DiVA Digitala Vetenskapliga Arkivet


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1 DVA Dgtala Vetenskaplga Arkvet Ths s a book chapter publshed n Hghperformance scentfc computng: algorthms and applcatons (ed Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe, B; Saad, Y; Saed, F) Ctaton for the publshed paper: Kjelgaard Mkkelsen, Carl Chrstan 0 The explct Spke algorthm: Iteratve soluton of the reduced system In Hghperformance scentfc computng: algorthms and applcatons, Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe, B; Saad, Y; Saed, F (ed), p  London: Sprnger The orgnal publcaton s avalable at wwwsprngerlnkcom
2 The explct SPIKE algorthm: Iteratve soluton of the reduced system Carl Chrstan Kjelgaard Mkkelsen Department of Computng Scence and HPCN Umeå Unversty Sweden Dedcated to Ahmed Sameh on the occason of hs 0th brthday Summary The explct SPIKE algorthm apples to narrow banded lnear systems whch are strctly dagonally domnant by rows The parallel bottleneck s the soluton of the socalled reduced system whch s block trdagonal and strctly dagonally domnant by rows The reduced system can be solved teratvely usng the truncated reduced system matrx as a precondtoner In ths paper we derve a tght estmate for the ualty of ths precondtoner Introducton A matrx A = [a j] R n n s dagonally domnant by rows f : X a j a j If the neualty s sharp, then A s strctly dagonally domnant by rows If A s nonsngular and dagonally domnant by rows or f A s strctly dagonally domnant by rows, then a 0, and the domnance factor ɛ gven by ɛ = max ( P j aj a s well defned The matrx A has lower bandwdth b l f a j = 0 for > j + b l and upper bandwdth b u f a j = 0 for j > + b u If b = max{b l, b u} n, then we say that A s narrow banded Every suare banded matrx can be parttoned as a block trdagonal matrx wth suare dagonal blocks, e )
3 Carl Chrstan Kjelgaard Mkkelsen A C A = B Cm, () B m A m only the dmenson of each dagonal block must be bounded from below by b In partcular, we do not have to choose the same dmenson for each dagonal block, even n the exceptonal case where b dvdes n The SPIKE algorthms are desgned to solve banded systems on a parallel machne The central dea was ntroduced by Sameh and Kuck [] who consdered the trdagonal case and Chen, Kuck and Sameh [] who studed the trangular case Lawre and Sameh [] appled the algorthm to the symmetrc postve defnte case whle Dongarra and Sameh [] consdered the dagonally domnant case Polzz and Sameh [8, 9] ntroduced the truncated SPIKE algorthm for systems whch are strctly dagonally domnant by rows Recently, Manguoglu, Sameh, and Schenk [] have combned PARDISO wth SPIKE n the PSPIKE package The explct SPIKE algorthm by Dongarra and Sameh [] can be used to solve narrow banded lnear systems whch are strctly dagonally domnant by rows The algorthm extends naturally to systems whch are block trdagonal Moreover, the analyss s smplfed f we focus on the number of dagonal blocks, rather than the bandwdth of the matrx In Secton we state the explct SPIKE algorthm for systems whch are block trdagonal and strctly dagonally domnant by rows The parallel bottleneck s the soluton of a reduced system whch s block trdagonal and strctly dagonally domnant by rows The reduced system can be solved teratvely usng the man block dagonal as a precondtoner We derve a tght estmate for the ualty of ths precondtoner n Secton Ths s a specal case of a more general theorem by Mkkelsen [] The explct SPIKE algorthm In ths secton we state the explct SPIKE algorthm for systems whch are block trdagonal and strctly dagonally domnant by rows The valdty and the basc analyss of the algorthm hnges on the followng lemma Lemma Let G = ˆE, D, F be a matrx such that ˆD, E, F s strctly dagonally domnant by rows wth domnance factor ɛ Then G s row euvalent to a unue matrx K = ˆU, I, V Moreover, the matrx ˆU, V satsfes ˆU, V ɛ Proof Mkkelsen and Manguoglu [] contans an elementary proof Now consder the soluton of a block trdagonal lnear system Ax = f on a parallel machne wth p processors Gven a small tolerance δ > 0, we shall now seek an approxmaton y, such that the forward error satsfes
