Journal of Computational and Applied Mathematics. Breakdown-free version of ILU factorization for nonsymmetric positive definite matrices

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1 Journal of Computatonal and Appled Mathematcs 230 (2009) Contents lsts avalable at ScenceDrect Journal of Computatonal and Appled Mathematcs ournal homepage: wwwelsevercom/locate/cam Breakdown-free verson of ILU factorzaton for nonsymmetrc postve defnte matrces A Rafe, F Toutounan Department of Mathematcs Ferdows Unversty of Mashhad, Mashhad, P O Box , Iran a r t c l e n f o a b s t r a c t Artcle hstory: Receved 20 August 2007 Receved n revsed form 18 July 2008 MSC: 65F10 Keywords: Implct precondtoner Sparse matrces RIF RIF p In ths paper a new ILU factorzaton precondtoner for solvng large sparse lnear systems by teratve methods s presented The factorzaton whch s based on A- borthogonalzaton process s well defned for a general postve defnte matrx Numercal experments llustratng the performance of the precondtoner are presented A comparson wth the well known precondtoner RIF p of Benz and Tůma s also ncluded 2009 Elsever BV All rghts reserved 1 Introducton In ths paper we consder the soluton of lnear systems of the form Ax = b, where the coeffcent matrx A R n n s large, sparse and nonsymmetrc postve defnte (NSPD), and b s a gven rght hand sde vector usng precondtoned conugate gradent-type methods Suppose that A admts the factorzaton A = LDU, where L, U T are unt lower trangular matrces and D s a dagonal matrx If L and Ū T are sparse unt lower trangular matrces approxmatng (n some sense) the matrces L and U T, respectvely, and D s a nonsngular dagonal matrx approxmatng D, then we say that matrx M wth M = L DŪ A, s an ncomplete LU (ILU) factorzaton precondtoner for matrx A The transformed lnear systems or AM 1 u = b, M 1 u = x, M 1 Ax = M 1 b, have the same soluton as system (1) and seem to be better-condtoned than the orgnal system (1) to solve It s wellknown that an ncomplete factorzaton of a general matrx A may fal due to the occurrence of zero pvots, regardless of (1) (2) (3) (4) (5) Correspondng author Tel: E-mal addresses: rafeam@gmalcom (A Rafe), toutoun@mathumacr (F Toutounan) /$ see front matter 2009 Elsever BV All rghts reserved do:101016/cam

2 700 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) whether A admts the LU factorzaton or not Moreover, trouble (n the form of numercal nstabltes) can be expected n the presence of small pvots In the SPD case, a breakdown occurs when a non postve pvot s encountered A number of remedes have been proposed to deal wth the problem of breakdown, partcularly n the SPD case Sgnfcant progress for the SPD case and the general case has been made n recent years Some of these technques can be found n [1 8] In [3], Benz and Tůma have presented a safe and relable method, called RIF (robust ncomplete factorzaton) for computng an approxmate factorzaton A L D LT, wth L unt lower trangular matrx and D a dagonal matrx Unlke the standard Cholesky factorzaton, ther technque s based on an nherently stable A-orthogonalzaton process and s therefore applcable to general SPD matrces Ths s smply the Gram Schmdt process wth respect to the nner product generated by the SPD matrx In [9], Rezgh and Hossen have ntroduced an ILU precondtoner for NSPD matrces by usng the conugate Gram Schmdt process They have shown that by applyng the A-bconugaton algorthm to both A and A T matrces, t s possble to obtan an approxmate factorzaton of a NSPD matrx The am of our paper s to present a new breakdown-free