Based on the preconditioned MHSS (PMHSS) and generalized PMHSS (GPMHSS) methods, a double-parameter GPMHSS (DGPMHSS) method for solving a class of complex symmetric linear systems from Helmholtz equation is presented. A parameter region of the convergence for DGPMHSS method is provided. From practical point of view, we have analyzed and implemented inexact DGPMHSS (IDGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical examples are reported to confirm the efficiency of the proposed methods.
1. Introduction
Let us consider the following form of the Helmholtz equation:
(1)-Δu+σ1u+iσ2u=f,
where σ1,σ2 are real coefficient functions and u satisfies Dirichlet boundary conditions in D=[0,1]×[0,1].
Using the finite difference method to discretize the Helmholtz equation (1) with both σ1 and σ2 strictly positive leads to the following system of linear equations:
(2)Ax=b,A∈ℂn×n,x,b∈ℂn,
where A=W+iT∈ℂn×n and i=-1 with real symmetric matrices W,T∈ℝn×n satisfying -W⪯T≺W under certain conditions. Throughout the paper, for ∀B,C∈ℝn×n, B≺C means that B-C is symmetric negative definite and B⪯C means that B-C is symmetric negative semidefinite.
Systems such as (2) are important and arise in a variety of scientific and engineering applications, including structural dynamics [1–3], diffuse optical tomography [4], FFT-based solution of certain time-dependent PDEs [5], lattice quantum chromodynamics [6], molecular dynamics and fluid dynamics [7], quantum chemistry, and eddy current problem [8, 9]. One can see [10, 11] for more examples and additional references.
Based on the specific structure of the coefficient matrix A, one can verify that the Hermitian and skew-Hermitian parts of the complex symmetric matrix A, respectively, are
(3)H=12(A+A*)=W,S=12(A-A*)=iT.
Obviously, the above Hermitian and skew-Hermitian splitting (HSS) of the coefficient matrix A is in line with the real and imaginary parts splitting of the coefficient matrix A. Based on the HSS method [12], Bai et al. [2] skillfully designed the modified HSS (MHSS) method to solve the complex symmetric linear system (2). To generalize the concept of this method and accelerate its convergence rate, Bai et al. in [3, 13] designed the preconditioned MHSS (PMHSS) method. It is noted that MHSS and PMHSS methods can efficiently solve the linear system (2) with W≻0 and T⪰0.
Recently, Xu [14] proposed the following GPMHSS method for solving the complex symmetric linear systems (2) with -W⪯T≺W and it is described in the following.
The GPMHSS Method. Let x(0)∈ℂn be an arbitrary initial guess. For k=0,1,2,… until the sequence of iterates {x(k)}k=0∞ converges, compute the next iterate x(k+1) according to the following procedure:
(4)(αV+W-T)x(k+1/2)=(αV-i(W+T))x(k)+(1+i)b,(αV+W+T)x(k+1)=(αV+i(W-T))x(k+1/2)+(1-i)b,
where α is a given positive constant and V∈ℝn×n is a prescribed symmetric positive definite matrix.
Theoretical analysis in [14] shows that the GPMHSS method converges unconditionally to the unique solution of the complex symmetric linear system (2). Numerical experiments are given to show the effectiveness of the GMHSS method.
In this paper, based on the asymmetric HSS and generalized preconditioned HSS methods in [15, 16], a natural generalization for the GMHSS iteration scheme is that we can design a double-parameter GMHSS (DGPMHSS) iteration scheme for solving the complex symmetric linear systems (2) with -W⪯T≺W. That is to say, the DGPMHSS iterative scheme works as follows.
The DGPMHSS Method. Let x(0)∈ℂn be an arbitrary initial guess. For k=0,1,2,… until the sequence of iterates {x(k)}k=0∞ converges, compute the next iterate x(k+1) according to the following procedure:
(5)(αV+W-T)x(k+1/2)=(αV-i(W+T))x(k)+(1+i)b,(βV+W+T)x(k+1)=(βV+i(W-T))x(k+1/2)+(1-i)b,
where α is a given nonnegative constant, β is a given positive constant, and V∈ℝn×n is a prescribed symmetric positive definite matrix.
Just like the GPMHSS method (4), both matrices αV+W-T and βV+W+T are symmetric positive definite. Hence, the two linear subsystems in (5) can also be effectively solved either exactly by a sparse Cholesky factorization or inexactly by a preconditioned conjugate gradient scheme. Theoretical analysis shows that the iterative sequence produced by DGPMHSS iteration method converges to the unique solution of the complex symmetric linear systems (2) for a loose restriction on the choices of α and β. The contraction factor of the DGPMHSS iteration can be bounded by a function, which is dependent only on the choices of α and β, the smallest eigenvalues of matrices V-1(W-T) and V-1(W+T).
