Numer Algor https://doi.org/10.1007/s11075-018-0520-4 ORIGINAL PAPER
A note on block diagonal and block triangular preconditioners for complex symmetric linear systems Li-Dan Liao1
· Guo-Feng Zhang1,2
Received: 25 November 2017 / Accepted: 27 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this note, the additive block diagonal preconditioner (Bai et al., Numer. Algorithms 62, 655–675 2013) and the block triangular preconditioner (Pearson and Wathen, Numer. Linear Algebra Appl. 19, 816–829 2012) are further studied and optimized, respectively. The eigenvalue properties of these two preconditioned matrices are analyzed by new way and an expression of the quasi-optimal parameter is derived. Particularly, when W − T or T − W is symmetric positive semidefinite, the exact eigenvalue bounds are obtained which are tighter than the state of the art. At last, numerical experiments are presented to show the effectiveness of the two proposed optimized preconditioners. Keywords Preconditioner · Complex symmetric problem · Optimal parameter · Condition number Mathematics Subject Classification (2010) 65F10 · 65F50 · 65W05
This work was supported by the National Natural Science Foundation of China (No. 11771193) and Fundamental Research Funds for the Central Universities (No. lzujbky-2017-it56). Guo-Feng Zhang
gf
[email protected] Li-Dan Liao
[email protected] 1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
2
Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu Province, People’s Republic of China
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1 Introduction Consider the following complex symmetric linear system [1, 16], Au ≡ (W + iT )u = c,
(1) √
where A ∈ Cn×n , u = (x + iy) ∈ Cn , c = (f + ig) ∈ Cn , i = −1 is the imaginary unit, x, y, f, g ∈ Rn , W ∈ Rn×n is symmetric positive definite, T ∈ Rn×n is symmetric positive semidefinite. The linear system (1) arises in many important large-scale application problems, such as wave propagation, distributed control problems, structural dynamics, FFT-based solution of certain time-dependent PDEs, molecular scattering, and lattice quantum chromo dynamics [14, 15, 17, 18, 20, 21]. For more details, readers can see [6, 22] and references therein. The linear system (1) can be rewritten into the following symmetric form: W T y g ¯ A z¯ ≡ = ≡ b, (2) T −W x −f which avoids involving complex arithmetic; meanwhile, the coefficient matrix in (2) becomes double in size. In order to solve the linear system (1) or (2) efficiently, many iteration methods and preconditioning techniques have been developed in the past few years (see [2–6, 9–13, 24] and the references therein). Among which, the PMHSS preconditioner proposed in [11] has been paid much attention. However, the PMHSS preconditioner is a nonsymmetric matrix, which may require considerable memory and storage in computations. Therefore, by further simplifying and modifying the PMHSS preconditioner, Bai et al. proposed an additive block diagonal (ABD) preconditioner for solving (2) [12]. In addition, the block triangular preconditioner proposed in [19] also has gained a lot of attention, because it gave a more efficient Schur complement approximation. In this paper, we first further study the ABD preconditioner in [12]. The spectral properties of the ABD preconditioned matrix are analyzed and an explicit expression of the quasi-optimal parameter is obtained. Then, we propose a new block triangular preconditioner, which is an improved one of the block triangular preconditioner proposed √ in [19]. Particularly, when W T or T W , we obtain a practical parameter α ∗ = 2 − 1, which makes the block diagonal and block triangular preconditioned matrices have tighter spectrum than the state of the art. Here and in the sequel, for any matrices B and C, the notation B C(B C) means that B − C is symmetric positive definite (symmetric positive semidefinite). Finally, some numerical experiments are given to validate the theoretical results and to illustrate the effectiveness of these two improved preconditioners. The remainder of this paper is organized as follows. In Section 2, the block diagonal preconditioner and the block triangular preconditioner are discussed with W 0 and T 0; the strategy for selecting quasi-optimal parameter is demonstrated. In Section 3, when W 0 and T W , the optimized block diagonal and block triangular preconditioners are used to solve an equivalent linear system. In Section 4, numerical experiments are demonstrated. Finally, in Section 5, some concluding remarks are given.