method called RIF p NS, to construct an approxmate factorzaton for any NSPD matrx The new method s based on the A-bconugaton process but t only use matrx A Ths paper s organzed as follows In Secton 2 we brefly revew A-bconugaton algorthm and dscuss the mplementaton of RIF and RIF p precondtoners In Secton 3 we descrbe the new technques RIF NS and RIF p NS for computng an approxmate factorzaton of the form (3) These technques are applcable to any postve defnte matrx The resultng precondtoners are relable (pvot breakdown can not occur) and effectve at reducng the number of teratons Secton 4 s devoted to the numercal experments that are done by usng some test matrces In Secton 5 some remarks and conclusons are presented 2 Reformulaton of A-bconugaton algorthm The A-bconugaton algorthm that s presented by Benz and Tůma n [10] s an algorthm that uses matrces A and A T to construct the unt upper trangular matrx Z, dagonal matrx D, and unt upper trangular matrx W n the nverse factorzaton A 1 = ZD 1 W T It has been shown that for lmted class of matrces, e, H-matrces, no dvson by zero wll occur durng the run of A- bconugaton algorthm In [11,12] the authors have shown that by a reformulaton of the A-bconugate algorthm no dvson by zero wll occur for broad range of matrces, e, postve defnte matrces The relable verson of the AINV algorthm based on the reformulaton of A-bconugaton algorthm s as follows: Algorthm 1 (Reformulaton of the A-bconugaton algorthm) 1 For = 1,, n Do: 2 w (0) = e, z (0) = e 3 EndDo 4 For = 1, 2,, n Do: 5 v = Aw (1), u = A T z (1) 6 q (1) = (w (1) ) T v, p (1) 7 If = n goto 13 8 For = + 1,, n Do: = (w (1) 9 q (1) 10 w () = w (1) ( q (1) q (1) ) T v, p ) (1) = (z (1) ) T u = (z (1) ) T u w (1), z () = z (1) ( ) p (1) p (1) z (1) 11 EndDo 12 EndDo 13 Let z := z (1) ; w = w (1) and p = p (1) for 1 n p p Return Z = [z 1,, z n ], W = [w 1,, w n ], and D = 0 0 p n From (2), (6), and the fact that the factorzaton of the form (2) s unque, we have Z 1 = U = D 1 W T A, W T = L = AZD 1 (7) In [13], we have shown that for 1 n the followng two relatons hold p (1) = e T Az(1) = z T Az(1) = w T Ae, (6) (8)

3 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) and q (1) = e T AT w (1) Therefore, by usng relatons (7) (9) we obtan = w T AT w (1) = z T AT e, (9) u = p(1) p (1), l = q(1) q (1) (10) Hence the L and U factors of A = LDU factorzaton can be obtaned as a by-product of the A-bconugaton process at no extra cost Based on relaton (10) the followng algorthm can be gven Algorthm 2 0 Set L = U = I 1 For = 1,, n Do: 2 w (0) = e, z (0) = e 3 EndDo 4 For = 1, 2,, n Do: 5 v = Aw (1), u = A T z (1) 6 q (1) = (w (1) ) T v, p (1) 7 If = n goto 14 8 For = + 1,, n Do: 9 q (1) 10 l = q(1) 11 w () = (w (1) ) T v, p (1) q (1) = w (1), u = p(1) ( q (1) q (1) 12 EndDo 13 EndDo 14 Let p = p (1) for 1 n p (1) = (z (1) ) T u = (z (1) ) T u ) w (1), z () = z (1) ( ) p (1) p (1) z (1) p p Return L = (l ) 1 < n, U = (u ) 1 < n and D = 0 0 p n The ncomplete factorzaton based on A-bconugaton process needs two drop tolerances, one for ncomplete A- bconugaton process, to be appled to the Z and W factors, and a second one to be appled to the entres of L and U matrces The latter s smply post-fltraton that s once a column of L and a row of U have been computed Removng small entres from the columns and rows of L and U factors, lead to some degradaton of the convergence rate of teratve methods, but ths s often compensated by savng n the computatonal work obtaned from havng sparser L and U factors As mentoned n [3], two varants of ncomplete