This paper is organized as follows. In Section 2, we study the convergence properties of the DGPMHSS method. In Section 3, we discuss the implementation of DGPMHSS method and the corresponding inexact DGPMHSS (IDPGMHSS) iteration method. Numerical examples are reported to confirm the efficiency of the proposed methods in Section 4. Finally, we end the paper with concluding remarks in Section 5.
2. Convergence Analysis for the DGPMHSS Method
In this section, the convergence of the DGPMHSS method is studied. The DGPMHSS iteration method can be generalized to the two-step splitting iteration framework. The following lemma is required to study the convergence rate of the DGPMHSS method.
Lemma 1 (see [<xref ref-type="bibr" rid="B4">13</xref>]).
Let A∈ℂn×n, A=Mi-Ni(i=1,2) be two splittings of A, and let x(0)∈ℂn be a given initial vector. If {x(k)} is a two-step iteration sequence defined by
(6)M1x(k+1/2)=N1x(k)+b,M2x(k+1)=N2x(k+1/2)+b,k=0,1,…, then
(7)x(k+1)=M2-1N2M1-1N1x(k)+M2-1(I+N2M1-1)b,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhk=0,1,….
Moreover, if the spectral radius ρ(M2-1N2M1-1N1)<1, then the iterative sequence {x(k)} converges to the unique solution x*∈ℂn of system (1) for all initial vectors x(0)∈ℂn.
Applying this lemma to the DGPMHSS method, we get convergence property in the following theorem.
Theorem 2.
Let W^=R-T(W-T)R-1,T^=R-T(W+T)R-1, and V=RTR, where R∈ℝn×n is a prescribed nonsingular matrix. Let A=W+iT∈ℂn×n be defined in (2), α a nonnegative constant, and β a positive constant. Then the iteration matrix Mα,β of GPMHSS method is
(8)Mα,β=(βV+W+T)-1(βV+i(W-T))×(αV+W-T)-1(αV-i(W+T)),
which satisfies
(9)ρ(Mα,β)≤maxλi∈λ(W^)β2+λi2α+λimaxμi∈μ(T^)α2+μi2β+μi,
where λ(W^) and μ(T^), respectively, are the spectral sets of the matrices W^ and T^.
In addition, let μmin and λmin, respectively, be the smallest eigenvalues of the matrices W^ and T^. If
(10)α2+μmin2-μmin≤β<α2+2αλmin,
then the DGPMHSS iteration (5) converges to the unique solution of the linear system (2).
Proof.
Let
(11)M1=αV+W-T,N1=αV-i(W+T),M2=βV+W+T,N2=βV+i(W-T).
Obviously, αV+W-T and βV+W+T are nonsingular for any nonnegative constants α or positive constants β. So formula (8) is valid.
The iteration matrix Mα,β is similar to
(12)M^α,β=(βV+i(W-T))(αV+W-T)-1×(αV-i(W+T))(βV+W+T)-1.
That is,
(13)M^α,β=RT(βI+iR-T(W-T)R-1)×RR-1(αI+R-T(W-T)R-1)-1R-T×RT(αI-iR-T(W+T)R-1)×RR-1(βI+R-T(W+T)R-1)-1R-T=RT(βI+iR-T(W-T)R-1)×(αI+R-T(W-T)R-1)-1×(αI-iR-T(W+T)R-1)×(βI+R-T(W+T)R-1)-1R-T=RT(βI+iW^)(αI+W^)-1×(αI-iT^)(βI+T^)-1R-T.
Therefore, we have
(14)ρ(Mα,β)=ρ(M^α,β)=ρ(RT(βI+iW^)(αI+W^)-1hh×(αI-iT^)(βI+T^)-1R-T)=ρ((βI+iW^)(αI+W^)-1(αI-iT^)(βI+T^)-1)≤∥(βI+iW^)(αI+W^)-1∥2×∥(αI-iT^)(βI+T^)-1∥2.
Since W∈ℝn×n and T∈ℝn×n, respectively, are symmetric positive definite and symmetric positive semidefinite and R is a nonsingular matrix, W^ and T^ are symmetric positive definite and symmetric positive semidefinite, respectively. Therefore, there exist orthogonal matrices S1,S2∈ℝn×n such that
(15)S1TW^S1=ΛW^,S2TT^S2=ΛT^,
where ΛW^=diag(λ1,λ2,…,λn) and ΛT^=diag(μ1,μ2,…,μn) with λi>0(1≤i≤n) and μi≥0(1≤i≤n) being the eigenvalues of the matrices W^ and T^, respectively.