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2 The block diagonal and block triangular preconditioners Consider the following block diagonal and block triangular preconditioners W + αT 0 PGBD (α) = 0 W + αT and
PBT (α) =
W 0 , T −S(α)
(3)
(4)
respectively, where S(α) := (W + αT )W −1 (W + αT ) is an approximation of the Schur complement W + T W −1 T . In fact, the preconditioner (3) is a variant of the additive block diagonal (ABD) preconditioner [12]; here, the ABD preconditioner is of the form 1 W + α1 T 0 αW + T 0 PABD (α) = =α . = αP GBD 1 0 αW + T 0 W + αT α It is seen that when ω = 0, β = 1, M = W , and K = T , the block diagonal preconditioner proposed in [23] reduces to the ABD preconditioner; however, ω cannot be zero and β is smaller than 1 in general. The reason why we take the preconditioner PGBD (α) into consideration is that PGBD (α) can make the eigenvalues of the preconditioned matrix be close to 1 and away from 0, which is expected to be better than the preconditioner PABD (α). We also note that if α = 1, the preconditioner (4) reduces to the block triangular preconditioner in [19]. Denote G(α) := PGBD (α)−1 A and H := (W + αT )−1 W . By (2) and (3), we can derive 1 H (I − H ) α G(α) = 1 −H α (I − H ) and
G2 (α) =
H2 +
1 (I α2
− H )2
0
0 . H 2 + α12 (I − H )2
(5)
From (5), we know that the eigenvalues of the matrix G2 (α) are the same as those of H 2 + α12 (I − H )2 , which means that the eigenvalues of G2 (α) are of the form λ2 (H ) + α12 (1 − λ(H ))2 , where λ(H ) is an eigenvalue of H . Next, we consider the eigenvalues clustering of the preconditioned matrix G(α). Utilizing the techniques similar to those in [2, 5, 23], we define δ(G) :=
λup (G) , λlow (G)
where λup (G) and λlow (G) denote the upper and lower bounds of eigenvalues of the matrix G, respectively. Then, we can choose α by minimizing the value of δ(G(α)) and obtain the following results. Theorem 1 Let W 0, T 0, α be a positive constant number. Assume that vmax is the maximum eigenvalue of the matrix W −1 T , f is an eigenvalue of matrix G2 (α),
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where the G2 (α) is defined in (5). Then, we have f > 0 and the upper and lower bounds of eigenvalues of the matrix G2 (α) satisfy ⎧ 2 ⎨ 1 + vmax 1 , 0 < α < α0 , and flow = 2 fup = (αvmax + 1)2 ⎩ α +1 1, α ≥ α0 , vmax . Moreover, δ(G2 (α)) is minimized when α = α0 and 2 1 + vmax +1 δ(G2 (α0 )) = 1 + α02 < 2. with α0 =
Proof Noting that W 0 and T 0, it is easy to verify that all eigenvalues of W −1 T are real and non-negative. Obviously, H can also be rewritten as H = (I + αW −1 T )−1 ; then, it is not hard to obtain that f =
1 + v2 , (αv + 1)2
(6)
where v is an eigenvalue of W −1 T . On one hand, we can verify that the following inequality 1 1 + v2 ≥ 2 2 (αv + 1) α +1 1 + v2 ≤ 1 holds for any 0 ≤ v ≤ (αv + 1)2 2 1 + vmax = . Therefore, we have completed (αvmax + 1)2
always holds. On the other hand, if α ≥ α0 , vmax , else if 0 < α < α0 , then fup
the proof of the first part. As shown above, when α ≥ α0 , the upper bound of eigenvalues of the matrix G2 (α) is 1; otherwise, it is larger than 1. Therefore, δ(G2 (α)) is minimized in the interval [α0 , +∞]. Since the lower bound of eigenvalues of the matrix G2 (α) is given by α 21+1 , which is decreasing with α. Therefore, δ(G2 (α)) is minimized at α = α0 , i.e., δ(G2 (α)) ≤ δ(G2 (α0 )) = 1 + α02 < 2. Thus, we complete the proof. According to the results in Theorem 1, we have the following spectral propertities for the preconditioned matrix PGBD (α)−1 A . Theorem 2 Assume that the conditions of Theorem 1 are satisfied, then the eigen−1 values of
the block diagonal preconditioned matrix PGBD (α) A are of the form 2 vj + 1 , where vj (j = 1, 2, · · · , n) are the eigenvalues of W −1 T . In λ=± (αvj + 1)2 addition, if α = α0 , then we have
1 1 −1 ∪ sp(PGBD (α0 ) A ) ⊆ − 1, − , 1 , α02 + 1 α02 + 1 where sp(•) denotes the spectrum of a matrix.
