 1/2 ) = sup (B 1/2 AB 1/2 y, y) E. y R n (y, y) E When t s ssumed tht B 1/2 y = e,wehve ρ(b 1 (Ae, e) E A) = sup. e R n (Be, e) E Therefore we obtn the equvlence (Ae, e) E c(be, e) E ρ(b 1 A) c. (3.2) Lemm 3.2. Let A FF be strongly dgonlly domnnt, tht s, j δ ( F) j F, j wth some fxed, pre-defned δ>0. Then the followng nequlty holds ρ(a 1 FF D FF) 1/δ. Proof. As (3.3), then λ + 1 j δ λ, j F, j where λ s ny egenvlue of the mtrx DFF 1 A FF. Now we cn drw the concluson tht λ δ. Otherwse, by (3.4) we wll deduce det( λi + DFF 1 A FF) 0. Ths wll contrdct tht λ s the egenvlue of DFF 1 A FF. Thus, ρ(a 1 FF D FF) 1/δ s strghtforwrd. Lemm 3.3. Let A > 0 nd A FF be strongly dgonlly domnnt. (u,v) E,F s the Euclden nner product for the F component, nd (u,v) 1,F = (A FF u,v) E,F. Then the followng estmte holds: I FF DFF 1 A FF 1,F < 1. Proof. The postve defnteness of A ensures the followng computton. I FF DFF 1 A FF 2 1,F = mx (A FF(I FF DFF 1 A FF)x,(I FF DFF 1 A FF)x) E,F x 1,F =1 = mx x 1,F =1 (A 1/2 FF ρ((i FF D 1 FF A FF) 2 ). (I FF DFF 1 A FF) T A FF (I FF DFF 1 A FF)x, A 1/2 FF x) (3.3) (3.4)

4 338 Z. Hung, P. Sh / Appled Mthemtcs Letters 20 (2007) And ρ((i FF DFF 1 A FF) 2 )<1then follows from the dgonl domnnce of A FF. Hence, we hve I FF DFF 1 A FF 1,F < 1. Now we gve the mn convergence theorem. Theorem 3.1. Assume A > 0, nd ρ(d 1 A) η. LetA FF be strongly dgonlly domnnt mtrx, nd the nterpolton opertor Ī FC stsfy e F Ī FC e C 2 0,F β 1 e 2 1. Then we hve the followng estmte for Ĩ FC whch s one step of the fully relxed Jcob nterpolton of Ī FC : e F Ĩ FC e C 2 0,F β 2 e 2 1. Proof. e F Ĩ FC e C 2 0,F ρ(a 1 FF D FF) e F Ĩ FC e C 2 1,F 2 δ ( e F + A 1 FF A FCe C 2 1,F + Ĩ FC + A 1 FF A FCe C 2 1,F ) 2 δ ( e F + A 1 FF A FCe C 2 1,F + I FF DFF 1 A FF 2 1,F I FC + A 1 FF A FCe C 2 1,F ) 2(1 + ηβ 1) e 2 1 δ = β 2 e 2 1. Remrk. The generl AMG convergence result s gven n [11]. Obvously, the bove mn theorem presents smller nd elegnt upper bound. Lemm 3.4. Let A = ( j ) n n be wekly dgonlly domnnt mtrx, tht s, t = j j 0, = 1, 2,...,n. Then for n rbtrry e = (e F, e C ) T we hve A FC e C 2 M(A CC e C, e C ), (3.5) n whch M = mx F j C j. Proof. Let s = n j=1 j = t + 2 j + j