4 The explct SPIKE algorthm: Iteratve soluton of the reduced system ˆ A f = A () C () f () B () () C B () A () C () f () f () B () A () C () B () () C B () A () C () f () f () B () A () C () B () C () B () A () f () Fg The SPIKE parttonng for p = processors x y δ x We assume that A has m = p dagonal blocks and we assgn consecutve block rows to each processor The case of p = s llustrated n Fgure If A s strctly dagonally domnant by rows, then we can predvde wth the man block dagonal n order to obtan an euvalent lnear system Sx = g The case of p = s dsplayed n Fgure It s from the narrow columns or spkes protrudng from the man dagonal that the orgnal algorthm has derved t name The matrx S s called the SPIKE matrx; the vector g s called the modfed rght hand sde By Lemma, S I ɛ <, so S s strctly dagonally domnant by rows The euatons wthn a sngle block row of each the man parttons lnes form a reduced system Rx r = g r whch can be solved ndependently The general structure of the reduced system s gven n Fgure Once the reduced system has been solved, the soluton of the orgnal system can be retreved by backsubsttuton Specfcally, we have x (j) = g (j) U (j) x (j ) V (j) x (j+),, j p, () where U (), V (p), x (0), and x (p+) are undefned and should be taken as zero Suppose for the moment that we have somehow solved the reduced system wth a small normwse relatve forward error, say,
5 Carl Chrstan Kjelgaard Mkkelsen I V () g () I V () g () U () I V () g () ˆ S g = U () I V () g () U () I g () U () I g () Fg The SPIKE matrx correspondng to p = processors ˆ R gr = I V () g () U () I V () g () U () I V () U () I g () () g V (p ) U (p ) I V (p ) g (p ) U (p) I g (p) Fg The general structure of the reduced system x r y r δ x r In vew of euaton () t s natural to partton y r conformally wth x r, e y r = (x () and defne a vector y R n usng Then y (j) = g (j) x (j) U (j) y (j ) y (j) and t follows mmedately that T () T, x,, x (p ) h = T, x (p) T ) T V (j) y (j+),, j p U (j) ", V (j) x (j ) y (j ) x (j+) y (j+) #,
6 The explct SPIKE algorthm: Iteratve soluton of the reduced system ˆ T gr = I V () g () U () I g () I V () g () U () I g () I U (p) I g (p) V (p ) g (p ) Fg The structure of the truncated reduced system x y ɛ x r y r ɛδ x r δ x It s clear that we must solve the reduced system accurately n order to acheve a small forward normwse relatve error We now consder the soluton of the reduced system The reduced system matrx R s block trdagonal and strctly dagonally domnant by rows The neualty R I ɛ < s nherted from the SPIKE matrx S Freuently, but not unversally, the off dagonal blocks are nsgnfcant and can be dropped Ths phenomenon s exploted heavly n the truncated SPIKE algorthm by Polzz and Sameh [8, 9] Let T denote the man block dagonal of R, see Fgure Mkkelsen and Manguoglu [] showed that T R ɛ when A s banded and strctly dagonally domnant by rows In ths paper we consder the sgnfcance of the off dagonal blocks relatve to the man block dagonal To ths end we defne an auxlary matrx B by B = T (T R) Now, let x tr be the soluton of the truncated reduced system Then T x tr = g r T (x r x tr) = (R (T R))x r g r = (Rx r g r) (T R))x r = (T R)x r from whch t mmedately follows, that f x r 0, then x tr x r x r B We have already understood the need to solve the reduced system wth a forward normwse relatve error of at most δ If B δ, then we smply drop the off dagonal blocks and approxmate x r wth x tr If B > δ, then we can solve the
7 Carl Chrstan Kjelgaard Mkkelsen reduced system teratvely usng the man block dagonal as a precondtoner If we use the statonary teraton where x (0) r = 0, then T x () r and we can stop the teraton whenever = (T R)x ( ) r + g r, =,,, x r x () r B x r B δ In the next secton we establsh a tght upper bound on the central parameter B The SPIKE and the PSPIKE packages both apply BCG, rather than the statonary teraton Nevertheless, the sze of B remans an nterestng ueston The man result Our purpose s to establsh Theorem Theorem The auxlary matrx B satsfes B ɛ, where s the number of dagonal blocks assgned to each processor and eualty s possble We shall reduce the problem of provng Theorem to a sngle applcaton of the followng theorem Theorem (Mkkelsen []) Let G k be a representaton of k consecutve block rows of a block trdagonal matrx A whch s strctly dagonally domnant by rows wth domnance factor ɛ, e G k = B k A k C k B A C B 0 A 0 C 0 B A C Then G k s row euvalent to a unue matrx K k of the form B k A k C k
