3.9 COLETA DE DADOS E ANÁLISES
3.9.2 Procedimentos estatísticos
3
Depth(km)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Distance(km)
1028 1728 2428 3128 3828 4528
m/s
Figure 2.8: Marmousi 2: P-wave velocity model
2.2.4.2 The 2D Helmholtz equation
Consider the 2D Helmholtz equation (1.17) which is recast in the 2D approximation. An explosive sourcesmodeled by a Delta function at the position(x0, z0)m
s(x, z) =δ(x−x0)δ(z−z0).
Perfectly Matched Layers (PML) are used at the boundaries to mimic infinite domain wave propagation.
The discretization of this problem is performed using the second-order staggered-grid finite-difference method (see Chapter 1 Section 1.5). Attenuation is accounted for using the Kolsky model (Kolsky, 1956) through the relation (2.27).
2.2.4.3 Numerical results
Several experiments are performed to illustrate the performance of the CGMN method for solving the 2D Helmholtz equation on the Marmousi 2 model together with the preconditioning strategies. For each experiment, the starting point x(0) is set to0. The relaxation parameter used for the Kaczmarz projections is set to1.2which is the optimal value selected from several experiments. The stopping criterion is set to10−4on the relative residual
||Ax(k)−b||
||b|| <10−4.
The explosive Dirac source is located at the middle top of the domain atx0= 8500m andz0 = 100m.
The Marmousi2model, presented in Figure 2.8 is given in a2801×13601grid points withd= 1.25 meter space step length. This model is filtered and then sampled for several frequencies f = {5,10,15,20}. We use10grid points per wavelength to ensure the accuracy of the second-order finite-difference scheme. NP M L = 10grid points are used in each PML. The damping coefficient inside the PML is set tocP M L = 1500. Table 2.2 summarizes the size of the models used for each frequency.
Representative pressure wavefields solution of this problem are presented at the end of this section at the frequencies 5 and 20 Hz in Figures 2.11 and 2.12.
f h Nz Nx Nu
5 30 145 709 120285 10 15 289 1417 444033 15 10 433 2125 971685 20 7.5 577 2833 1703241
Table 2.2: Discritization of the Marmousi 2 model (Nx, Nz) following the second-order finite-difference scheme and number of unknownsNu. 10 grid points per minimum wavelength are used.
CGMN In Figure 2.9, a complexity analysis of the CGMN method is presented. The number of iterations is plotted as a function of the mean size of one dimension
N =p NxNz
on a log-log scale. The black solid line represents the linear increase of the computational complexity Niter = N. The solid red curve indicates the dependence between Niter andN. A standard linear regression, indicated by the dotted line, is performed on the solid red curve.
102.6 102.8 103 103.4
103.6 103.8 104
N Niter
Linear increase CGMN
Slope
Figure 2.9: Complexity analysis of the CGMN method for the frequency domain wave propagation problem in the 2D heterogeneous acoustic model Marmousi2. The number of iterations are plotted as a function of the average number of grid points by dimensionNon a log-log scale ( ). The blue line ( ) represents the line regression.
The complexity of the CGMN method is approximately given by Niter = 50N0.68.
2.2 Preconditioning strategy for the CARP-CG method Therefore, following the analysis presented in Chapter 1 Section 1.6, the total computational complexity of the CGMN method for 2D modeling with one source is
O(Niter×N2) =O(N2.68).
This sub-linear complexity of the number of iterations with respect to N indicates the good perfor-mances of the CGMN method. This complexity is an improvement compared toO(N3)performed by the time-domain modeling strategy.
In Figure 2.10, the relative residuals of the CGMN method are plotted as a function of the iteration number for each frequency described in Table 2.2. The rate of convergence is very fast until the norm of the relative residual reaches10−2. From then on, the rate of convergence decreases. The number of
0 1,000 2,000 3,000
10−4 10−3 10−2 10−1 100
Iteration
||Ax(k)−b|| ||b||
5Hz 10Hz 15Hz 20Hz
Figure 2.10: Convergence histories of the CGMN method for the frequency-domain acoustic wave sim-ulations using the Marmousi2P-wave velocity model. The relative residuals are plotted as a function of the iteration number during the CGMN runs.
iterations and the computation time performed to solve the linear systems are summarized in Table 2.3.
f Nu NiterCGM N Time(s)
5 120285 2764 44.3
10 444033 4038 263.5 15 971685 5407 778.6 20 1703241 6878 1786.9
Table 2.3: Number of iterations performed by the CGMN method applied to the Helmholtz problem using the Marmousi2P-wave velocity model.
0
Figure 2.11: Marmousi2acoustic pressure wavefield (real part) solution at5Hz.
0
Figure 2.12: Marmousi2acoustic pressure wavefield (real part) solution at20Hz.
CGMN with ILUT and AINV preconditioners In this paragraph, the results of the CGMN method with the preconditioning techniques ILUT and AINV are presented. Both preconditioners are computed from the damped version of the matrixA. Damping is introduced in the medium by tuning the quality factor qatt. The ILUT preconditioner is computed with p = 10 and τ = 10−10. As discussed in the previous part, the AINV preconditioner is made sparse following two dropping strategies. The factorization process is performed by reducing the internal loop to the bandwidthν=Nz. A threshold τ = 10−2 is applied allowing to keep the preconditioner sparse with only9nonzero coefficients per columns.
