In: Advances in Computational Mechanics, M. Papadrakakis andB.H.V. Topping (eds.), Civil-Comp Ltd., Edinburgh, Scotland, 1994GRADIENT EIGENANALYSIS ONNESTED FINITE ELEMENTSLuca Bergamaschi, Giuseppe Gambolati, Giorgio Pini and Mario PuttiDipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversity of Padova - via Belzoni 7 - 35131 Padova - ItalyAbstractThe e�cient computation of the leftmost eigenpairs of thegeneralized symmetric eigenproblem Ax = �Bx by a de- ation accelerated conjugate gradient (DACG) methodmay be enhanced by an improved estimate of the ini-tial eigenvectors obtained with a multigrid (MG) typeapproach. The DACG algorithm essentially optimizesthe Rayleigh quotient in subspaces of decreasing size B-orthogonal to the eigenvectors previously computed bya preconditioned conjugate gradient (CG) scheme. TheDACG asymptotic rate of convergence may be shown tobe controlled by the relative separation of the eigenvaluebeing currently sought and the next higher one and canbe e�ectively accelerated by the use of various precondi-tioners taken from the family of the incomplete Choleskydecompositions of A. The initial rate may be amelioratedby providing an initial guess calculated on nested �niteelement (FE) grids of growing resolution.The overall algorithm has been applied to structuraleigenproblems de�ned over four nested FE grids. Theresults for the computation of the 40 smallest eigenpairsindicate that the asymptotic convergence is very much de-pendent on the actual eigenvalue distribution and may besubstantially improved by the use of appropriate and rela-tively inexpensive preconditioners. The nested iterations(NI) may lead to a marked reduction of the initial itera-tions on the �nest grid level where the solution is �nallyrequired. NI decreases the CPU time by a factor of 2.5.The performance of the NI-DACG method is very promis-ing and emphasizes the potential of this new approach inthe partial solution of symmetric positive de�nite eigen-problems of large and very large size.1 IntroductionThe partial eigensolution of large sparse symmetric pos-itive de�nite matrices is a problem of great interest inmany scienti�c and engineering applications that makeuse of �nite di�erence or �nite element models. Practicalexamples may be found in the �eld of dynamical anal-ysis of elastic structures [2], lightwave technology [25],and solution of large sets of di�erential equations via thespectral superposition technique [7]. Several approachesmay be used for the partial eigenanalysis of sparse ma-trices. Among others, the subspace iteration [2, 1], the

Lanczos method [18, 5], multigrid procedures [13, 14], andoptimization schemes [15, 16] may be cited. An e�cientoptimization iteration, the De ation-Accelerated Conju-gate Gradient (DACG), based on the minimization of theRayleigh quotient by preconditioned conjugate gradientsover subspaces of decreasing size, has recently been de-veloped [12]. This method achieves high computationale�ciency in particular for large matrices and when onlyfew eigenpairs are sought. In these cases, DACG becomescompetitive with respect to other solution procedures,such as the Lanczos scheme [11]. A vector and parallelversion of this procedure can be found in [22].Typical DACG convergence pro�les are characterizedby an initial phase, which may converge slowly, followedby an asymptotic convergence phase, theoretically relatedto the spectral condition number of the Hessian of theRayleigh quotient [4]. Persistence of the initial phasedepends mainly on the starting guess eigenvectors andmay represent the most time consuming part of the al-gorithm. The e�ciency of DACG can be increased bymeans of a Nested Iteration (NI) technique [10]. In thismethod, named NI-DACG, solutions obtained on a num-ber of increasingly �ner nested grids are used to providean accurate initial guess eigenvector at the �nest level,thus accelerating the initial convergence, and, therefore,increasing the overall performance of the scheme. In thispaper we present an application of NI-DACG to two sam-ple problems arising from the �nite element discretizationof the 2-D membrane equation and a 2-D structural prob-lem. The results obtained with NI-DACG show speed-upsranging from 1.6 to 6 in the various cases considered here.The outline of the paper is as follows. After a shortdescription of the DACG and NI-DACG algorithms, theconvergence to several of the leftmost eigenpairs is stud-ied numerically and the relative performance of NI-DACGand DACG is compared and discussed. An acceleration ofDACG, as proposed in [3], that guarantees optimal per-formance, is also considered. The numerical results showthat this acceleration is very e�ective for NI-DACG aswell.2 NI-DACG algorithmThe NI-DACG algorithm consists essentially of two mainmodules. The �rst implements the DACG iterativescheme for the partial solution of the generalized eigen-1