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Remark 1 From Theorems 1–2 above, some observations can be obtained: –
– –
The eigenvalues of the preconditioned matrix PGBD (α0 )−1 A are more clustered than that of PGBD (1)−1 A as α0 < 1, suggesting that the parameter α0 is meaningful to accelerate the convergence rate of the GMRES method. In [23], there is a similar expression for the optimal parameter, but it represents different meanings. Our results are feasible for the case with T being positive semidefinite. However, in [23], only the case with K being symmetric positive definite is considered, and if K is symmetric semidefinite, the results of Lemma 2 in [23] cannot be true. Define M(α) := S(α)−1 (W + T W −1 T ).
(7)
The following lemma will be useful to analyze the spectrum of the preconditioned matrix PBT (α)−1 A . Lemma 1 Assume that W 0 and T 0. Then, the eigenvalues of the matrix M(α) vj2 + 1 are of the form λ = , where vj (j = 1, 2, · · · , n) are eigenvalues of the (αvj + 1)2 matrix W −1 T . In addition, δ(M(α0 )) is bounded by 2, i.e., δ(M(α0 )) = 1 + α02 < 2. The eigenvalues of the matrix M(α0 ) are clustered in the interval:
1 ,1 , 1 + α02 where α0 is defined as in Theorem 1. Proof As W 0 and T 0, we have M(α) ∼ (I + αQ)−2 (I + Q2 ), 1
Q ∼ W −1 T ,
(8)
1
where Q := W − 2 T W − 2 . Therefore, from (8), we know that the eigenvalues of v2 + 1 M(α) are of the form , where v is an arbitrary eigenvalue of the matrix (αv + 1)2 W −1 T . In addition, from (6), we find that the eigenvalues of M(α) have the same expression as those of G2 (α), where G2 (α) is defined in (5). Therefore, from Theorem 1, it is easy to obtain the following results. Theorem 3 Assume that the conditions of Theorem 1 are satisfied. Then, the eigenvalues of the preconditioned matrix PBT (α)−1 A are 1 with multiplicity n, the remainvj2 + 1 ing eigenvalues are the same as those of the matrix M(α), i.e., λj = (αvj + 1)2 −1 with vj (j = 1, 2, · · · , n) being the eigenvalues of the matrix W T . Moreover,
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δ(PBT (α)−1 A ) is bounded by 1 + α02 , and the eigenvalues of the preconditioned matrix PBT (α0 )−1 A are clustered in the interval:
1 ,1 , 1 + α02 where α0 is defined as in Theorem 1. Proof By (4), it is easy to verify that PBT (α)−1 A =
I W −1 T 0 M(α)
,
where M(α) = S(α)−1 (W + T W −1 T ) defined in (7). Therefore, by Lemma 1, we obtain the results. Remark 2 From Theorem 3, we see that the eigenvalues of the preconditioned matrix PBT (α0 )−1 A are bounded by [ 1 2 , 1] ⊂ [ 12 , 1]; it is tighter than that of the block 1+α0
triangular preconditioner in [19], which means that the preconditioner PBT (α0 ) is expected to be better than the block triangular preconditioner in [19]. Remark 3 In [5], the author provided an alternative quasi-optimal parameter α1 =