Through simple calculations, we can get that
(16)ρ(Mα,β)≤maxλi∈λ(W^)|β+iλiα+λi|maxμi∈μ(T^)|α-iμiβ+μi|=maxλi∈λ(W^)β2+λi2α+λimaxμi∈μ(T^)α2+μi2β+μi,
which gives the upper bound for ρ(Mα,β) in (9). The next proof is similar to that of Theorem 1 in [17]. Here it is omitted.
The approach to minimize the upper bound is very important in theoretical viewpoint. However, it is not practical since the corresponding spectral radius of the iteration matrix Mα,β in (9) is not optimal. How to choose the suitable preconditioners and parameters for practical problem is still a great challenge.
3. The IDGPMHSS Iteration
In the DGPMHSS method, it is required to solve two systems of linear equations whose coefficient matrices are αV+W-T and βV+W+T, respectively. However, this may be very costly and impractical in actual implementation. To overcome this disadvantage and improve the computational efficiency of the DGPMHSS iteration method, we propose to solve the two subproblems iteratively [12, 18], which leads to IDGPMHSS iteration scheme. Its convergence can be shown in a similar way to that of the IHSS iteration method, using Theorem 3.1 of [12]. Since αV+W-T and βV+W+T are symmetric positive definite, some Krylov subspace methods (such as CG) can be employed to gain its solution easily. Of course, if good preconditioners for matrices αV+W-T and βV+W+T are available, we can use the preconditioned conjugate gradient (PCG) method instead of CG for the two inner systems that yields a better performance of IDGPMHSS method. If either αV+W-T or βV+W+T (or both) is Toeplitz, we can use fast algorithms for solution of the corresponding subsystems [19].
4. Numerical Examples
In this section, we give some numerical examples to demonstrate the performance of the DGPMHSS and IDGPMHSS methods for solving the linear system (2). Numerical comparisons with the GPMHSS method are also presented to show the advantage of the DGPMHSS method.
In our implementation, the initial guess is chosen to be x(0)=0 and the stopping criteria for outer iterations is
(17)∥b-Ax(k)∥2∥b∥2≤10-6.
The preconditioner V used in GPMHSS method is chosen to be V=W-T. For the sake of comparing, the corresponding preconditioner V used in DGPMHSS method is chosen to be V=W-T. Since the numerical results in [2, 3] show that the PMHSS iteration method outperforms the MHSS and HSS iteration methods when they are employed as preconditioners for the GMRES method or its restarted variants [20], we just examine the efficiency of DGPMHSS iteration method as a solver for solving complex symmetric linear system (2) by comparing the iteration numbers (denoted as IT) and CPU times (in seconds, denoted as CPU(s)) of DGPMHSS method with GPMHSS method.
Example 3 (see [<xref ref-type="bibr" rid="B10">5</xref>, <xref ref-type="bibr" rid="B15">21</xref>–<xref ref-type="bibr" rid="B22">23</xref>]).
Consider the following form of the Helmholtz equation:
(18)-Δu+σ1u+iσ2u=f,
where σ1,σ2 are real coefficient functions and u satisfies Dirichlet boundary conditions in D=[0,1]×[0,1]. The above equation describes the propagation of damped time harmonic waves. We take H to be the five-point centered difference matrix approximating the negative Laplacian operator on a uniform mesh with mesh size h=1/(m+1). The matrix H∈ℝn×n possesses the tensor-product form H=Bm⊗I+I⊗Bm with Bm=h-2·tridiag(-1,2,-1)∈ℝm×m. Hence, H is an n×n block-tridiagonal matrix, with n=m2. This leads to the complex symmetric linear system (2) of the form
(19)[(H+σ1I)+iσ2I]x=b.
In addition, we set σ1=100 and the right-hand side vector b to be b=(1+i)A1, with 1 being the vector of all entries equal to 1. As before, we normalize the system by multiplying both sides by h2.
As is known, the spectral radius of the iteration matrix may be decisive for the convergence of the iteration method. The spectral radius corresponding to the iteration method is necessary to consider. The comparisons of the spectral radius of the two different iteration matrices derived by GPMHSS and DGPMHSS methods with different mesh size are performed in Tables 1, 2, 3, and 4. In Tables 1–4, we used the optimal values of the parameters α and β, denoted by αG* for GPMHSS method, and αD*, βD* for DGPMHSS method. These parameters are obtained experimentally with the least spectral radius for the iteration matrices of the two methods.
