Numerische Mathematik

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1 Numer. Math. (00) 93: Digital Object Identifier (DOI) /s Numerische Mathematik Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems Qiya Hu 1,, Jun Zou, 1 Institute of Computational Mathematics and Scientific Engineering Computing, Chinese Academy of Sciences, Beijing , People s Republic of China; hqy@lsec.cc.ac.cn Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People s Republic of China; zou@math.cuhk.edu.hk Received December 8, 1999 / Revised version received September 8, 001 / Published online March 8, 00 c Springer-Verlag 00 Summary. In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms. Mathematics Subject Classification: 65F10, 65N0 1 Introduction The purpose of this paper is to study some nonlinear inexact Uzawa algorithms for solving the following system of linear equations The work of this author was partially supported by a grant from The Institute of Mathematical Sciences, CUHK The work of this author was partially supported by Hong Kong RGC Grants CUHK 49/00P and CUHK 444/01P

2 334 Q. Hu, J. Zou (1.1) ( )( ) ( ) AB x f B t =, 0 y g where A is a symmetric positive definite n n matrix, and B is an n m matrix (m n). We assume that the (n + m) (n + m) coefficient matrix ( ) AB M = B t 0 is nonsingular, which is equivalent to the positive definiteness of the Schur complement matrix (1.) C = B t A 1 B. Linear systems such as (1.1) are called saddle-point problems, which may arise from finite element discretizations of Stokes equations and Maxwell equations [7, 11, 15], mixed finite element formulations for second order elliptic problems [, 7] or from Lagrange multiplier methods for optimization problems [1,16], for the parameter identification and domain decomposition problems [1,17] and [19]. In recent years, there is a rapidly increasing literature which is concerned with preconditioned iterative methods for solving the indefinite system of equations like (1.1), see [3,5,6,8,14,0 ]. In particular, inexact Uzawatype algorithms have attracted wide attention, see [3,5,8,14,1] and the references therein. The common nice properties of these Uzawa-type algorithms are that they have minimal memory requirements, and are easy to implement. Let  and Ĉ be two positive definite matrices, which are assumed to be the preconditioners of the matrices A and C = B t A 1 B respectively. And let R l denote the usual l-dimensional Euclidean space. For any l l positive definite matrix G, weuse x G to denote the G-induced norm, namely x G =(Gx, x) 1/ for all x R l. Then the standard inexact Uzawa algorithm can be described as follows (cf. [5] and [14]): Algorithm 1.1 (inexact Uzawa). Given x 0 R n and y 0 R m, the sequence {(x i,y i )} R n R m is defined for i =1,,, by (1.3) (1.4) x i+1 = x i +  1 [f (Ax i + By i )], y i+1 = y i + Ĉ 1 (B t x i+1 g). There are some earlier versions of the above algorithm, e.g., see [3] and [1]. The existing convergence results indicate that these algorithms are convergent by assuming some good knowledge of the spectrum of the preconditioned matrices  1 A and Ĉ 1 C or under some proper scalings of the

3 Two new variants of nonlinear inexact Uzawa algorithms 335 preconditioners  and Ĉ. This preprocessing may not be easy to achieve in some applications. To avoid the estimate on the minimal generalized eigenvalues of  with respect to A, a practical way is to replace the inner iteration (1.3) of the standard inexact Uzawa algorithm by some nonlinear iteration, for example, the PCG iteration. The detailed convergence properties of the resulting algorithm can be found in [5] and [14]. However, some improvements on the convergence results achieved so far seem necessary (see the numerical examples given in [5] and Sect. below). Moreover, a good knowlwdge on the spectrum of the preconditioned Schur complement is needed for a proper scaling of the preconditioner to guarantee the convergence of the global inexact Uzawa iteration, see Sect. for the details. To avoid the estimate of the generalized eigenvalues of Ĉ with respect to B t  1 B, the Uzawa-type algorithm proposed in [3] introduced a PCG algorithm as an inner iteration of (1.4). It was proved that the Uzawa-type algorithm converges under the assumptions that each inner iteration teminates with a certain accuracy and the basic iteration for the preconditioned system  1 A is convergent. The preconditioned minimal residual method studied in [0] and [] is always convergent, but needs some good knowledge of the smallest eigenvalues of the matrices  1 A and Ĉ 1 (B t  1 B) to achieve a practical convergence rate. Without a good apriori estimate on the smallest eigenvalues of these two preconditioned matrices, the condition number of the (global) preconditioned system may be still very large even if the condition numbers of the matrices  1 A and Ĉ 1 (B t  1 B) are small (cf. []). In this case, the convergence of this iterative method may be slow (see Sect. 4). The purpose of this paper is twofold. By means of a new technique, we first give an essential improvement on the convergence results of Bramble- Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. We will show our algorithms are always convergent without any assumptions on the spectra of the preconditioners Ĉ and Â. The outline of the remainder of the paper is as follows. We give a new convergence result for a known nonlinear inexact Uzawa algorithm in Sect., and present two new algorithms and analyse their convergence rates in Sects. 3 and 4. Finally, we apply the proposed new algorithms for solving the Stokes problem and a linear system of purely algebraic equations in Sect. 5.