vmin + vmax . 2 + 1) − v 2 1 + (vmax + 1)(vmin min vmax
It is easy to prove that sp(M(α1 )) ⊆ [
1 , 1+α12
sp(PBT (α1 )−1 A ) ⊆ [
γ 2 ] and 1 , γ 2 ] ∪ {1}, 1 + α12
(9)
where vmin denotes the minimum eigenvalue of W −1 T and γ =
vmin + vmax
. 2 +1 2 vmin vmax + 1 + vmax vmin
Define λ(G)max , λ(G)min where λ(G)max and λ(G)min are the maximum and minimum eigenvalues of G, respectively. If vmin = 0, then α1 > α0 and γ < 1. By Theorem 3 and (9), we can obtain K (PBT (α0 )−1 A ) = 1 + α02 < 1 + α12 = K (PBT (α1 )−1 A ). K (G) :=
Therefore, PBT (α0 ) should be better than PBT (α1 ). However, we know that the parameter α0 is related to the maximum eigenvalue of W −1 T , which is impractical to implement when n is large. Fortunately, we find that
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if W T , then vmax ≤ 1, which also can result in α0 ≤ Theorem 1, we have the following results.
√
2 − 1. By the analysis in
Theorem 4 Under the conditions of Theorem 1, if W T , then the eigenvalues of PGBD (α ∗ )−1 A are contained in the intervals √ √ 2+2 2+2 [−1, − ]∪[ , 1]; 2 2 the eigenvalues of PBT (α ∗ )−1 A are located in the interval √ 2+2 [ , 1], 4 where √ α ∗ := 2 − 1.
3 Preconditioners for the case with W 0 and T W Given W 0 and T W , we know that multiplying the matrix I −I
W + T 0 and T − W 0. Thus, by I I
on both sides of the linear system (2), we obtain the following linear system: T W y g−f A z¯ ≡ = ≡ b˜ x −f − g T −W
(10)
with := W + T W
and
T := T − W.
Next, we apply the proposed block diagonal preconditioner (3) and the block triangular preconditioner (4) to the linear system (10), and denote the preconditioners as GBD (α) and P BT (α), respectively. Here, P GBD (α) and P BT (α) have the following P form: + α T W 0 W 0 (11) PGBD (α) := + α T , PBT (α) := T − 0 W S(α) + α T). + α T)W −1 (W with S(α) := (W −1 T is less As W − T = 2W 0, we know that the maximum eigenvalue of W than 1. Similar to the discussions in Theorem 4, we can obtain the following results. Theorem 5 Assume that W 0, T is a symmetric matrix that makes T − W 0. Then, we have √ √ 2 2 ∗ −1 ∪ ,1 (12) sp(PGBD (α ) A ) ⊆ − 1, − 2 2
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and BT (α ) sp(P
∗ −1
where α ∗ =
√
A) ⊆
2 + √2 4
,1 ,
(13)
2 − 1.
Remark 4 Comparing with the related results in [2, 12, 19], if T W , we see from GBD (α ∗ )−1 A (12) and (13) that the spectral bounds for the preconditioned matrix P are better than those in [12]. The eigenvalue distributions for the preconditioned BT (α ∗ )−1 Aare much tighter than that in [2, 19]. These indicate that the two matrix P proposed preconditioners are expected to be better than the state of the art (see [7, 8]).