problem Ax = �Bx (1)where A and B are sparse positive de�nite symmetric ma-trices of dimension N . The second is an implementationof the NI scheme and is intended to improve the initialguess used by DACG. The full scheme can be generallyapplied to the partial solution of (1), with matrices A andB arising from the �nite element integration of partial dif-ferential equations. We focus our attention on linear tri-angular �nite elements, so that nested grids can be easilygenerated and used in the de�nition of NI. Examples ofnested triangulations are shown in Figure 1 for a circularand a rectangular domain.2.1 DACG schemeThe DACG scheme can be de�ned as follows. Let �1 ��2 : : : � �N be the eigenvalues and v1;v2 : : : ;vN the cor-responding eigenvectors of (1). Assume that the �rst jleftmost eigenpairs have been already evaluated. The iter-ation for the calculation of the N�j-th smallest eigenpairproceeds as is shown below:1. start with an initial eigenvector estimate x0 suchthat V Tj Bx0=0, where Vj is the matrix formedby the previously calculated j eigenvectors: Vj =[vN ; : : : ;vN�j+1]; initialize the iteration index k tozero2. if k = 0 (�rst iteration), then set �k = 0, otherwise�k is calculated following either [21] and [9]:�k := �(1)k = �pTk�1AK�1gkpTk�1Apk�1or [23] and [12]:�k := �(2)k = gTkK�1(gk � gk�1)gTk�1K�1gk�1where K�1 = (LLT )�1, L is the pointwise incom-plete Cholesky factor of A [19, 17], andgk = 2xTkBxk [Axk �R(xk)Bxk]is the gradient of the Rayleigh quotient R(xk) =�xkTAxk� = �xkTBxk�3. calculate the vector ~p by:~pk = K�1gk + �kpk�14. evaluate pk by B-orthogonalizing vector ~pk with re-spect to Vj via a Gram-Schmidt B-orthogonalizationprocess5. compute the coe�cient �k as the largest root ofthe second order algebraic equation (see [21], [20],and [8]): a�2k + b�k + c = 0 (2)

wherea = �pTkApk� �pTkBxk�� �pTkAxk� �pTkBpk�b = �xTkBxk� �pTkApk�� �xTkAxk� �pTkBpk�c = �xTkBxk� �pTkAxk�� �xTkAxk� �pTkBxk�that provides the minimization of the Rayleigh quo-tient R(xk + �kpk) with respect to the parameter�k6. let ~xk+1 = xk + �kpk; the new approximate eigen-vector xk+1 is evaluated by B-normalization of ~xk+17. if k is smaller than the maximum number of iterationsIMAX and the relative residual calculated using ei-ther the last eigenvalue update, erj;k+1, or the lasteigenvector update, rrj;k+1, is greater or equal thanthe prescribed tolerance, i.e.:erj;k+1 = jq(xk+1)� q(xk)jq(xk) � TOL1or rrj;k+1 = kAxk � q(xk)Bxkk2kAxkk2 � TOL2then go back to step 2; otherwise, if erj;k+1 < TOL1or rrj;k+1 < TOL2, then q(xk+1) and xk+1 are theN � j-th eigenvalue and the corresponding eigenvec-tor of equation (1).2.2 NI schemeLet 1; 2; : : : ;L be a sequence of nested triangulations.We assume ` = 1 to be the coarsest level, and ` = L the�nest. Starting from ` = 2, each grid is obtained fromthe previous one by subdivision of each triangular elementinto four triangles, by connecting the midpoints of eachside of a triangle (Figure 2). The NI algorithm, appliedin combination with DACG, reads:for ` = 1; : : : ; L do:1. calculate the solution (�`;x`) of A`x =�B`x at level ` by iterating with DACGuntil convergence is achieved (A` and B`are the sti�ness and capacity matrices atlevel `)2. interpolate the eigenvector x` to `+1 toobtain x`+13. use x`+1 as initial guess for iteration 1 atlevel `+ 1The �nal solution is obtained at the �nest level L.The transfer mechanism of the eigenvectors from acoarser mesh, `�1, to the �ner, `, is obtained usinglinear interpolation formulae. For a sequence of circularand rectangular nested grids the interpolated values ofselected components of the j-th smallest eigenvector vj̀(1 � j � p) at level ` for the circular grid (Figure 2a) are:�vj̀(4`) = v`�1j (1`�1) + v`�1j (4`�1)2�vj̀(1`) = v`�1j (1`�1)�vj̀(10`) = v`�1j (3`�1)2

