Research Article Parallel Rayleigh Quotient Optimization with FSAI-Based Preconditioning

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1 Alied Mathematics Volume 2012, Article ID , 14 ages doi: /2012/ Research Article Parallel Rayleigh Quotient Otimization with FSAI-Based Preconditioning Luca Bergamaschi, Angeles Martínez, and Giorgio Pini Deartment of Mathematical Methods and Models for Scientific Alications, University of Padova, Via Trieste 63, Padova, Italy Corresondence should be addressed to Luca Bergamaschi, Received 2 November 2011; Revised 1 February 2012; Acceted 3 February 2012 Academic Editor: Massimiliano Ferronato Coyright q 2012 Luca Bergamaschi et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. The resent aer describes a arallel reconditioned algorithm for the solution of artial eigenvalue roblems for large sarse symmetric matrices, on arallel comuters. Namely, we consider the Deflation-Accelerated Conjugate Gradient DACG algorithm accelerated by factorizedsarse-aroximate-inverse- FSAI- tye reconditioners. We resent an enhanced arallel imlementation of the FSAI reconditioner and make use of the recently develoed Block FSAI-IC reconditioner, which combines the FSAI and the Block Jacobi-IC reconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these tye of reconditioners is cometitive with resect to the available ublic arallel hyre ackage, eseciallyin thecomutationofa fewof theleftmosteigenairs. The arallel DACG code accelerated by FSAI is written in MPI-Fortran 90 language and exhibits good scalability u to one thousand rocessors. 1. Introduction The comutation by iterative methods of the s artial eigensectrum of the generalized eigenroblem: Au λbu, 1.1 where A, B R n n are large sarse symmetric ositive definite SPD matrices, is an imortant and difficult task in many alications. It has become increasingly widesread owing to the develoment in the last twenty years of robust and comutationally efficient schemes and corresonding software ackages. Among the most well-known aroaches for the imortant class of symmetric ositive definite SPD matrices are the imlicitly restarted Arnoldi method equivalent to the Lanczos technique for this tye of matrices 1, 2,

2 2 Alied Mathematics the Jacobi-Davidson JD algorithm 3, and schemes based on reconditioned conjugate gradient minimization of the Rayleigh quotient 4, 5. The basic idea of the latter is to minimize the Rayleigh Quotient q x xt Ax x T Bx. 1.2 in a subsace which is orthogonal to the reviously comuted eigenvectors via a reconditioned CG-like rocedure. Among the different variants of this technique we chose to use the Deflation-Accelerated Conjugate Gradient DACG scheme 4, 6 which has been shown to be cometitive with the Jacobi Davidson method and with the PARPACK ackage 7. As in any other aroach, for our DACG method, the choice of the reconditioning technique is a key factor to accelerate and, in some cases even to allow for, convergence. To accelerate DACG in a arallel environment we selected the Factorized Sarse Aroximate inverse FSAI reconditioner introduced in 8. We have develoed a arallel imlementation of this algorithm which has dislayed excellent erformances on both the setu hase and the alication hase within a Krylov subsace solver The effectiveness of the FSAI reconditioner in the acceleration of DACG is comared to that of the Block FSAI-IC reconditioner, recently develoed in 12, which combines the FSAI and the Block Jacobi-IC reconditioners obtaining good results on a small number of rocessors for the solution of SPD linear systems and for the solution of large eigenroblems 13. We used the resulting arallel codes to comute a few of the leftmost eigenairs of a set of test matrices of large size arising from Finite Element discretization of geomechanical models. The reorted results show that DACG reconditioned with either FSAI or BFSAI is a scalable and robust algorithm for the artial solution of SPD eigenroblems. The arallel erformance of DACG is also comared to that of the ublicly available arallel ackage hyre 14 which imlements a number of reconditioners which can be used in combination with the Locally Otimal Block PCG LOBPCG iterative eigensolver 15. The results resented in this aer show that the arallel DACG code accelerated by FSAI exhibits good scalability u to one thousand rocessors and dislays comarable erformance with resect to hyre, secially when a low number of eigenairs is sought. The outline of the aer is as follows: in Section 2 we describe the DACG Algorithm; in Sections 3 and 4 we recall the definition and roerties of the FSAI and BFSAI reconditioners, resectively. Section 5 contains the numerical results obtained with the roosed algorithm in the eigensolution of very large SPD matrices of size u to almost 7 million unknowns and nonzeros. A comarison with the hyre eigensolver code is also included. Section 6 ends the aer with some conclusions. 2. The DACG Iterative Eigensolver and Imlementation The DACG algorithm sequentially comutes the eigenairs, starting from the leftmost one λ 1, u 1. To evaluate the jth eigenair, j>1, DACG minimizes the Rayleigh Quotient RQ in a subsace orthogonal to the j 1 eigenvectors reviously comuted. More recisely, DACG minimizes the Rayleigh Quotient: q z zt Az z T z, 2.1