In the experiment (2.13), the influence of the attenuation on the quality of the preconditioner for the f = 10Hz problem is investigated. The experiment is performed with the ILUT preconditioner but sim-ilar behavior is obtained with the AINV preconditioner. The CGMN method performs4038iterations to solve the Helmholtz equation applied to the Marmousi2P-wave velocity model for the frequency f = 10 Hz. Figure 2.13 shows how the CGMN method performs when the ILUT preconditioner is computed based on a damped version ofA, which is denoted byG. When a small damping is applied, the quality of the preconditioner is very poor. The preconditioned method performs even more itera-tions than CGMN. However, by introducing a stronger damping, the preconditioned method becomes efficient. The minimum number of iterations Niter= 1205 performed by the ILUT-preconditioned CGMN method is reached forqatt= 1.1. Such behavior is due to the fact that the preconditioner needs to be computed based on a diagonal dominant matrix. If the damping is not strong enough, the sparse approximation of the ILUT preconditioner does not compute a good a approximation of theG−1. In the following, both preconditioner are computed based on the damped version ofAusingqatt= 1.1.
Figure 2.14 shows the convergence of the relative residual for the given frequencies. For each fre-quency, the comparison between the convergence histories of CGMN, CGMN with ILUT and CGMN with AINV shows the AINV preconditioner reduces at best the number of iterations. A “super” conver-gence is observed compared to CGMN.
2.2 Preconditioning strategy for the CARP-CG method
0 1 2 3 4
2,000 4,000 6,000 8,000
qatt
Niter
CGMN+ILUT CGMN
Figure 2.13: Effect of the quality factor on the quality of the preconditioner. The number of iterations Niter performed by the preconditioned CGMN method is plotted as a function the quality factorqatt ( ). The Helmholtz problem at the frequencyf = 10 Hz is considered for this experiment. The ILUT preconditioner is computed withp = 10, τ = 10−10 by varying the quality factor qatt. The number of iterations performed by the CGMN method without preconditioning ( ) isNiter= 4038.
Table (2.4) summarizes the number of iterations and the computation time in seconds performed by the CGMN method and the preconditioned CGMN method using the ILUT and AINV precondi-tioners. The number of iterations is reduced by a factor going from3.6 to6.0 when using the ILUT preconditioner. The computation time is reduced by a factor 1.5 up to 2.2.
The AINV preconditioner clearly performs better than the ILUT preconditioner for the reduction in the number of iterations. A factor8.6up to9.2is obtained. A reduction in the computation time by a factor3.6up to3.9is obtained.
f Nu NiterCGM N Time(s) NiterILU T Time(s) NiterAIN V Time(s)
5 120285 2764 44.3 460 20.1 310 12.3
10 444033 4038 263.5 790 134.1 470 70.8
15 971685 5407 778.6 1350 500.2 620 215.1
20 1703241 6878 1786.9 1800 1189.2 750 449.7
Table 2.4: Number of iterations and computation time performed by the preconditioned CGMN method for the solution of the 2D Helmholtz problem using the Marmousi2P-wave velocity model. CGMN is preconditioned with the ILUT and the AINV preconditioners.NiterCGM N denotes the number of iterations performed by the CGMN method without preconditioning. NiterILU T denotes the number of iterations performed by the CGMN method using the ILUT preconditioner. NiterAIN V denotes the number of iterations performed by the CGMN method using the AINV preconditioner. The computation time is presented in the column “Time” in seconds.
A complexity analysis of the CGMN method using the ILUT and AINV preconditioners is presented in Figure 2.15. For the ILUT preconditioner, we have
NiterILU T = 0.9N1.04.
Despite the gain in computation time, the complexity of the CGMN seems worsen for high frequencies.
0 200 400 600 800 1,0001,2001,400 10−4
10−3 10−2 10−1 100
||Ax(k)−b|| ||b||
(a) 5 Hz
0 500 1,000 1,500 2,000 10−4
10−3 10−2 10−1 100
(b) 10 Hz
0 500 1,000 1,500 2,000 2,500 10−4
10−3 10−2 10−1 100
Iteration number
||Ax(k)−b|| ||b||
(c) 15 Hz
0 1,000 2,000 3,000
10−4 10−3 10−2 10−1 100
Iteration number
(d) 20 Hz
Figure 2.14: Convergence histories of the CGMN method for the resolution of the Helmholtz problem using the 2D Marmousi2P-wave velocity model. CGMN without preconditioned is plotted with ( ), with the ILUT preconditioner ( ) and with the AINV preconditioner ( ). The ILUT preconditioner is computed withp = 10,τ = 10−10 andqatt = 1.1. The AINV preconditioner is computed on the same damped matrix as for the ILUT preconditioner. The thresholdτ = 10−2so that the preconditioner only has9non-zero values per column.
A complexity analysis of the CGMN method using the AINV preconditioner gives NiterAIN V = 6.2N0.66.
The latter clearly indicates the reduction in the complexity of the preconditioned method. Both coeffi-cientsαandβas inα Nβare reduced. In comparison with the CGMN method without preconditioning, the exponent is sensitively the same but the constant multiplying the complexity is drastically reduced by a factor 8.1.
The above experiments show that the second preconditioning strategy provides clearly better results.
Therefore, the approach which consists in solving the linear system AP2y=b, x=P2y,
using the CGMN or CARP-CG method shall be preferred in the following.
2.2 Preconditioning strategy for the CARP-CG method
102.6 102.8 103 103
104
N Niter
CGMN ILUT AINV Linear increase
Figure 2.15: Complexity analysis of the CGMN method with and without preconditioning (ILUT, AINV) for the frequency domain wave propagation problem in the 2D heterogeneous acoustic model Marmousi2.
2.2.5 A revisited sparse approximate inverse preconditioner for the frequency-domain