` = 1 (a) ` = 2` = 4 ` = 3

` = 1 (b) ` = 2` = 3 ` = 4Figure 1: Typical nested triangulations used in (a) circular and (b) rectangular domains.

(a) AAAAAA������AAAAAA������AAAAAAAAAAAA�������

�������������� AAAAAAAAA ���������AAAAAAA

AA 1`�12`�1

3`�14`�15`�1

6`�17`�1

1` 2` 3`4`5`6` 7` 8` 9` 10`11`12`13`14`15`16 1̀7` 18` 19`(b)

������������

���������������������

���������������

3`�1 6`�1 9`�12`�1 5`�1 8`�11`�1 4`�1 7`�15` 10` 15` 20` 25`4` 9` 14` 19` 24`3` 8` 13` 18` 23`2` 7` 12` 17` 22`1` 6` 11` 16` 21`

Figure 2: Sample node numbering at two levels ` and `� 1 in (a) circular and (b) rectangular triangulations.3

while for the rectangular grid (Figure 2b) take on theform:�vj̀(7`) = �v`�1j (1`�1) + v`�1j (2`�1) + v`�1j (5`�1) + v`�1j (4`�1)� =4�vj̀(12`) = v`�1j (4`�1) + v`�1j (5`�1)2�vj̀(13`) = v`�1j (5`�1)3 Mathematical Models and Nu-merical ExperimentsThe �rst sample problem (test case I) deals with the eigen-solution of the membrane equation:@2h@x2 + @2h@y2 + �h = 0 (3)where h, the vertical displacement from the rest position,is an eigenfunction of (3) and � the corresponding eigen-value. Equation (3) is de�ned on a circular domain ofradius R = 1000 m and is solved with a zero displace-ment condition at the external boundary.The second example (test case II) describes the vibra-tion modes of a two-dimensional rectangular structure:@2ux@x2 + 12(1� �)@2ux@y2 + 12(1 + �) @2uy@x@y + �ux = 0(4)@2uy@y2 + 12(1� �)@2uy@x2 + 12(1 + �) @2ux@x@y + �uy = 0Here ux and uy indicate the x and y components ofthe displacement vector, and � is the Poisson ratio. Theboundary conditions are as follows:� zero displacement for ux on the right and left bound-aries� uy = 0 on the lower and right boundaries.� a zero stress condition on the top boundary.The �nite element discretization of these two problemsformally leads to equation (1), with matrices A and Bbeing the sti�ness and capacity matrices, respectively. Weused triangular �nite elements with linear basis functions,TOL1=10�8 and TOL2=5� 10�3.3.1 Convergence of NI-DACGNI-DACG, with the two di�erent choices of the coe�cient�k, was applied for the calculation of 40 leftmost eigen-pairs on the two test problems. Four nested grids wereused for both problems, as is shown in Figure 1. The sti�-ness and capacity matrices for test case I have dimensionN = 128 for ` = 1, N = 496 for ` = 2, N = 1952 for ` = 3,andN = 7744 for the �nest level ` = 4. In test case II ma-trices A and B have dimensionN = 410; 1458; 5474; 21186for ` = 1; 2; 3; 4, respectively. Note that these latter ma-trices have a size that is twice the number of nodes of thecorresponding mesh at the di�erent levels, because in thisproblem there are two unknowns per node.