3 Alied Mathematics 3 Choose tolerance ε,setu 0. DO j 1,s 1 Choose x 0 such that U T x 0 0; set k 0, β 0 0; 2 x A 0 Ax 0,γ x T 0 xa 0,η xt 0 x 0,q 0 q x 0 γ/η, r 0 x A 0 q 0x 0 ; 3 REPEAT 3.1 g k q x k 2/η r k ; 3.2 g M k Mg k; 3.3 IF k>0thenβ k g T k gm k gm k 1 /gt k 1 gm k 1 ; 3.4 k g M k β k k 1 ; 3.5 k k U U T k 3.6 A k A k, 3.7 α k argmin t {q x k t k } ηd γb Δ /2 bc ad, with a T k xa k,b T k A k,c T k x k, d T k k, Δ ηd γb 2 4 bc ad γa ηc ; 3.8 x k 1 x k α k k, x A k 1 xa k α k A k ; 3.9 γ γ 2aα k bα 2 k, η η 2cα k dα 2 k ; 3.10 q k 1 q x k 1 γ/η; 3.11 k k 1; 3.12 r k x A k q kx k UNTIL q k 1 q k /q k 1 <tol; 4 λ j q k, u j x k / η, U U, u j. END DO Algorithm 1: DACG Algorithm. where z x U j ( U T j x ), U j [ u 1,...,u j 1 ], x R n. 2.2 The first eigenair λ 1, u 1 is obtained by minimization of 2.1 with z x U 1. Indicating with M the reconditioning matrix, that is, M A 1,thes leftmost eigenairs are comuted by the conjugate gradient rocedure 6 described in Algorithm 1. The schemes relying on the Rayleigh quotient otimization are quite attractive for arallel comutations; however reconditioning is an essential feature to ensure ractical convergence. When seeking for an eigenair λ j, u j it can be roved that the number of iterations is roortional to the square root of the condition number ξ j κ H j of the Hessian of the Rayleigh quotient in the stationary oint u j 4. It turns out that H j is similar to A λ j I M which is not SPD. However, H j oerates on the orthogonal sace sanned by the revious eigenvectors, so that the only imortant eigenvalues are the ositive ones. In the non-reconditioned case i.e., M I we would have κ ( H j ) λ N λ j 1 λ j. 2.3 where in the ideal case M A 1, we have κ ( H j ) λ j ξ j λ N. λ j 1 λ j λ j 1 λ j 2.4

4 4 Alied Mathematics Therefore, even though A 1 is not the otimal reconditioner for A λ j I, however, if M is a good reconditioner of A then the condition number κ H j will aroach ξ j. 3. The FSAI Preconditioner The FSAI reconditioner, initially roosed in 8, 16, has been later develoed and imlemented in arallel by Bergamaschi and Martínez in 9. Here, we only shortly recall the main features of this reconditioner. Given an SPD matrix A the FSAI reconditioner aroximately factorizes its inverse as a roduct of two sarse triangular matrices as A 1 W T W. 3.1 The choice of nonzeros in W is based on a sarsity attern which in our work may be the same as à d where à is the result of refiltration 10 of A, that is, droing of all elements below of a threshold arameter δ. The entries of W are comuted by minimizing the Frobenius norm of I WL, where L is the exact Cholesky factor of A, without forming exlicitly the matrix L. The comuted W is then sarsified by droing all the elements which are below a second tolerance arameter ε. The final FSAI reconditioner is therefore related to the following three arameters: δ, refiltration threshold; d, ower of A generating the sarsity attern we allow d {1, 2, 4} in our exeriments ; ε, ostfiltration threshold Parallel Imlementation of FSAI-DACG We have develoed a arallel code written in FORTRAN 90 and which exloits the MPI library for exchanging data among the rocessors. We used a block row distribution of all matrices A, W, andw T, that is, with comlete rows assigned to different rocessors. All these matrices are stored in static data structures in CSR format. Regarding the reconditioner comutation, we stress that any row i of matrix W of FSAI reconditioner is comuted indeendently of each other, by solving a small SPD dense linear system of size n i equal to the number of nonzeros allowed in row i of W. Some of the rows which contribute to form this linear system may be nonlocal to rocessor i and should be received from other rocessors. To this aim we imlemented a routine called get extra rows which carries out all the row exchanges among the rocessors, before starting the comutation of W, which roceed afterwards entirely in arallel. Since the number of nonlocal rows needed by each rocessor is relatively small we chose to temorarily relicate these rows on auxiliary data structures. Once W is obtained a arallel transosition routine rovides every rocessor with its art of W T. The DACG iterative solver is essentially based on scalar and matrix-vector roducts. We made use of an otimized arallel matrix-vector roduct which has been develoed in 17 showing its effectiveness u to 1024 rocessors. 4. Block FSAI-IC Preconditioning The Block FSAI-IC reconditioner, BFSAI-IC in the following, is a recent develoment for the arallel solution to Symmetric Positive Definite SPD linear systems. Assume that D is an arbitrary nonsingular block diagonal matrix consisting of n b equal size blocks.