The experimentally optimal parameters and the corresponding spectral radii for the iteration matrices of GPMHSS and DGPMHSS with V=W-T and m=8.
σ2
10
50
80
100
GPMHSS
αG*
1.1
1.5
2.2
2
ρ(MαG*)
0.5009
0.5222
0.5667
0.6274
DGPMHSS
αD*
1.1
1.5
2.2
2
βD*
1
0.9
0.8
0.8
ρ(MαD*,βD*)
0.5001
0.4986
0.5012
0.4901
The experimentally optimal parameters and the corresponding spectral radii for the iteration matrices of GPMHSS and DGPMHSS with V=W-T and m=16.
σ2
10
50
80
100
GPMHSS
αG*
1.1
1.5
2.2
1.8
ρ(MαG*)
0.5010
0.5230
0.5697
0.6337
DGPMHSS
αD*
1.1
1.5
2.2
1.8
βD*
1
1
0.9
0.9
ρ(MαD*,βD*)
0.5004
0.5063
0.5235
0.5097
The experimentally optimal parameters and the corresponding spectral radii for the iteration matrices of GPMHSS and DGPMHSS with V=W-T and m=24.
σ2
10
50
80
100
GPMHSS
αG*
1.1
1.5
2.2
1.6
ρ(MαG*)
0.5011
0.5234
0.5708
0.6360
DGPMHSS
αD*
1.1
1.5
2.2
1.8
βD*
1
1
1
1
ρ(MαD*,βD*)
0.5005
0.5081
0.5289
0.5091
The experimentally optimal parameters and the corresponding spectral radii for the iteration matrices of GPMHSS and DGPMHSS with V=W-T and m=32.
σ2
10
50
80
100
GPMHSS
αG*
1.1
1.5
2.2
1.7
ρ(MαG*)
0.5011
0.5235
0.5714
0.6372
DGPMHSS
αD*
1.1
1.5
2.2
1.8
βD*
1
1
1
1
ρ(MαD*,βD*)
0.5005
0.5089
0.5310
0.5139
From Tables 1–4, one can see that with the mesh size creasing, the trend of the experimentally optimal parameter of the GPMHSS and DGPMHSS methods is relatively stable. In Tables 1–4, we observe that the optimal spectral radius of DGPMHSS method is still smaller than those of the GPMHSS method, which implies that the DGPMHSS method may outperform the GPMHSS method. To this end, we need to examine the efficiency of the GPMHSS and DGPMHSS methods for solving the systems of linear equations Ax=b, where A is described.
In Tables 5, 6, 7, and 8, we list the numbers of iteration steps and the computational times for GPMHSS and DGPMHSS iteration methods using the optimal parameters in Tables 1–4. In Tables 5–8, “RES” denotes the relative residual error.
RES, CPU(s), and IT for PMHSS and GPMHSS with V=W-T and m=8.
σ2
10
50
80
100
GPMHSS
RES
9.818e-7
9.8917e-7
9.1469e-7
6.306e-7
CPU(s)
0.015
0.016
0.016
0.016
IT
20
21
24
30
DGPMHSS
RES
9.0916e-7
7.0211e-7
4.9072e-7
6.5183e-7
CPU(s)
0.015
0.015
0.015
0.013
IT
20
19
18
17
RES, CPU(s), and IT for PMHSS and GPMHSS with V=W-T and m=16.
σ2
10
50
80
100
GPMHSS
RES
9.8506e-7
9.4015e-7
8.9127e-7
9.4832e-7
CPU(s)
0.031
0.046
0.046
0.063
IT
20
21
24
29
DGPMHSS
RES
9.3425e-7
7.2018e-7
7.4847e-7
8.9905e-7
CPU(s)
0.031
0.032
0.031
0.031
IT
20
20
20
19
RES, CPU(s), and IT for PMHSS and GPMHSS with V=W-T and m=24.
σ2
10
50
80
100
GPMHSS
RES
9.8907e-7
9.6059e-7
9.7633e-7
7.5427e-7
CPU(s)
0.094
0.094
0.11
0.125
IT
20
21
24
29
DGPMHSS
RES
9.5055e-7
9.5856e-7
7.8461e-7
7.6081e-7
CPU(s)
0.078
0.078
0.094
0.094
IT
20
20
21
20
RES, CPU(s), and IT for PMHSS and GPMHSS with V=W-T and m=32.