factorzaton based on A-bconugaton process, called RIF (wthout post-fltraton) and RIF p (wth post-fltraton), can be obtaned If a drop tolerance parameter T 1 s used to sparsfy the newly updated vectors w () and z () after step 11 of Algorthm 2 by droppng those entres that are less than T 1, then RIF precondtoner wll be obtaned If n addton, a drop tolerance T 2 s used to drop the small entres of L and U factors after step 10 (post-fltraton) of Algorthm 2, then RIF p precondtoner wll be obtaned It should be mentoned that, the other versons of RIF and RIF p precondtoners called NRIF and NRIF p, were presented n [9] To construct NRIF and NRIF p precondtoner step 11 of Algorthm 2 was replaced by w () = w (1) l w (1), z () = z (1) u z (1) The most mportant ssue to be mentoned s that, snce for every, the vectors z (1) ) T A T z (1) computaton, then for 1 n the numbers p (1) = (z (1) and q (1) and w (1) ) T Aw (1) are nonzero by = (w (1) are all postve, provded that the orgnal matrx A s postve defnte Thus n exact arthmetc, the RIF and RIF p precondtoners exst and no dvson by zero wll occur durng ther constructon

4 702 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) RIF NS method Note that, n Algorthm 2, matrx A T generates vectors {u } n =1 at step 5 These vectors are needed for generatng numbers p (1), for 1 n We have shown that [13], by usng the followng proposton, t s possble to obtan the numbers p (1) wthout usng matrx A T Proposton Let p (1) s and q (1) s be the scalars produced by Algorthm 1, then for 1 n we have and p (1) q (1) 1 = a k=1 1 = a k=1 p (k1) p (k1) k q (k1) q (k1) k q (k1) p (k1) From (10) and (11), we mmedately observe that p (1) 1 = a k=1 u k q (k1) (11) (12) (13) Based on (13), a transpose-free verson of Algorthm 2 can be stated as follows: Algorthm 3 0 Set L = U = I 1 Let w (0) = e ; 1 n 2 For = 1, 2,, n Do: 3 v = Aw (1) 4 q (1) = (w (1) ) T v 5 p (1) = q (1) 6 If = n goto 15 7 For = + 1,, n 8 q (1) 9 l = q(1) q (1) 10 p (1) 11 u = p(1) p (1) 12 w () = (w (1) ) T v = a 1 k=1 u kq (k1) = w (1) ( q (1) q (1) 13 EndDo 14 EndDo 15 Let p = p (1) for 1 n )w (1) p p Return L = (l ) 1 < n, U = (u ) 1 < n and D = 0 0 p n In exact arthmetc the above algorthm, as Algorthm 2, s applcable for nonsymmetrc postve defnte matrces and no dvson by zero wll occur Of course, nstabltes due to postve but extremely small pvots may occur n fnte precson and a thresholdng technque may stll be necessary to guard aganst such possbltes As prevous secton, we can use drop tolerances T 1 and T 2 and obtan two varants of ncomplete factorzaton based on A-bconugaton process The RIF NS precondtoner wll obtan by ncorporatng a sngle droppng strategy wth drop tolerance T 1 for only sparsfyng vector w () after step 12 of Algorthm 3 The RIF p NS precondtoner wll obtan by usng a droppng strategy wth drop tolerance T 1 for sparsfyng vectors w () after step 12 of Algorthm 3 and also by usng a droppng strategy wth drop tolerance T 2 for removng small elements of L and U factors after steps 9 and 11 of ths algorthm