Figure 3 and Figure 4 give the leftmost eigenvalues ofproblems I and II, respectively, at the four discretizationlevels. Figures 5 and 6 provide the three dimensional plotsof the 1-st, 5-th, 25-th, and 30-th smallest eigenvectors onthe four circular nested grids, while Figures 7 and 8 showthe 5th and 20th eigenvectors on the rectangular grids.Note in Figures 7 and 8 that the eigenvectors have beendivided into two parts, ux and uy, providing the horizontaland the vertical vibration mode, respectively.As was expected the convergence of the eigenpairs onthe grids of increasing resolution is faster for the leftmostones. In test case II the smallest six eigenvalues convergefor ` = 2, and already at ` = 1 their estimates is quitegood (Figure 4). This is not the case in problem I (Fig-ure 3), where we may note a slower convergence.Similar behavior can be inferred from the three-dimensional plots of the eigenvectors over the di�erentgrids. Convergence is faster for the smallest eigenvectorsand slower for the highest ones, especially in problem I(Figures 5 and 6). Actually, the di�erences between theeigenvectors calculated at di�erent levels can be muchpronounced, as can be seen for the 30th eigenvector ofFigure 6, where even at the �nest level ` = 4 convergencehas not yet been achieved. It is worth noticing that theconvergence pro�les of ux and uy may also di�er as ispointed out by the outcome of Figure 8. In general, con-vergence is slowlier at the rightmost extreme of the partialeigeninterval, as was also observed with traditional itera-tive methods for the eigensolution [6, 24].3.2 Comparative Performance of DACGvs. NI-DACGIn this section we compare the performance of DACG andNI-DACG for the solution of the two sample problems.All numerical computations are performed in double pre-cision arithmetic on an IBM RISC-6000 and the CPUtimes (s) required by DACG and NI-DACG in the calcu-lation of the p smallest eigenpairs, p = 1; 5; 10; 20; 30; 40,for various experiments with two �k choices are shown inTables 1, 2, 3, and 4. The NI-DACG CPU times at eachlevel ` include the time needed to achieve convergence atthe coarser levels as well, while the number of iterationsrefers to the current level only. The NI-DACG tables alsogive the values of the speed-ups S` at each level `. In thepresent analysis the speed-up is de�ned as the ratio be-tween the CPU times required by NI-DACG and DACG,with a consistent selection of �k. Note that for ` = 1 thesame initial guess is used for both DACG and NI-DACGand therefore the performance of the two schemes is thesame.The DACG results reveal that the CPU time for eacheigenpair to converge at each level grows more than lin-early with the grid size N (Tables 1 and 3). As is shownin a recent theoretical analysis [4], the use of �k = �(2)kgenerally improves the performance of DACG, in particu-lar for the higher eigenpairs. Typical DACG convergencepro�les for problem I are shown in Figure 9 giving the log-arithm of the norm of the relative residual rrj;k vs. thenumber of iterations (N = 7744).4

6*10-6

1*10-5

3*10-5

6*10-5

1*10-4

3*10-4

6*10-4

1*10-3

3*10-3

6*10-3

1*10-2

6*10-6 1*10-5 3*10-5 6*10-5 1*10-4 3*10-4 6*10-4 1*10-3 3*10-3 6*10-3 1*10-2

6*10-6 1*10-5 3*10-5 6*10-5 1*10-4 3*10-4 6*10-4 1*10-3 3*10-3 6*10-3 1*10-2

6*10-6

1*10-5

3*10-5

6*10-5

1*10-4

3*10-4

6*10-4

1*10-3

3*10-3

6*10-3

1*10-2

N = 7744, ` = 4N = 1952, ` = 3N = 496, ` = 2N = 128, ` = 1

Figure 3: Distribution of the 40 leftmost eigenvalues at di�erent grid levels for test case I (circular domain).

