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Title
Abstract
Figures
Experimental Results
Case: Modified Bratu Problem
Method
Case: Infiltration of the Vadose Zone
Conclusions
References
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Mathematics
Applied Mathematics

Krylov-secant methods for solving systems of nonlinear equations

Profile image of Hector KlieHector Klie

Abstract

We present a novel way of reusing the Krylov information generated by GMRES for solving the linear system arising within a Newton method. Our approach departs from the theory of ,ecant preconditioners developed by Martinez and then combines secant updates of the Hessenberg matrix generated by the Arnoldi process in GMRES, the Richardson iteration and limited memory quasi-Newton compact representations to generate descent directions for each Newton step. The proposed method allows to reflect-without explicitly computing them-secant updates of the Jacobian matrix that lead us to skip GM RES for the benefit of satisfying the Dembo-Eisenstat-Steihaug condition. Hence, the resulting method turns out to be computationally more economical than traditional inexact Newton implementations. Computational experiments reveal the suitability of this approach for large scale problems in several application contexts.

Figures (14)
This means that the eigenvalue information in which the Richardson iteration is based corresponds to the form above. Therefore, to apply the steps (33) and (34) we have to make sure that the Jacobian operator is somehow consistent with the relaxation parameters. 4.2. The role of preconditioners in GMRES. So far, our method has been described under the assumption that we have not employed a preconditioner in GMRES. It is not difficult to realize that, if a preconditioner is used, the update (26) rather than reflecting a secant update of the Jacobian, reflects a secant update of the Jacobian times its preconditioner (i.e. assuming right preconditioning). In other words, (27) would become
This means that the eigenvalue information in which the Richardson iteration is based corresponds to the form above. Therefore, to apply the steps (33) and (34) we have to make sure that the Jacobian operator is somehow consistent with the relaxation parameters. 4.2. The role of preconditioners in GMRES. So far, our method has been described under the assumption that we have not employed a preconditioner in GMRES. It is not difficult to realize that, if a preconditioner is used, the update (26) rather than reflecting a secant update of the Jacobian, reflects a secant update of the Jacobian times its preconditioner (i.e. assuming right preconditioning). In other words, (27) would become
Total number of GMRES iterations for inezact Newton, Broyden, modified Broyden and the Hybri Krylov-Secant method. The quantities tn parentheses indicate the number of Richardson tteration employed by the HKS method. TABLE 2 We solve this problem in the unit square with Dirichlet boundary conditions. See, e.g., Glowinski, Keller and Reinhart proposed problem in [33] for a detailed description. In this work, the problem is discretized by a block-centered finite-difference scheme and no upwinding was used for the convective coefficient. The linear system generated by the Newton step becomes harder as a and \ grow. A Block Jacobi (with 8 almost even blocks) preconditioner was used for the Richardson iteration, except where indicated in the tables. A Newton tolerance of 1 x 10~° was considered for these experiments and the linear solution tolerances were computed by means of equations (14) and (15). The values of the coefficients in the model equation were chosen as \ = 50 and a = 30. Both the Broyden and modified Broyden methods yield a lower number of accumu-
Total number of GMRES iterations for inezact Newton, Broyden, modified Broyden and the Hybri Krylov-Secant method. The quantities tn parentheses indicate the number of Richardson tteration employed by the HKS method. TABLE 2 We solve this problem in the unit square with Dirichlet boundary conditions. See, e.g., Glowinski, Keller and Reinhart proposed problem in [33] for a detailed description. In this work, the problem is discretized by a block-centered finite-difference scheme and no upwinding was used for the convective coefficient. The linear system generated by the Newton step becomes harder as a and \ grow. A Block Jacobi (with 8 almost even blocks) preconditioner was used for the Richardson iteration, except where indicated in the tables. A Newton tolerance of 1 x 10~° was considered for these experiments and the linear solution tolerances were computed by means of equations (14) and (15). The values of the coefficients in the model equation were chosen as \ = 50 and a = 30. Both the Broyden and modified Broyden methods yield a lower number of accumu-
computer architectures. Fic. 1. Number of iterations taken by linear solver vs. Number of Mflops.
computer architectures. Fic. 1. Number of iterations taken by linear solver vs. Number of Mflops.
Fic. 2. Logo of residual norms vs. Number of iterations taken by linear solver. Forcing terms were chosen according to (14) and (15) Figure 1 shows the number of floating point operations for all four methods as a function of the total number of linear iterations. The faster processing time of an adequate implementation of HKS becomes apparent because it consistently yields a much lower operation count for the same number of linear iterations than any of the other methods. All savings in computational cost of the HKS method discussed throughout the paper are combined in this operation count.