5 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) Table 1 Convergence results for teratve methods wthout precondtonng Matrx n NZ It-QMR It-BCG It-GMRES(10) Reference hor Ď Ď Ď MatrxMarket [14] cdde MatrxMarket [14] pde MatrxMarket [14] sherman MatrxMarket [14] sherman MatrxMarket [14] add MatrxMarket [14] add MatrxMarket [14] pde MatrxMarket [14] fdap MatrxMarket [14] saylr MatrxMarket [14] cavty MatrxMarket [14] cavty Ď Ď Ď MatrxMarket [14] raat Ď Ď Ď UFCollecton [15] raat Ď UFCollecton [15] raefsky UFCollecton [15] raefsky UFCollecton [15] epb Ď 1324 Ď UFCollecton [15] pol UFCollecton [15] Snce for k < the elements u k are used to compute p (1) n lne 10 of Algorthm 3, the drop tolerance T 2 should be chosen small snce large value leads to the degradaton of the convergence rate of teratve methods 4 Numercal experments In ths secton we focus our attenton on comparson between our new precondtoner RIF p NS and the well-known RIF p precondtoner All the tests were run on a PC wth CPU 3 GHz(full) and 100 GB of RAM and all the codes were wrtten n Matlab Results of rght precondtoned QMR, BCGSTAB and GMRES(10) methods usng both RIF p and RIF p NS precondtoners are presented n Tables 3 5 The results of unprecondtoned methods and matrx propertes are lsted n Table 1 In ths table n means dmenson of the matrx, NZ stands for the number of nonzero entres of the matrx, and It-QMR, It-BCG and It-GMRES(10) denote the number of teratons of QMR, BCGSTAB and GMRES(10) methods, respectvely The nonsymmetrc test matrces are taken from the Unversty of Florda sparse matrx collecton [15] and Matrx Market collecton [14] The test matrces were not reordered and no scalng was used The ntal vector for all tests s selected zero vector and the rght hand sde vector s b = Ae, where e = [1,, 1] T The stoppng crteron r k , was used, where r k = b Ax k s the kth terated resdual of the lnear system to be solved In ths table a dagger (Ď) ndcates that no convergence s acheved after teratons In Table 2, propertes of the precondtoners are lsted densty means densty = NZ( L) + NZ(Ū), NZ(A) where NZ(X) s the number of nonzero entres of matrx X In ths table Prcosts denotes the number of arthmetc operatons (Flops) for constructng the precondtoner dvded by NZ(A) For all matrces we have used the drop tolerances T 1 = 01 and T 2 = 001 to construct both precondtoners In Tables 3 5, Itr stands for the number of teratons, Itcosts denotes the number of Flops needed for the teraton phase dvded by NZ(A) and Tcosts s the total number of arthmetc operatons, e, Tcosts = Prcosts + Itcosts In Tables 3 5 a dagger (Ď) means that no convergence s acheved after 1000 teratons In the computaton of the two precondtoners whenever the pvot element was less than we replaced t by 01 Ths case only happened for matrces saylr3, cavty05, and cavty06 Table 2 shows that for T 1 = 01 and T 2 = 001, the nonzero densty of RIF p NS s close to that of RIF p precondtoner Thus the comparson of the number of teratons and the Prcosts for RIF p and RIF p NS precondtoners wll be meanngful Table 2 also shows that, for all matrces except epb0 the Prcosts of RIF p NS precondtoner s better than that of RIF p precondtoner Results of Tables 3 5 show that the two precondtoners gve smlar results from the pont of vew of the rate of convergence Table 3 shows that for all matrces except cavty06, raat04, and epb0, the total costs (Tcosts) of applyng the QMR method wth RIF p NS precondtoner are less than the total costs of applyng QMR method wth RIF p precondtoner The results of Table 4 are smlar to those of Table 3, except for cavty05 for whch Tcosts of RIF p precondtoner s also smaller than that of RIF p NS precondtoner Table 5 shows that for most problems RIF p NS

6 704 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) Table 2 Propertes of mplct precondtoners Matrx RIF p RIF p NS densty Prcosts densty Prcosts hor cdde pde sherman sherman add add pde fdap saylr cavty cavty raat raat raefsky raefsky epb pol Table 3 Results of rght precondtoned QMR Matrx RIF p RIF p NS Itr Itcosts Tcosts Itr Itcosts Tcosts hor-131 Ď Ď Ď cdde pde sherman sherman add add pde fdap saylr cavty cavty raat raat raefsky raefsky epb pol Table 4 Results of rght precondtoned BCGSTAB Matrx RIF p RIF p NS Itr Itcosts Tcosts Itr Itcosts Tcosts hor-131 Ď Ď Ď cdde pde sherman sherman add add pde fdap saylr cavty cavty raat raat raefsky raefsky2 Ď Ď Ď epb pol