5

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

N = 21186, ` = 4N = 5474, ` = 3N = 1458, ` = 2N = 410, ` = 1

Figure 4: Distribution of the 40 leftmost eigenvalues at di�erent grid levels for test case II (rectangular domain).

6

Figure 5: Leftmost eigenvectors 1 and 5 calculated at levels ` = 1; 2; 3; 4 for test case I (circular domain).7

Figure 6: Leftmost eigenvectors 25 and 30 calculated at levels ` = 1; 2; 3; 4 for test case I (circular domain).8

E-vector 5 (ux)

E-vector 5 (ux)

E-vector 5 (uy)

E-vector 5 (uy)

N=205, =1

N=729, =2

E-vector 5 (ux)

E-vector 5 (ux)

E-vector 5 (uy)

E-vector 5 (uy)

N=2737, =3

N=10593, =4

Figure 7: Leftmost eigenvector 5, components ux and uy, calculated at levels ` = 1; 2; 3; 4 for test case II (rectangulardomain).9

E-vector 20 (ux)

E-vector 20 (ux)

E-vector 20 (uy)

E-vector 20 (uy)

N=205, =1

N=729, =2

E-vector 20 (ux)

E-vector 20 (ux)

E-vector 20 (uy)

E-vector 20 (uy)

N=2737, =3

N=10593, =4

Figure 8: Leftmost eigenvector 20, components ux and uy, calculated at levels ` = 1; 2; 3; 4 for test case II (rectangulardomain).10

Table 1: CPU times (s) and cumulative number of iterations for the convergence of the DACG algorithm for thecalculation of the p smallest eigenpairs for test case I with (a) �k = �(1)k and (b) �k = �(2)k .(a)` = 1, N = 128 ` = 2, N = 496 ` = 3, N = 1952 ` = 4, N = 7744p time # iter. time # iter. time # iter. time # iter.1 0.02 (16) 0.15 (26) 0.93 (40) 7.09 (77)5 0.11 (62) 0.76 (132) 5.79 (248) 46.15 (483)10 0.43 (227) 1.61 (267) 9.71 (405) 76.66 (775)20 0.70 (376) 3.71 (563) 23.11 (904) 143.08 (1354)30 1.24 (634) 7.55 (1071) 38.61 (1412) 244.77 (2143)40 1.69 (817) 10.16 (1365) 61.42 (2099) 340.62 (2805)(b)` = 1, N = 128 ` = 2, N = 496 ` = 3, N = 1952 ` = 4, N = 7744p time # iter. time # iter. time # iter. time # iter.1 0.02 (13) 0.17 (29) 1.14 (51) 8.64 (94)5 0.11 (60) 0.65 (114) 5.54 (236) 39.86 (418)10 0.26 (137) 1.21 (207) 9.17 (379) 67.23 (675)20 0.48 (239) 2.31 (356) 17.32 (663) 122.32 (1145)30 0.82 (381) 3.76 (532) 25.91 (927) 189.80 (1650)40 1.20 (516) 5.46 (709) 36.50 (1216) 251.39 (2051)Table 2: CPU times (s), cumulative number of iterations, and speed-ups S` for the convergence of NI-DACG for thecalculation of the p smallest eigenpairs for test case I with (a) �k = �(1)k and (b) �k = �(2)k .(a)` = 2, N = 496 ` = 3, N = 1952 ` = 4, N = 7744p time # iter. S2 time # iter. S3 time # iter. S41 0.06 (8) 2.50 0.27 (12) 3.44 4.45 (44) 1.595 0.54 (75) 1.41 1.58 (42) 3.66 9.68 (78) 4.7710 1.15 (116) 1.40 3.22 (75) 3.02 19.31 (148) 3.9720 2.37 (238) 1.57 10.60 (288) 2.18 56.80 (381) 2.5230 6.55 (713) 1.15 24.56 (593) 1.57 109.69 (659) 2.2340 9.14 (949) 1.11 38.87 (929) 1.58 172.51 (968) 1.97(b)` = 2, N = 496 ` = 3, N = 1952 ` = 4, N = 7744p time # iter. S2 time # iter. S3 time # iter. S41 0.09 (13) 1.88 0.33 (10) 3.45 4.29 (43) 2.015 0.43 (52) 1.51 1.45 (39) 3.82 10.47 (76) 3.8110 0.87 (90) 1.15 2.89 (71) 3.17 17.39 (135) 3.8620 1.81 (175) 1.28 8.53 (224) 2.03 34.49 (212) 3.5430 3.73 (351) 1.01 15.93 (374) 1.63 63.44 (349) 3.9940 5.47 (474) 1.00 24.03 (531) 1.52 98.34 (499) 2.56