4 Numerical experiments In this section, we will give some examples to verify the correctness of the theoretical analysis given in Sections 2 and 3. We are also interested in the effectiveness of the proposed preconditioners. All tests are performed in MATLAB R2013a with machine precision 2.2 × 10−16 on a personal computer with 3.20-GHz CPU (Intel(R) Core(TM) i5-3470), 8.00 GB RAM. All runs are started from the zero vector, and the iteration is terminated once the current residual in the first stage satisfies −6 or the number of the prescribed iteration k k RES := r max = 1000 is r0 < 10 exceeded. We denote the number of iteration steps as “IT”, and the elapsed CPU time in seconds as “CPU”. Example 1 ([2, 9, 10]) We consider the complex Helmholtz equation −p + σ1 p + iσ2 p = f, where σ1 and σ2 are real coefficient functions and p satisfies Dirichlet boundary conditions in D = [0, 1] × [0, 1]. We discretize the problem with finite differences on a m × m grid with mesh size h = 1/(m + 1). This leads to a system of linear equations ((K + σ1 I ) + iσ2 I )u = c, where K = I ⊗ Vm + Vm ⊗ I is the discretization of − by means of centered differences, Vm = h−2 tridiag(−1, 2, −1) ∈ Rm×m . The right-hand side vector c is taken to be c = (1 + i)Al, with l being the vector of all entries equal to 1. In this example, we take m = 16, 32, 64, 128, 256, 512 for six test sizes, and we set the parameters σ1 = 100 and σ2 = 100. For this example, it is easy to verify W − T = K 0. Therefore, we consider applying the preconditioner PBT (α) to accelerate the GMRES method and adopting the preconditioner PGBD (α) to accelerate the MINRES method, √ respectively. Furthermore, by the results of Theorem 4, if we take α = α ∗ = 2 − 1, the proposed preconditioners PGBD (α ∗ ) and PBT (α ∗ ) are expected to achieve satisfactory results. To investigate the efficiency of the proposed preconditioners, we compare the results
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with those of the preconditioners PPMHSS [10], PPEA [19], and PABD [12]. Here, the preconditioners PPMHSS , PPEA , and PABD are of the form α + 1 I −I αW + T 0 PPMHSS := , 0 αW + T I I 2α W +T 0 , PABD := 0 W +T W 0 PPEA := T − S with S := (W + T )W −1 (W + T ). The numerical results for Example 1 are listed in Table 1. From Table 1, we can find that all tested preconditioned Krylov subspace methods can efficiently converge to the unique solution. Both the preconditioners PGBD (α ∗ ) and PBT (α ∗ ) outperform the preconditioners PPMHSS , PPEA , and PABD with respect to the CPU times. The block triangular preconditioner PBT (α ∗ ) performs the best compared with the other preconditioners in terms of the iteration steps. The block diagonal preconditioner PGBD (α ∗ ) is the best one since it costs the least CPU times. We have also observed that the iteration steps of the PGBD (α ∗ )-preconditioned MINRES method and the PBT (α ∗ )-preconditioned GMRES method are h-independent, i.e., both the preconditioners PGBD (α ∗ ) and PBT (α ∗ ) are robust and effective with respect to the mesh size h. In order to further validate the correctness of the theoretical results and the effectiveness of the preconditioners PGBD (α ∗ ) and PBT (α ∗ ), the eigenvalue distributions for the preconditioned matrices are drawn in Fig. 1. It can be observed that the preconditioned matrix PGBD (α ∗ )−1 A has more clustered eigenvalue distributions than −1 that of PABD A . Moreover, the eigenvalue bounds for PGBD (α ∗ )−1 A described in Fig. 1 are almost consistent with the theoretical values given in Theorem 4. This phenomenon also holds for the block triangular preconditioner PBT (α ∗ ), which indicates Table 1 Numerical results for Example 1 Pre