5 Alied Mathematics 5 Let S L and S BD be a sarse lower triangular and a dense block diagonal nonzero attern, resectively, for an n n matrix. Even though not strictly necessary, for the sake of simlicity assume that S BD consists of n b diagonal blocks with equal size m n/n b and let D R n n be an arbitrary full-rank matrix with nonzero attern S BD. Consider the set of lower block triangular matrices F with a rescribed nonzero attern S BL and minimize over F the Frobenius norm: D FL F, 4.1 where L is the exact lower Cholesky factor of an SPD matrix A. A matrix F satisfying the minimality condition 4.1 for a given D is the lower block triangular factor of BFSAI-IC. Recalling the definition of the classical FSAI reconditioner, it can be noticed that BFSAI-IC is a block generalization of the FSAI concet. The differentiation of 4.1 with resect to the unknown entries F ij, i, j S BL, yields the solution to n indeendent dense subsystems which, in the standard FSAI case, do not require the exlicit knowledge of L. The effect of alying F to A is to concentrate the largest entries of the reconditioned matrix FAF T into n b diagonal blocks. However, as D is arbitrary, it is still not ensured that FAF T is better than A in an iterative method, so it is necessary to recondition FAF T again. As FAF T resembles a block diagonal matrix, an efficient technique relies on using a block diagonal matrix which collects an aroximation of the inverse of each diagonal block B ib of FAF T. It is easy to show that F is guaranteed to exist with SPD matrices and B ib is SPD, too 12. Using an IC decomosition with artial fill-in for each block B ib and collecting in J the lower IC factors, the resulting reconditioned matrix reads J 1 FAF T J T WAW T 4.2 with the final reconditioner M W T W F T J T J 1 F. 4.3 M in 4.3 is the BFSAI-IC reconditioner of A. For its comutation BFSAI-IC needs the selection of n b and S L. The basic requirement for the number of blocks n b is to be larger than or equal to the number of comuting cores. From a ractical viewoint, however, the most efficient choice in terms of both wall clock time and iteration count is to kee the blocks as large as ossible, thus imlying n b. Hence, n b is by default set equal to. By distinction, the choice of S L is theoretically more challenging and still not comletely clear. A widely acceted otion for other aroximate inverses, such as FSAI or SPAI, is to select the nonzero attern of A d for small values of d on the basis of the Neumann series exansion of A 1. Using a similar aroach, in the BFSAI construction we select S L as the lower block triangular attern of A d. As the nonzeros located in the diagonal blocks are not used for the comutation of F a larger value of d, say 3 or 4, can still be used.