11

Table 3: CPU times (s) and cumulative number of iterations for the convergence of the DACG algorithm for thecalculation of the p smallest eigenpairs for test case II with (a) �k = �(1)k and (b) �k = �(2)k .(a)` = 1, N = 410 ` = 2, N = 1458 ` = 3, N = 5474 ` = 4, N = 21186p time # iter. time # iter. time # iter. time # iter.1 0.10 (15) 0.55 (22) 3.86 (40) 40.56 (110)5 0.65 (93) 3.97 (153) 29.50 (283) 238.75 (633)10 3.82 (553) 39.50 (1531) 253.49 (2512) 1428.14 (3657)20 6.94 (958) 61.79 (2318) 371.23 (3586) 2088.89 (5214)30 11.95 (1554) 79.12 (2866) 476.10 (4452) 2673.85 (6461)40 17.33 (2143) 107.80 (3736) 605.21 (5442) 3414.41 (7931)(b)` = 1, N = 410 ` = 2, N = 1458 ` = 3, N = 5474 ` = 4, N = 21186p time # iter. time # iter. time # iter. time # iter.1 0.11 (15) 0.53 (21) 4.02 (42) 31.32 (84)5 0.58 (80) 3.78 (157) 28.96 (295) 268.03 (706)10 1.71 (233) 14.22 (543) 106.21 (1052) 1014.03 (2585)20 3.38 (432) 24.71 (900) 193.57 (1839) 1717.76 (4230)30 5.43 (652) 34.23 (1186) 267.98 (2439) 2270.01 (5400)40 7.46 (846) 46.59 (1534) 352.49 (3067) 2965.53 (6768)Table 4: CPU times (s), cumulative number of iterations, and speed-ups S` for the convergence of NI-DACG for thecalculation of the p smallest eigenpairs for test case II with (a) �k = �(1)k and (b) �k = �(2)k .(a)` = 2, N = 1458 ` = 3, N = 5474 ` = 4, N = 21186p time # iter. S2 time # iter. S3 time # iter. S41 0.31 (8) 1.77 1.41 (11) 2.74 6.84 (14) 5.935 1.91 (45) 2.08 9.16 (70) 3.22 47.92 (97) 4.9810 22.35 (713) 1.77 129.43 (1055) 1.99 509.89 (956) 2.8020 38.89 (1185) 1.59 213.74 (1658) 1.74 906.67 (1675) 2.3030 63.45 (1811) 1.25 331.31 (2415) 1.44 1245.68 (2124) 2.1540 96.99 (2671) 1.11 510.91 (3536) 1.18 1798.89 (2831) 1.90(b)` = 2, N = 410 ` = 3, N = 5474 ` = 4, N = 21186p time # iter. S2 time # iter. S3 time # iter. S41 0.30 (7) 1.77 1.42 (11) 2.83 6.91 (14) 4.535 1.75 (42) 2.16 9.40 (70) 3.08 48.45 (97) 5.5310 9.43 (285) 1.51 64.17 (515) 1.66 251.27 (454) 4.0420 17.91 (507) 1.38 114.16 (871) 1.70 520.57 (946) 3.3030 27.54 (728) 1.24 175.44 (1271) 1.53 801.88 (1383) 2.8340 41.16 (1051) 1.13 257.83 (1776) 1.37 1162.21 (1898) 2.55