nd j = { j, f j < 0, 0, f j > 0, + j = { j, f j < 0, 0, f j > 0. (3.6) For ny e = (e F, e C ) T,wehve (Ae, e) = j e e j = 1 ( j )(e e j ) 2 + s e 2 2 = 1 j 2 (e e j ) 2 1 j + 2 (e e j ) 2 + s e 2 = 1 j 2 (e e j ) 2 + j + (2e 1 2 (e e j ) 2 ) + t e 2 j 1 j 2 (e e j ) j + 2 (2e + 2e 2 j (e e j ) 2 ) + j = 1 ( j 2 (e e j ) 2 + ) j + (e + e j ) 2 + t e 2 j j ( j (e e j ) 2 + ) j + (e + e j ) 2 + t e 2. F j C j C F t e 2

5 Z. Hung, P. Sh / Appled Mthemtcs Letters 20 (2007) Let e F = 0 n the bove equton. Then we hve (A CC e C, e C ) = (Ae, e) ( j e2 j + ) j + e2 j = j e 2 j. (3.7) F j C j C F j C On the other hnd, employng Schwrz s nequlty, we cn estmte A FC e C 2 = ( )2 j e j ( j e2 j + ) j + e2 j M(A CC e C, e C ), F j C F j C j C where M = mx F j C j. Then we present the other mn result. Theorem 3.2. Let A m be symmetrc postve defnte nd wekly dgonlly domnnt mtrces. Suppose tht there re constnts M 1 nd M 2 such tht (A m e m, e m ) M 1 (e m, e m ) nd (A m FF em F, em F ) M 2(e m F, em F ) for em = (e m F, em C )T. If the pproxmte nverse (Â m FF ) 1 stsfes I (Â m FF ) 1 A m FF Fr μ, n whch μ s constnt, then the nequlty (3.5) holds for the nterpolton nd β 3 = M 1 2 ( M 1 Θ + μθ Φ),where Φ = ρ(a m CC )ρ((am ) 1 ), Θ = ρ(dg(a m )). Proof. For ny e m = (e m F, em C )T,wehve e m I m m+1 em+1 0 = e m F + (Ãm FF ) 1 A m FC em C 0,F e m F + (Am FF ) 1 A m FC em C 0,F + (A m FF ) 1 A m FC em C (Ãm FF ) 1 A m FC em C 0,F. By the hypotheses nd Lemm 3.4, we cn obtn (3.8) nd e m F + (Am FF ) 1 A m FC em C 2 0,F = (Dm FF (em F + (Am FF ) 1 A m FC em C ), (em F + (Am FF ) 1 A m FC em C )) Θ((A m FF ) 1 (A m FF em F + Am FC em C ), (Am FF ) 1 (A m FF em F + Am FC em C )) (3.9) Θ 2 Ae 2 Θ M 1 2 e 2 1 ; (A m FF ) 1 A m FC em C (Ãm FF ) 1 A m FC em C 2 0,F = (Dm FF (((Am FF ) 1 (Ã m FF ) 1 )A m FC em C ), ((Am FF ) 1 (Ã m FF ) 1 )A m FC em C ) Θ (((A m FF ) 1 (Ã m FF ) 1 )A m FC em C ) 2 Combnng (3.8) (3.10), we rrvet Θ I F (Ã m FF ) 1 A m FF 2 Fr (Am FF ) 1 A m FC em C 2 Θμ 2 (A m FF ) 1 A m FC em C 2 Θμ2 2 A m FC em C 2 Θ 2 μ 2 2 (A m CC em C, em C ) ΦΘ 2 μ 2 2 (A m e m, e m ) = ΦΘ 2 μ 2 2 e m 2 1. (3.10) ( Θ M1 + Θμ ) 2 Φ e m 2 1, e m I m m+1 em M 2 2 nd ths completes the proof.