8 The explct SPIKE algorthm: Iteratve soluton of the reduced system K k = k I V(k) k I V (k) 0 I V (k) 0 I V (k) k Z (k) = I V (k) k where the spkes decay exponentally as we move towards the man block row Specfcally, f we defne " # and then V (k) V (k), 0 < < k, h Z (k) 0 = 0, V (k) 0 Z (k) ɛ k, 0 < k Proof The exstence and unueness of K k follows mmedately from Lemma The central neualty can be establshed usng the well orderng prncple The detals can be found n a report by Mkkelsen and Kågström [] We now move to prove the estmate gven by Theorem It s straghtforward to verfy that eualty s acheved for matrces A gven by euaton () where B = O k, A = I k, C = ɛi k, and O k s the k by k zero matrx, I k s the k by k dentty matrx and ɛ < In order to prove the general neualty t suffces to consder the nteracton between two neghborng parttons Ths follows mmedately from the propertes of the nfnty norm Let G k be a compact representaton of k block rows drawn from the orgnal matrx A, e G k = B k A k C k B A C B A C B k A k C k and let H k be a compact representaton of the correspondng rows of the assocated SPIKE matrx Then G k H k and H k has the form ()
9 8 Carl Chrstan Kjelgaard Mkkelsen U k I V k U I V H k = () U I V U k I V k Our task s to show that the auxlary matrx Z k gven by»»» Z Z I V U 0 Z k = = Z Z U I 0 V () satsfes Z k ɛ k, k =,,, We contnue to reduce H k usng row operatons We repartton H k n order to focus our attenton on the two central block rows, e U k I V k U I V G k U I V U k I V k Then we predvde wth the central by block matrx and obtan U k I V k Z I Z G k Z I Z U k I V k and t s clear that there exsts a matrx K k such that G k K k and
10 The explct SPIKE algorthm: Iteratve soluton of the reduced system 9 and the matrx Z k satsfes U I V U I V U I V K k = U I V U I V U I V ()» U V Z k = () U V At ths pont we have reduced the problem of provng Theorem to a straghtforward applcaton of Theorem Concluson The explct SPIKE algorthm by Dongarra and Sameh [] extends naturally to systems whch are block trdagonal and strctly dagonally domnant by rows Moreover, the analyss of the method s smplfed by focusng on the number of dagonal blocks rather than the bandwdth The parallel bottleneck remans the soluton of the reduced system Rx r = g r whch s strctly dagonally domnant and block trdagonal The sgnfcance of the off dagonal blocks can be measured usng the auxlary matrx B gven by B = T (T R) = I T R, where T denotes the man block dagonal of R If B s suffcently small, then we can gnore the off dagonal blocks and approxmate x r wth the soluton of the truncated reduced system T x tr = g r In general, we can solve the reduced system teratvely usng the man block dagonal T as a precondtoner and the convergence rate s controlled by the sze of B Our man contrbuton s Theorem whch establshes a tght upper bound on B References SC Chen, DJ Kuck and A Sameh, Practcal parallel band trangular system solvers, ACM Trans Math Software,, (98), pp 0 JJ Dongarra and A Sameh, On some parallel banded system solvers, Parallel Comput, vol (98), pp  DH Lawre and A Sameh, The computaton and communcaton complexty of a parallel banded system solver, ACM Trans Math, Software, vol 0 (98), pp 89
11 0 Carl Chrstan Kjelgaard Mkkelsen M Manguoglu, A Sameh and O Schenk, PSPIKE: A Parallel Hybrd Sparse Lnear System Solver, In Proceedngs of EUROPAR009, LNCS 0 (009), pp C C K Mkkelsen and M Manguoglu, Analyss of the truncated SPIKE algorthm, SIMAX, vol 0, no (008), pp 009 C C K Mkkelsen and Bo Kågström, Analyss of ncomplete cyclc reducton for narrow banded and strctly dagonally domnant lnear systems Tech Rep UMINF 0, Department of Computng Scence, Umeå Unversty (0) Submtted to PPAM0 A Sameh and DJ Kuck, On stable parallel lnear systems solvers, J ACM, vol (98), pp E Polzz and A Sameh, A parallel hybrd banded system solver: The SPIKE algorthm, Parallel Comput, vol (00), pp 9 9 E Polzz and A Sameh, SPIKE: A parallel envronment for solvng banded lnear systems, Comput Fluds, vol (00), pp 0
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