12

0 100 200 300 400 500 600

Iteration #

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

rrj

4th eigenvalue

9th eigenvalue

Figure 9: Typical convergence pro�les of DACG for problem I (N = 7744)The results provided in Tables 2 and 4 show that thee�ciency of DACG is signi�cantly improved by the use ofnested iterations, with speed-ups ranging from 1.6 to 6.Using �k = �(2)k further enhances the NI-DACG perfor-mance. Note that the speed-ups decrease as the numberp of sought eigenpairs increases. This fact re ects theconvergence behavior of the partial eigenspectrum overthe di�erent nested grids, as is shown in Figures 3 and 4,where the rightmost eigenvalues converge slowlier versus` than the leftmost ones.The slower convergence of the eigenpairs for test caseI is consistent with the smaller speed-ups obtained withNI-DACG. However, the reduced NI-DACG performancein problem I can also be accounted for by the fact thatthe nested grids are not uniform. For this reason, thesimple linear interpolation scheme used to transfer theeigenvectors from grids ` to ` + 1 may not be su�cientenough to guarantee a good initial guess at the �ner level,and higher order inter-grid transfer algorithms should bestudied.4 ConclusionsIn this paper we have analyzed the performance of opti-mization methods based on the conjugate gradient tech-nique for the partial solution of generalized eigenproblemsarising from the �nite element integration of structuralequations. The basic method, DACG, uses the conjugategradient iteration to minimize the Rayleigh quotient onappropriate de ated vector spaces. DACG is accelerated

by a nested iteration (NI) algorithm, providing solutionson grids of decreasing resolution which are used as initialDACG guesses over the next �ner grids. The performanceof DACG and NI-DACG is compared on two sample prob-lems with N = 7744 and 21186.From the numerical results, the following points areworth summarizing.1. The DACG number of iterations and CPU time ob-tained with �k = �(2)k are always smaller than thoseobtained with �k = �(1)k , as is anticipated in [4].2. NI-DACG improves the convergence of DACG by aCPU time factor that varies from a minimum of 1.6to a maximum of 5.9.3. The NI-DACG performance is relatively lower in testcase I, where the nested grids are not uniform. Thisbehavior may be accounted for by both the low ac-curacy of the linear inter-grid eigenvector transfer al-gorithm and the fact that the spectral convergenceover the grids in problem I is slower than in problemII.Acknowledgments. This work has been supported inpart by the Italian GNIM-CNR and \Fondi MURST40%".13