7 A Rafe, F Toutounan / Journal of Computatonal and Appled Mathematcs 230 (2009) Table 5 Results of rght precondtoned GMRES(10) Matrx RIF p RIF p NS Itr Itcosts Tcosts Itr Itcosts Tcosts hor-131 Ď Ď Ď cdde pde sherman sherman add add pde fdap saylr cavty cavty Ď Ď Ď raat04 Ď Ď Ď Ď Ď Ď raat raefsky raefsky2 Ď Ď Ď epb0 Ď Ď Ď pol precondtoner s cheaper (n terms of total costs) than RIF p precondtoner and for some matrces (lke cavty06 and raat04) there s no convergence wth RIF p NS precondtoner From the above dscusson we can conclude that RIF p NS precondtoner s a robust and effectve precondtoner for teratve methods 5 Concluson In ths paper a breakdown-free varant of an ILU precondtoner for NSPD matrces, based on the reformulaton of A- bconugaton algorthm, was presented The new precondtoner, called RIF p NS can be computed wthout usng matrx A T We observe that RIF p NS precondtoner s applcable for nonsymmetrc postve defnte matrces and no breakdown wll occur durng ts constructon Numercal experments show that RIF p NS precondtoner s as effectve as RIF p precondtoner at reducng number of teratons of teratve methods Numercal results ndcate that the nonzero denstes of RIF p NS and RIF p precondtoners are about the same and for most of problems RIF p NS precondtoner s cheaper (n terms of precondtonng costs and total costs) than RIF p precondtoner We can conclude that RIF p NS precondtoner can be a useful tool for solvng large and sparse NSPD lnear systems References [1] O Axelsson, LYu Kolotlna, Dagonally compensated reducton and related precondtonng methods, Numer Lnear Algebra Appl 1 (1994) 155 [2] MA Az, A Jennngs, A robust ncomplete Cholesky-conugate algorthm, Int J Numer Methods Eng 20 (1984) 949 [3] M Benz, M Tůma, A robust ncomplete factorzaton precondtoner for postve defnte matrces, Numer Lnear Algebra Appl 10 (2003) [4] DS Kershaw, The ncomlpete Cholesky conugste gradent method for the teratve soluton of systems of lnear equatons, J Comput Phys 26 (1978) 43 [5] TA Manteuffel, An ncomplete factorzaton technque for postve defnte lnear systems, Math Comp 34 (1980) 473 [6] M Tsmenetsky, A new precondtonng technque for solvng large sparse lnear systems, Lnear Algebra Appl (1991) [7] LN Trefethen, D Bau III, Numercal Lnear Algebra, SIAM, Phladelpha, PA, 1997 [8] HA Van der Vorst, Iteratve soluton methods for certan sparse lnear systems wth a nonsymmetrc matrx arsng from PDE-problems, J Comput Phys 44 (1981) 1 [9] M Rezgh, SM Hoseen, An ILU precondtoner for nonsymmetrc postve defnte matrces by usng the conugate Gram Schmdt process, J Comput Appl Math 188 (2006) [10] M Benz, M Tůma, A sparse approxmate nverse precondtoner for nonsymmetrc lnear systems, SIAM J Sc Comput 19 (1998) [11] M Benz, JK Cullum, M Tůma, Robust approxmate nverse precondtonng for the conugate gradent method, SIAM J Sc Comput 22 (2000) [12] SA Kharchenko, LYu Kolotlna, AA Nkshn, AYu Yeremn, A robust AINV-type method for constructng sparse approxmate nverse precondtoners n factored form, Numer Lnear Algebra Appl 8 (2001) [13] A Rafe, F Toutounan, New breakdown-free varant of AINV method for nonsymmetrc postve defnte matrces, J Comput Appl Math 219 (1) (2008) [14] Matrx Market page [15] Unversty of Florda Sparse Matrx Collecton web page