6 6 Alied Mathematics Though theoretically not necessary, three additional user-secified arameters are worth introducing in order to better control the memory occuation and the BFSAI-IC density: 1 ε is a ostfiltration arameter that allows for droing the smallest entries of F. In articular, F ij is neglected if F ij <ε f i 2, where f i is the ith row of F; 2 ρ B is a arameter that controls the fill-in of B ib and determines the maximum allowable number of nonzeros for each row of B ib in addition to the corresonding entries of A. Quite obviously, the largest ρ B entries only are retained; 3 ρ L is a arameter that controls the fill-in of each IC factor L ib denoting the maximum allowable number of nonzeros for each row of L ib in addition to the corresonding entries of B ib. An OenMP imlementation of the algorithms above is available in Numerical Results In this section we examine the erformance of the arallel DACG reconditioned by both FSAI and BFSAI in the artial solution of four large-size sarse eigenroblems. The test cases, which we briefly describe below, are taken from different real engineering mechanical alications. In detail, they are as follows. i FAULT-639 is obtained from a structural roblem discretizing a faulted gas reservoir with tetrahedral finite elements and triangular interface elements 19. The interface elements are used with a enalty formulation to simulate the faults behavior. The roblem arises from a 3D discretization with three dislacement unknowns associated to each node of the grid. ii PO-878 arises in the simulation of the consolidation of a real gas reservoir of the Po Valley, Italy, used for underground gas storage uroses for details, see 20. iii GEO-1438 is obtained from a geomechanical roblem discretizing a region of the earth crust subject to underground deformation. The comutational domain is a boxwithanarealextentof50 50 km and 10 km dee consisting of regularly shaed tetrahedral finite elements. The roblem arises from a 3D discretization with three dislacement unknowns associated to each node of the grid 21. iv CUBE-6091 arises from the equilibrium of a concrete cube discretized by a regular unstructured tetrahedral grid. Matrices FAULT-639 and GEO-1438 are ublicly available in the University of Florida Sarse Matrix Collection at htt:// In Table 1 we reort sizes and nonzeros of the four matrices together with three of the most significant eigenvalues for each roblem. The comutational erformance of FSAI is comared to the one obtained by using BFSAI as imlemented in 12. The comarison is done evaluating the number of iterations n iter to converge at the same tolerance, the wall clock time in seconds T rec and T iter for the reconditioner comutation, and the eigensolver to converge, resectively, with the total time T tot T rec T iter. All tests are erformed on the IBM SP6/5376 cluster at the CINECA Centre for High Performance Comuting, equied with IBM Power6 rocessors at 4.7 GHz

7 Alied Mathematics 7 Table 1: Size, number of nonzeros, and three reresentative eigenvalues of the test matrices. Size Nonzeros λ 1 λ 10 λ N FAULT ,802 28,614, PO ,355 38,847, GEO ,437,960 63,156, CUBE ,091, ,800, with 168 nodes, 5376 comuting cores, and 21 Tbyte of internal network RAM. The FSAI- DACG code is written in Fortran 90 and comiled with -O4 -q64 -qarch=wr6 -qtune=wr6 -qnoia -qstrict -bmaxdata:0x otions. For the BFSAI-IC code only an OenMP imlementation is resently available. To study arallel erformance we will use a strong scaling measure to see how the CPU times vary with the number of rocessors for a fixed total roblem size. Denote with T the total CPU elased times exressed in seconds on rocessors. We introduce a relative measure of the arallel efficiency achieved by the code, S, which is the seudo seedu comuted with resect to the smallest number of rocessors used to solve a given roblem. Accordingly, we will denote by E the corresonding efficiency: S T T, E S T T FSAI-DACG Results In this section we reort the results of our FSAI-DACG imlementation in the comutation of the 10 leftmost eigenairs of the 4 test roblems. We used the exit test described in the DACG algorithm see Algorithm 1 with tol The results are summarized in Table 2. As the FSAI arameters, we choose δ 0.1, d 4, and ε 0.1 for all the test matrices. This combination of arameters roduces, on the average, the best or close to the best erformance of the iterative rocedure. Note that the number of iterations does not change with the number of rocessors, for a fixed roblem. The scalability of the code is very satisfactory in both the setu stage reconditioner comutation and the iterative hase BFSAI-IC-DACG Results We resent in this section the results of DACG accelerated by the BFSAI-IC reconditioner for the aroximation of the s 10 leftmost eigenairs of the matrices described above. Table 3 rovides iteration count and total CPU time for BFSAI-DACG with different combinations of the arameters needed to construct the BFSAI-IC reconditioner for matrix PO-878 and using from 2 to 8 rocessors. It can be seen from Table 3 that the assessment of the otimal arameters, ε, ρ B,andρ L, is not an easy task, since the number of iterations may highly vary deending on the number of rocessors. We chose in this case the combination of arameters roducing the second smallest total time with 2, 4, 8 rocessors. After