References[1] K. J. Bathe, Finite Element Procedures in En-gineering Analysis, Prentice-Hall, Englewood Cli�s,1982.[2] K. J. Bathe and E. Wilson, Solution methods foreigenvalue problems in structural dynamics, Int. J.Numer. Methods Eng., 6 (1973), pp. 213{226.[3] L. Bergamaschi, G. Gambolati, and G. Pini,Spectral analysis of large �nite element problems byoptimization methods, J. Shock and Vibr., 1 (1994),pp. 529{540.[4] L. Bergamaschi, G. Gambolati, and G. Pini,Convergence analysis of conjugate gradient methodsfor the partial symmetric eigenproblem, technical re-port, Depth. of Mathematical Methods for the Ap-plied Sciences, University of Padua, February 1996.[5] J. Cullum and R. A. Willoughby, Lanczos Al-gorithms for Large Symmetric Eigenvalue Computa-tions, vol. 1. Theory, Birkh�auser, Boston, 1985.[6] I. Fried, Optimal gradient minimization schemefor �nite element eigenproblems, J. Sound Vibr., 20(1972), pp. 333{342.[7] G. Gambolati, On time integration of groundwa-ter ow equations by spectral methods, Water Resour.Res., 29 (1993), pp. 1257{1267.[8] , Solution of large scale eigenvalue problems,in Solving Large Scale Problems in Mechanics: TheDevelopment and Application of Computational So-lution Methods, M. Papadrakakis, ed., New York,1993, John Wiley, pp. 125{156.[9] G. Gambolati and L. Bergamaschi, Partialeigensolution for transient groundwater ow equa-tions, in ICCMWR IX. Vol. 1: Numerical Meth-ods in Water Resources, T. R. Russell, R. E. Ewing,C. A. Brebbia, W. G. Gray, and G. F. Pinder, eds.,Southampton, 1992, Computational Mechanics andElsevier Applied Sciences, pp. 175{192.[10] G. Gambolati, G. Pini, and M. Putti, Nestediterations for symmetric eigenproblems, SIAM J. Sci.Comput., 16 (1995), pp. 173{192.[11] G. Gambolati and M. Putti, A comparison ofLanczos and optimization methods in the partial so-lution of sparse symmetric eigenproblems, Int. J. Nu-mer. Methods Eng., 37 (1994), pp. 605{621.[12] G. Gambolati, F. Sartoretto, and P. Flo-rian, An orthogonal accelerated de ation techniquefor large symmetric eigenproblems, Comp. MethodsApp. Mech. Eng., 94 (1992), pp. 13{23.[13] W. Hackbusch, On the computation of approxi-mate eigenvalues and eigenfunctions of elliptic op-erators by means of a multi-grid method, SIAM J.Num. Anal., 16 (1979), pp. 201{215.

[14] , Multi-Grid Methods and Applications,Springer-Verlag, New York, 1985.[15] M. R. Hestenes and W. Karush, A methodof gradients for the calculation of the characteristicroots and vectors of a real symmetric matrix, J. Res.Nat. Bur. Standard, 47 (1951), pp. 45{61.[16] , Solution of Ax = �Bx, J. Res. Nat. Bur. Stan-dard, 47 (1951), pp. 471{478.[17] D. S. Kershaw, The incomplete Cholesky-conjugategradient method for the iterative solution of sys-tems of linear equations, J. Comp. Phys., 26 (1978),pp. 43{65.[18] C. Lanczos, An iteration method for the solutionof the eigenvalue problem of linear di�erential andintegral operators, J. Res. Nat. Bur. Standard, 45(1950), pp. 255{282.[19] J. A. Meijerink and H. A. van der Vorst,An iterative solution method for linear systems ofwhich the coe�cient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148{162.[20] M. Papadrakakis, Solution of the partial eigen-problem by iterative methods, Int. J. Numer. MethodsEng., 20 (1984), pp. 2283{2301.[21] A. M. Perdon and G. Gambolati, Extremeeigenvalues of large sparse matrices by Rayleigh quo-tient and modi�ed conjugate gradients, Comp. Meth-ods App. Mech. Eng., 56 (1986), pp. 251{264.[22] G. Pini and F. Sartoretto, Vector and parallelcodes for large sparse eigenproblems, Supercomputer,50 (1992), pp. 29{39.[23] E. Polak, Computational Methods in Optimization:A Uni�ed Approach, Academic Press, New York,1971.[24] A. Ruhe, Computation of eigenvalues and eigenvec-tors, in Sparse Matrix Techniques, V. A. Barker, ed.,Berlin, 1977, Springer-Verlag, pp. 130{184.[25] M. Zoboli and P. Bassi, The �nite ele-ment method for anisotropic optical waveguides,in Anisotropic and Nonlinear Optical Waveguides,C. Someda and G. Stegeman, eds., Amsterdam, 1992,Elsevier, pp. 77{116.

14