PPMHSS
PPEA PABD PGBD (α ∗ ) PBT (α ∗ )
m
α IT CPU IT CPU IT CPU IT CPU IT CPU
16
32
64
128
256
512
1 14 0.0484 7 0.0081 14 0.0310 8 0.0026 4 0.0030
1 15 0.0120 7 0.0094 16 0.0093 10 0.0064 5 0.0073
1 16 0.0479 8 0.0374 16 0.0349 10 0.0242 5 0.0279
1 16 0.2110 8 0.2249 16 0.2277 10 0.1571 5 0.1701
1 16 1.4033 8 1.2626 14 1.1333 8 0.7281 5 0.9130
1 16 8.5582 8 6.8467 14 6.3046 8 4.1423 5 4.9901
Numer Algor P GBD (
0.1
=0.41421)
P ABD (
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.1
=1)
-0.1 -1
-0.8
-0.6
-0.4
-0.2
P BT (
0.1
0
0.2
0.4
0.6
0.8
1
-1
-0.8
-0.6
-0.4
=0.41421)
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.1
0
P PEA (
0.1
0.08
-0.2
0.2
0.4
0.6
0.8
1
0.4
0.6
0.8
1
=1)
-0.1 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Fig. 1 The eigenvalue distributions of the preconditioned matrices PGBD (α)−1 A and PBT (α)−1 A for Example 1 for 16 × 16 grids
√ that our theoretical analysis is credible, and the quasi-optimal parameter α ∗ = 2−1 does indeed improve the efficiency of the ABD and block triangular preconditioners in [12] and [19], respectively. Example 2 ([2, 9, 10]) Consider the iteration solution of linear system of the form √ √ (3 − 3) (3 + 3) K+ I +i K + I u = c, τ τ where τ is the time step-size, K = I ⊗ Vm + Vm ⊗ I with Vm = h−2 tridiag(−1, 2, −1) ∈ Rm×m , and the mesh size h = 1/(m + 1). The right-hand side vector c with its j -th entry cj being given by cj =
(1 − i)j , j = 1, 2, · · · , n. τ (j + 1)2 √
In this example, we take τ = h, then (T − W ) = 2 τ 3 I 0. Thus, as discussed in BT (α ∗ ) to solve GBD (α ∗ ) and P Section 3, we consider adopting the preconditioners P the linear system (10). To see the effectiveness of the two proposed preconditioners,
Numer Algor Table 2 Numerical results for Example 2 Pre
m 16
PPMHSS
PABD PPEA GBD (α ∗ ) P BT (α ∗ ) P
32
64
128
256
512 1
α
1
1
1
1
1
IT
8
9
10
11
11
11
CPU
0.0046
0.0084
0.0381
0.1796
1.1056
7.1175
IT
10
12
12
13
12
12
CPU
0.0030
0.0074
0.0287
0.1967
1.0130
5.6015
IT
5
6
7
7
7
7
CPU
0.0045
0.0079
0.0357
0.2094
1.1500
6.2636
IT
9
9
9
9
9
9
CPU
0.0033
0.0057
0.0225
0.1469
0.8015
4.4187
IT
4
4
4
4
4
4
CPU
0.0035
0.0066
0.0256
0.1516
0.8175
4.3192
we also list the numerical results for that of the preconditioners PPMHSS , PPEA , and PABD as comparison. The numerical results for Example 2 are reported in Table 2. GBD (α ∗ ) and P BT (α ∗ ) behave From Table 2, we see that the preconditioners P much better than the preconditioner PPMHSS both in terms of the iteration steps and CPU times. The performance of the preconditioned MINRES with the preGBD (α ∗ ) is much better than with the preconditioner PABD , and the conditioner P BT (α ∗ )-preconditioned GMRES method is better than that of the performance of P PPEA -preconditioned GMRES method. Moreover, it is observed that the iteration GBD (α ∗ ) and P BT (α ∗ ) remain constant for different steps for the preconditioners P GBD (α ∗ ) and P BT (α ∗ ) are also problem size, which implies that the preconditioners P efficient and robust.
5 Conclusions We have proposed two improved block diagonal and block triangular preconditioners for solving complex symmetric linear systems. The eigenvalue distributions of the proposed preconditioned matrices are analyzed and a quasi-optimal parameter is derived. Furthermore, if W T or T W , we obtain the parameter-free eigenvalue bounds for the corresponding preconditioned matrices, which are tighter than the state of the art. Finally, we give two experiments to show full robustness and effectiveness of the two improved preconditioners. Acknowledgments The authors are very thankful to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.
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