8 8 Alied Mathematics Table 2: Number of iterations, timings, and scalability indices for FSAI-DACG in the comutation of the 10 leftmost eigenairs of the four test roblems. FAULT-639 PO-878 GEO-1437 CUBE-6091 iter T rec T iter T tot S iter T rec T iter T tot S 16 E E 4 intensive testing for all the test roblems, we selected similarly the otimal values which are used in the numerical exeriments reorted in Table 4: i FAULT-639: d 3, ε 0.05, ρ B 10, ρ L 60, ii PO-878 d 3, ε 0.05, ρ B 10, ρ L 10, iii GEO-1438: d 3, ε 0.05, ρ B 10, ρ L 50, iv CUBE-6091: d 2, ε 0.01, ρ B 0, ρ L 10. The user-secified arameters for BFSAI-IC given above rovide evidence that it is imortant to build a dense reconditioner based on the lower nonzero attern of A 3 excet for CUBE-6091, which is built on a regular discretization with the aim at decreasing the number of DACG iterations. Anyway, the cost for comuting such a dense reconditioner aears to be almost negligible with resect to the wall clock time needed to iterate to convergence. We recall that, resently, the code BFSAI-IC is imlemented in OenMP, and the results in terms of CPU time are significant only for 8. For this reason the number of iterations

9 Alied Mathematics 9 Table 3: Performance of BFSAI-DACG for matrix PO-878 with 2 to 8 rocessors and different arameter values d ρ B ε ρ L iter T tot iter T tot iter T tot Matrix Table 4: Number of iterations for BFSAI-DACG in the comutations of the 10 leftmost eigenairs. n b FAULT PO GEO CUBE reorted in Table 4 is obtained with increasing number of blocks n b and with 8 rocessors. This iteration number accounts for a otential imlementation of BFSAI-DACG under the MPI or hybrid OenMP-MPI environment as the number of iterations deends only on the number of blocks, irresective of the number of rocessors. The only meaningful comarison between FSAI-DACG and BFSAI-DACG can be carried out in terms of iteration numbers which are smaller for BFSAI-DACG for a small number of rocessors. The ga between FSAI and BFSAI iterations reduces when the number of rocessors increases Comarison with the LOBPCG Eigensolver Provided by hyre In order to validate the effectiveness of our reconditioning in the roosed DACG algorithm with resect to already available ublic arallel eigensolvers, the results given in Tables 2 and 4 are comared with those obtained by the schemes imlemented in the hyre software ackage 14. The Locally Otimal Block Preconditioned Conjugate Gradient method LOBPCG 15 is exerimented with, using the different reconditioners develoed in the hyre roject, that is, algebraic multigrid AMG, diagonal scaling DS, aroximate inverse ParaSails, additive Schwarz Schwarz, and incomlete LU Euclid. The hyre reconditioned CG is used for the inner iterations within LOBPCG. For details on the imlementation of the LOBPCG algorithm, see, for instance, 22. The selected reconditioner, ParaSails, is on its turn based on the FSAI reconditioner, so that the different

10 10 Alied Mathematics Table 5: Iterations and CPU time for the iterative solver of LOBPCG-hyre reconditioned by Parasails with different values of bl and 16 rocessors. Matrix bl 10 bl 11 bl 12 bl 15 iter T iter iter T iter iter T iter iter T iter FAULT PO GEO CUBE FSAI-DACG and ParaSails-LOBPCG erformances should be ascribed mainly to the different eigensolvers rather than to the reconditioners. We first carried out a reliminary set of runs with the aim of assessing the otimal value of the block size bl arameter, that is, the size of the subsace where to seek for the eigenvectors. Obviously it must be bl s 10. We fixed to 16 the number of rocessors and obtained the results summarized in Table 5 with different values of bl 10, 15. We found that, only in roblem CUBE-6091, a value of bl larger than 10, namely, bl 12, yields an imrovement in the CPU time. Note that we also made this comarison with different number of rocessors, and we obtained analogous results. Table 6 resents the number of iterations and timings using the LOBPCG algorithm in the hyre ackage. The LOBPCG wall clock time is obtained with the reconditioner allowing for the best erformance in the secific roblem at hand, that is, ParaSails for all the roblems. Using AMG as the reconditioner did not allow for convergence in three cases out of four, with the only excetion of the FAULT-639 roblem, in which the CPU timings were however very much larger than using ParaSails. All matrices have to be reliminarily scaled by their maximum coefficient in order to allow for convergence. To make the comarison meaningful, the outer iterations of the different methods are stoed when the average relative error measure of the comuted leftmost eigenairs gets smaller than 10 10, in order to obtain a comarable accuracy as in the other codes. We also reort in Table 6 the number of inner reconditioned CG iterations cgitr. To better comare our FSAI DACG with the LOBPCG method, we deict in Figure 1 the total CPU time versus the number of rocessor for the two codes. FSAI-DACG and LOBPCG rovide very similar scalability, being the latter code a little bit more erforming on the average. On the FAULT-639 roblem, DACG reveals faster than LOBPCG, irresective of the number of rocessors emloyed. Finally, we have carried out a comarison of the two eigensolvers in the comutation of only the leftmost eigenair. Differently from LOBPCG, which erforms a simultaneous aroximation of all the selected eigenairs, DACG roceeds in the comutation of the selected eigenairs in a sequential way. For this reason, DACG should be the better choice, at least in rincile, when just one eigenair is sought. We investigate this feature, and the results are summarized in Table 7. We include the total CPU time and iteration count needed by LOBPCG and FSAI-DACG to comute the leftmost eigenair with 16 rocessors. For the LOBPCG code we reort only the number of outer iterations. The arameters used to construct the FSAI reconditioner for these exeriments are as follows: 1 FAULT-639. δ 0.1, d 2, ε 0.05, 2 PO-878. δ 0.2, d 4, ε 0.1,