6 340 Z. Hung, P. Sh / Appled Mthemtcs Letters 20 (2007) Remrk. Ths theorem gves the AMG convergence from the pproxmte nverse ngle, whch wll help for constructng more prctcl nterpolton opertors. Acknowledgements Ths project ws supported by Ntonl Nturl Scence Foundton of Chn ( , ), Chn Postdoctorl Scence Foundton ( ), nd K.C. Wong Educton Foundton, Hong Kong ( ). The uthors thnk Prof. Q. Chng for hs mportnt comments, nd the frst uthor fnshed some of the work of ths pper when nvted by Prof. Q. Du to vst Hong Kong Unversty of Scence nd Technology. And we wsh to thnk the revewers for mny vluble comments whch contrbuted substntlly to ths work. References [1] A. Brndt, S. McCormck, J. Ruge, Algebrc multgrd (AMG) for utomtc multgrd solutons wth pplcton to gedetc computtons, Report, Insttute for Computtonl Studes, Fort Collns, CO, [2] A. Brndt, Algebrc multgrd theory: The Symmetrc Cse, n: Prelmnry Proceedngs for the Interntonl Multgrd Conference, Copper Mountn, Colordo, [3] J. Ruge, K. Stüben, Algebrc multgrd, n: S.F. McCormck (Ed.), Multgrd Methods, vol. 4, SIAM, Phldelph, [4] K. Stüben, Algebrc multgrd (AMG): Experences nd comprsons, Appl. Mth. Comput. 13 (1983) [5] Z.H. Hung, Q.S. Chng, Multgrd solver bsed on the defect equton, Chnese J. Comput. Phys. 18 (5) (2001) [6] M. Brezn, A.J. Clery, R.D. Flgout, V.E. Henson, J.E. Jones, T.A. Mnteuffel, S.F. McCormck, J.W. Ruge, Algebrc multgrd bsed on element nterpolton (AMGe), SIAM J. Sc. Comput. 22 (2002) [7] T. Chrter, R.D. Flgout, V.E. Henson, J. Jones, T. Mnteuffel, S. McCormck, J. Ruge, P.S. Vsslevsk, Spectrl AMGe (ρamge), SIAM J. Sc. Comput. 25 (1) (2003) [8] V.E. Henson, P.S. Vsslevsk, Element-free AMGe: Generl lgorthms for computng nterpolton weghts n AMG, SIAM J. Sc. Comput. 23 (2) (2001) [9] Q. Du, Z.H. Hung, D.S. Wng, Mesh nd solver co-dptton n fnte element methods for nsotropc problems, Numer. Methods Prtl Dfferentl Equtons 21 (4) (2005) [10] Q.S. Chng, Y.S. Wong, et l., On the lgebrc multgrd method, J. Comput. Phys. 25 (1996) [11] Q.S. Chng, Z.H. Hung, Effcent lgebrc multgrd lgorthms nd ther convergence, SIAM J. Sc. Comput. 24 (2002) [12] Z.H. Hung, Q.S. Chng, Guss Sedel-type multgrd methods, J. Comput. Mth. 21 (4) (2003) [13] Q.S. Chng, Z.H. Hung, The development of multgrd methods, Chnese Sc. Abstr. 7 (5) (2001) [14] U. Trottenberg, et l., Multgrd, Acdemc Press, [15] Q.S. Chng, Z.H. Hung, A Combnton of lgebrc multgrd lgorthms wth the conjugte grdent technque, n: Recent Progress n Computtonl nd Appled PDEs, Kluwer Acdemc/Plenum Publshers, 2002, pp [16] G.Y. Le, Z.H. Hung, Estmton of norm nd condton number for error mtrce of hgh order ICCG method, Chnese J. Comput. Phys. 16 (3) (1999) [17] G.Y. Le, Z.H. Hung, Estmton of norm nd condton number for error mtrx of ICCG method, Chnese J. Comput. Phys. 13 (4) (1996) [18] Z.H. Hung, Q.S. Chng, An mproved lgorthm for the MG nterpolton opertor, Act Mthemtce Applcte Snc 26 (3) (2003) [19] Z.H. Hung, G.Y. Le, X.P. Lu, Hgh-order PCG method solvng complex systems, Chnese J. Comput. Phys. 17 (4) (2000) [20] S.C. Wu, G.Y. Le, X.Q. Lu, Z.H. Hung, Computng methods for the boundry poston of unconventonl ol nd gs reservor, n: Numercl Computton for Hgh Technologcl Reserch, The Press of Scentfc nd Technologcl Unversty of Ntonl Defence, 1995, pp