σ2
10
50
80
100
GPMHSS
RES
9.9181e-7
9.8976e-7
6.0829e-7
9.1273e-7
CPU(s)
0.172
0.172
0.203
0.235
IT
20
21
25
28
DGPMHSS
RES
9.5956e-7
5.5697e-7
5.6167e-7
5.5736e-7
CPU(s)
0.156
0.156
0.187
0.172
IT
20
20
22
21
From Tables 5–8, we see that the DGPMHSS method is the best among the two methods in terms of number of iteration steps and computational time. For the GPMHSS and DGPMHSS methods, the CPU's time grows with the problem size whereas the presented results in Tables 5–8 show that in all cases the DGPMHSS method is superior to the GPMHSS method. That is to say, under certain conditions, compared with the GPMHSS method, the DGPMHSS method may be given the priority for solving the complex symmetric linear system (W+iT)x=b with -W⪯T≺W.
As already noted, in the two-half steps of the DGPMHSS iteration, there is a need to solve two systems of linear equations, whose coefficient matrices are αV+W-T and βV+W+T, respectively. This can be very costly and impractical in actual implementation. We use the IDGPMHSS method to solve the systems of linear equations (2) in the actual implementation. That is, it is necessary to solve two systems of linear equations with αV+W-T and βV+W+T by using the IDGPMHSS iteration. It is easy to know that αV+W-T and βV+W+T are symmetric and positive definite. So, solving the above two subsystems, the CG method can be employed.
In our computations, the inner CG iteration is terminated if the current residual of the inner iterations satisfies
(20)∥p(j)∥2∥r(k)∥2≤0.1τ(k),∥q(j)∥2∥r(k)∥2≤0.1τ(k),
where p(j) and q(j) are, respectively, the residuals of the jth inner CG for αV+W-T and βV+W+T. r(k) is the kth outer IDGPMHSS iteration; τ is a tolerance.
Some results are listed in Tables 9 and 10 for m=24 and 32, which are the numbers of outer IDGPMHSS iteration (it.s), the average numbers (avg1) of inner CG iteration for αV+W-T and the average numbers (avg2) of CG iteration for βV+W+T.
Convergence results for the IDGPMHSS iteration with m=24.
(αG*,βG*)
σ2
τ=0.95
τ=0.9
τ=0.85
it.s
avg1
avg2
it.s
avg1
avg2
it.s
avg1
avg2
(1.1, 1)
10
51
20.9
18.2
68
26.1
23.7
84
30.4
28.3
(1.5, 1)
50
54
23.5
18.3
70
28.0
23.9
86
33.0
28.4
(2.2, 1)
80
57
25.0
20.1
72
30.0
25.3
88
34.2
30.0
(1.8, 1)
100
58
27.5
18.9
74
31.9
24.3
89
36.1
28.7
Convergence results for the IDGPMHSS iteration with m=32.
(αG*,βG*)(αG*,βG*)
σ2
τ=0.95
τ=0.9
τ=0.85
it.s
avg1
avg2
it.s
avg1
avg2
it.s
avg1
avg2
(1.1, 1)
10
66
27.1
23.6
90
34.2
31.1
111
40.0
37.2
(1.5, 1)
50
70
30.6
23.7
93
37.5
31.3
114
43.4
37.4
(2.2, 1)
80
75
35.0
24.2
97
41.1
31.8
118
47.0
37.8
(1.8, 1)
100
77
36.1
25.3
98
42.2
32.4
119
48.0
38.3
In our numerical computations, it is easy to find the fact that the choice of τ is important to the convergence rate of the IDGPMHSS method. According to Tables 9 and 10, the iteration numbers of the IDGPMHSS method generally increase when τ decreases. Meanwhile, the iteration numbers of the IDGPMHSS method generally increase when σ2 increases.
5. Conclusion
In this paper, we have generalized the GPMHSS method to the DGPMHSS method for a class of complex symmetric linear systems (W+iT)x=b with real symmetric matrices W,T∈ℝn×n satisfying -W⪯T≺W under certain conditions. Theoretical analysis shows that for any initial guess the DGPMHSS method converges to the unique solution of the linear system for a wide range of the parameters. Then, an inexact version has been presented and implemented for saving the computational cost. Numerical experiments show that DGPMHSS method and IDGPMHSS method are efficient and competitive.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for their helpful suggestions, which greatly improved the paper. The authors would like to thank Yan-Jun Liang, Ting Wang, Ming-Yue Shao, and Man-Man Wang for helpful discussions. This work is also supported by the National College Students' Innovative and Entrepreneurial Training Program (201310479069). This research was supported by NSFC (no. 11301009), Science and Technology Development Plan of Henan Province (no. 122300410316), and Natural Science Foundations of Henan Province (no. 13A110022).
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