11 Alied Mathematics 11 FAULT-639 PO-878 Total CPU time Total CPU time Number of rocessors Number of rocessors FSAI-DACG HYPRE a GEO-1439 FSAI-DACG HYPRE b CUBE Total CPU time Total CPU time Number of rocessors Number of rocessors FSAI-DACG HYPRE c FSAI-DACG HYPRE d Figure 1: Comarison between FSAI-DACG and LOBPCG-hyre in terms of total CPU time for different number of rocessors. 3 GEO δ 0.1, d 2, ε 0.1, 4 CUBE δ 0.0, d 1, ε These arameters differ from those emloyed to comute the FSAI reconditioner in the assessment of the 10 leftmost eigenairs and have been selected in order to roduce a reconditioner relatively chea to comute. This is so because otherwise the setu time would revail over the iteration time. Similarly, to comute just one eigenair with LOBPCG we need to setu a different value for cgitr, the number of inner iterations. As it can be seen from Table 7, in the majority of the test cases, LOBPCG takes less time to comute 2 eigenairs than just only 1. FSAI-DACG reveals more efficient than the best LOBPCG on roblems PO-878 and GEO On the remaining two roblems the slow convergence exhibited by DACG is robably due to the small relative searation ξ 1 between λ 1 and λ 2.

12 12 Alied Mathematics Table 6: Number of iterations, timings, and scalability of LOBPCG-hyre reconditioned by Parasails. FAULT-639 cgitr 5 PO-878 cgitr 30 GEO-1438 cgitr 30 CUBE-6091 iter T rec T iter T tot S iter T rec T iter T tot S 16 E E 4 Table 7: Performance of LOBPCG-hyre with Parasails and 16 rocessors in the comutation of the smallest eigenvalue using bl 1andbl 2 and FSAI-DACG. LOBPCG, bl 1 LOBPCG, bl 2 FSAI-DACG iter T tot iter T tot iter T tot FAULT PO GEO CUBE Conclusions We have resented the arallel DACG algorithm for the artial eigensolution of large and sarse SPD matrices. The scalability of DACG, accelerated with FSAI-tye reconditioners,