Fic. 2. Logo of residual norms vs. Number of iterations taken by linear solver. Forcing terms were chosen according to (14) and (15) Figure 1 shows the number of floating point operations for all four methods as a function of the total number of linear iterations. The faster processing time of an adequate implementation of HKS becomes apparent because it consistently yields a much lower operation count for the same number of linear iterations than any of the other methods. All savings in computational cost of the HKS method discussed throughout the paper are combined in this operation count.
Fig. 3. Logio of residual norms vs. Number of iterations taken by linear solver. No forcing term criterion was used. Linear tolerance: 1.e — 3. in HKS is competitive with the Newton method for acceptable tolerance values in engineering applications (i.e., 10-% or 10-4) and are either competitive of more efficient than the secant-type methods for much smaller values of the nonlinear tolerance (i.e., smaller than 107°).
Fig. 3. Logio of residual norms vs. Number of iterations taken by linear solver. No forcing term criterion was used. Linear tolerance: 1.e — 3. in HKS is competitive with the Newton method for acceptable tolerance values in engineering applications (i.e., 10-% or 10-4) and are either competitive of more efficient than the secant-type methods for much smaller values of the nonlinear tolerance (i.e., smaller than 107°).
Total and average number of GMRES iterations for inexact Newton, Broyden, modified Broyden and the HKS method for different preconditioners. The problem considered is a mesh of 20 x 20. The quantities in parentheses indicate the average of linear solver iterations for each Newton iteration.
Total and average number of GMRES iterations for inexact Newton, Broyden, modified Broyden and the HKS method for different preconditioners. The problem considered is a mesh of 20 x 20. The quantities in parentheses indicate the average of linear solver iterations for each Newton iteration.
Fia. 4. Logio of restdual norms vs. Number of Mflops. The results listed in Table 4 are most enlightening. The first column shows the number of nonlinear iterations taken by the four methods. The inexact Newton takes the lowest number of steps to converge and the secant-update methods exceed this number. The HKS scheme takes virtually the same number of nonlinear steps as either one of the secant methods (no claim about the fact that HKS takes one less). As mentioned above in connection with the graphs of Figures 2 and 3, the effect of the forcing term backtracking scheme manifests itself in the secondary effect of controlling the number of nonlinear iterations taken by both the Broyden-type and the HKS methods. However, as far as the analysis of the nonlinear iterations, this virtual agreement shows that the proposed secant update of the Hessenberg matrix in HKS maintains the same overall convergence properties for this methods as those observed in Broyden-type schemes.
Fia. 4. Logio of restdual norms vs. Number of Mflops. The results listed in Table 4 are most enlightening. The first column shows the number of nonlinear iterations taken by the four methods. The inexact Newton takes the lowest number of steps to converge and the secant-update methods exceed this number. The HKS scheme takes virtually the same number of nonlinear steps as either one of the secant methods (no claim about the fact that HKS takes one less). As mentioned above in connection with the graphs of Figures 2 and 3, the effect of the forcing term backtracking scheme manifests itself in the secondary effect of controlling the number of nonlinear iterations taken by both the Broyden-type and the HKS methods. However, as far as the analysis of the nonlinear iterations, this virtual agreement shows that the proposed secant update of the Hessenberg matrix in HKS maintains the same overall convergence properties for this methods as those observed in Broyden-type schemes.
Fic. 5. Number of Mflops erecuted to solve the problem vs. increasing problem sizes. The size N gives rise toa N x N unknowns.
Fic. 5. Number of Mflops erecuted to solve the problem vs. increasing problem sizes. The size N gives rise toa N x N unknowns.
Fic. 6. Water content distribution at first timestep.
Fic. 6. Water content distribution at first timestep.
Fic. 7. Water content distribution after 500 timesteps.
Fic. 7. Water content distribution after 500 timesteps.
Fic. 8. Hydraulic Conductivity coefficients at first timestep.
Fic. 8. Hydraulic Conductivity coefficients at first timestep.
Total and average number of GMRES/Richardson iterations for inexact Broyden, modified Broyden and the HKS method. The problems considered have mesh sizes of 6X6, 12X12 and18x18. The figures accumulate the iterations performed after 500 timesteps of simulation. The quantities in parentheses indicate the average number of linear solver iterations for each Newton iteration. Table 5 shows overall results from simulations on three discretization meshes, ie., 6 X 6, 12 x 12 and 18 x 18. As in the discussion of the previous section, the highlight of this table is the reduction in the number of GMRES iterations displayed TABLE 5
Total and average number of GMRES/Richardson iterations for inexact Broyden, modified Broyden and the HKS method. The problems considered have mesh sizes of 6X6, 12X12 and18x18. The figures accumulate the iterations performed after 500 timesteps of simulation. The quantities in parentheses indicate the average number of linear solver iterations for each Newton iteration. Table 5 shows overall results from simulations on three discretization meshes, ie., 6 X 6, 12 x 12 and 18 x 18. As in the discussion of the previous section, the highlight of this table is the reduction in the number of GMRES iterations displayed TABLE 5
Fic. 10. Accumulated Mflops for 500 timesteps of a 12 x 12 problem.
Fic. 10. Accumulated Mflops for 500 timesteps of a 12 x 12 problem.

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