13 Alied Mathematics 13 has been studied on a set of test matrices of very large size arising from real engineering mechanical alications. Our FSAI-DACG code has shown comarable erformances with the LOBPCG eigensolver within the well-known ublic domain ackage, hyre. Numerical results reveal that not only the scalability achieved by our code is roughly identical to that of hyre but also, in some instances, FSAI-DACG roves more efficient in terms of absolute CPU time. In articular, for the comutation of the leftmost eigenair, FSAI-DACG is more convenient in 2 roblems out of 4. Acknowledgment The authors acknowledge the CINECA Iscra Award SCALPREC 2011 for the availability of HPC resources and suort. References 1 P. Arbenz, U. Hetmaniuk, R. Lehoucq, and R. Tuminaro, A comarison of eigensolvers for large-scale 3D modal analysis using AMG-reconditioned iterative methods, International Journal for Numerical Methods in Engineering, vol. 64, no. 2, , R. B. Lehoucq and D. C. Sorensen, Deflation techniques for an imlicitly restarted Arnoldi iteration, SIAM Journal on Matrix Analysis and Alications, vol. 17, no. 4, , G. L. G. Sleijen and H. A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue roblems, SIAM Journal on Matrix Analysis and Alications, vol. 17, no. 2, , L. Bergamaschi, G. Gambolati, and G. Pini, Asymtotic convergence of conjugate gradient methods for the artial symmetric eigenroblem, Numerical Linear Algebra with Alications, vol. 4, no. 2, , A. V. Knyazev and A. L. Skorokhodov, Preconditioned gradient-tye iterative methods in a subsace for artial generalized symmetric eigenvalue roblems, SIAM Journal on Numerical Analysis, vol. 31, no. 4, , L. Bergamaschi, G. Pini, and F. Sartoretto, Aroximate inverse reconditioning in the arallel solution of sarse eigenroblems, Numerical Linear Algebra with Alications, vol.7,no.3, , L. Bergamaschi and M. Putti, Numerical comarison of iterative eigensolvers for large sarse symmetric ositive definite matrices, Comuter Methods in Alied Mechanics and Engineering, vol. 191, no. 45, , L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sarse aroximate inverse reconditionings. I. Theory, SIAM Journal on Matrix Analysis and Alications, vol. 14, no. 1, , L. Bergamaschi and A. Martínez, Parallel acceleration of Krylov solvers by factorized aroximate inverse reconditioners, in VECPAR 2004, M. Dayd et al., Ed., vol of Lecture Notes in Comuter Sciences, , Sringer, Heidelberg, Germany, L. Bergamaschi, A. Martínez, and G. Pini, An efficient arallel MLPG method for oroelastic models, Comuter Modeling in Engineering & Sciences, vol. 49, no. 3, , L. Bergamaschi and A. Martínez, Parallel inexact constraint reconditioners for saddle oint roblems, in Proceedings of the 17th International Conference on Parallel Processing (Euro-Par 11), R.N.E. Jeannot and J. Roman, Eds., vol. 6853, art 2, , Sringer, Bordeaux, France, 2011, Lecture Notes in Comuter Sciences. 12 C. Janna, M. Ferronato, and G. Gambolati, A block FSAI-ILU arallel reconditioner for symmetric ositive definite linear systems, SIAM Journal on Scientific Comuting, vol. 32, no. 5, , M. Ferronato, C. Janna, and G. Pini, Efficient arallel solution to large-size sarse eigenroblems with block FSAI reconditioning, Numerical Linear Algebra with Alications. In ress. 14 Lawrence Livermore National Laboratory, hyre user manual, software version Center for Alied Scientific Comuting CASC, University of California, A. V. Knyazev, Toward the otimal reconditioned eigensolver: locally otimal block reconditioned conjugate gradient method, SIAM Journal on Scientific Comuting, vol. 23, no. 2, , 2001.

14 14 Alied Mathematics 16 L. Yu. Kolotilina, A. A. Nikishin, and A. Yu. Yeremin, Factorized sarse aroximate inverse reconditionings. IV. Simle aroaches to rising efficiency, Numerical Linear Algebra with Alications, vol. 6, no. 7, , A. Martínez, L. Bergamaschi, M. Caliari, and M. Vianello, A massively arallel exonential integrator for advection-diffusion models, Comutational and Alied Mathematics, vol. 231, no. 1, , C. Janna, M. Ferronato, and N. Castelletto, BFSAI-IC OenMP imlementation, Release V1.0, January 2011, htt:// ferronat/software.html. 19 M. Ferronato, C. Janna, and G. Gambolati, Mixed constraint reconditioning in comutational contact mechanics, Comuter Methods in Alied Mechanics and Engineering, vol. 197, no , , N. Castelletto, M. Ferronato, G. Gambolati et al., 3D geomechanics in UGS rojects: a comrehensive study in northern Italy, in Proceedings of the 44th US Rock Mechanics Symosium, Salt Lake City, Utah, USA, P. Teatini, M. Ferronato, G. Gambolati, D. Bau, and M. Putti, An-throogenic Venice ulift by seawater uming into a heterogeneous aquifer system, Water Resources Research, vol. 46, Article ID W11547, 16 ages, A. V. Knyazev and M. E. Argentati, Imlementation of a reconditioned eigensolver using hyre, 2005, htt://math.ucdenver.edu/ rargenta/index files/re220.df.

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