4 336 Q. Hu, J. Zou An improved convergence result for the known nonlinear inexact Uzawa algorithm The main purpose of this section is to present an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a nonlinear inexact Uzawa algorithm. In this algorithm, the computation of the action of the inverse A 1 in (1.3) is replaced with that of an approximation to A 1 which results from applying a nonlinear iterative process for inverting A. Two examples of such approximations are from defining the approximate inverse by a PCG iteration or the operator which results from the application of a multigrid cycling algorithm with a nonlinear smoother (cf. [5]). We now describe the nonlinear inexact Uzawa algorithm, in which the nonlinear approximate inverse is defined by a map Ψ : R n R n.for φ R n, Ψ(φ) is an approximation to the solution ξ of (.1) Aξ = φ. The following assumption was made in [5] on the accuracy of the approximation: (.) Ψ(φ) A 1 φ A δ A 1 φ A, φ R n for some δ (0, 1). This is a natural assumption and can be satisfied by the approximate inverse associated with the PCG iteration and the approximate inverse defined by one sweep of a multigrid algorithm with conjugate gradient smoothing (cf. [5]). The nonlinear inexact Uzawa algorithm is defined as follows: Algorithm.1 (nonlinear inexact Uzawa). Given x 0 R n and y 0 R m, the sequence {(x i,y i )} R n R m is defined for i =1,,, by (.3) (.4) (.5) x i+1 = x i + Ψ(f (Ax i + By i )), y i+1 = y i + Ĉ 1 (B t x i+1 g). Under the assumption (.) and (1 γ)(ĉy,y) (Cy,y) (Ĉy,y), y Rm, it was proved in [5] that Algorithm.1 converges if (.6) δ< 1 γ 3 γ. And in this case the following estimate of the convergence rate holds: ( ) δ 1+δ ex i A + e y δ i Ĉ ρi 1+δ ex 0 A + e y 0 Ĉ

5 Two new variants of nonlinear inexact Uzawa algorithms 337 and ( ) δ e x i A (1 + δ)(1+δ) ρ i 1+δ ex 0 A + e y 0, Ĉ where e x i and ey i are the error functions e x i = x x i, e y i = y y i and ρ = δ + γ + (δ + γ) +4δ(1 γ). Generally speaking, the convergence condition (.6) is somehow not very satisfactory as it depends on the preconditioning parameter γ. The condition says that the evaluation of A 1 f i for some f i required in (.3) by the nonlinear iteration Ψ should be more accurate if the preconditioning of the Schur complement becomes worse (i.e., γ is close to 1). This seems not so reasonable. Indeed we find surprisingly that such a requirement on δ can be made completely independent of γ, namely the same results still hold when (.6) is replaced by δ<1/3 as shown below. To demonstrate such an essentially improved result, we will first establish the convergence rate for the residual f i = f (Ax i + By i )=Ae x i + Be y i, instead of the traditionally used error vector e x i as was done in [5]. This is an important observation which leads to our improved results. To do so, we define a new norm : v =( v 1 + v C) 1, v =(v1,v ) R n R m. With this norm we can show Theorem.1 Under the assumptions (.) and (.5), for δ< 1 3 we have (.7) E i+1 ρ E i, E i =( δa 1 fi, e y i ) Rn R m. The convergence rate ρ (< 1) can be estimated by (.8) (.9) ρ { δ + δ + δ for 0 γ 4δ 1+δ, γ(1+δ) δ+ [γ(1+δ) δ] +4δ 4δ for 1+δ { <γ<1 δ + δ + δ for 0 γ 4δ 1+δ, 1 1 (1 γ)(1 + δ) for 4δ 1+δ <γ<1.

6 338 Q. Hu, J. Zou Proof. The inequality (.8) was discovered and proved independently by Cheng [10] and the authors [18]. However, for reader s convenience, we outline the main idea of the proof. Let B t A 1 = U(Σ0 0)V t be the singular value decomposition of the matrix B t A 1. Define Q 0 = Σ 0 U t Ĉ 1 UΣ 0 and ( ) δ(i + Q0 ) δq0 F = δq0. (I Q 0 ) It can be verified directly that F (.10) max β σ(q 0 ) 1 δ β(1 + δ) + 1 δ β(1 + δ) +4δ, where σ(q 0 ) is the spectrum of the matrix Q 0. One can check that the matrix Q 0 has the same eigenvalues as the matrix Ĉ 1 C. Thus we have σ(q 0 ) [1 γ,1] by (.5). Then we can show ρ F, and (.8) follows from (.10). We next show (.9). When γ>4δ/(1 + δ), wehave γ(1 + δ) δ >δ >0. This with the condition that γ<1 leads to γ(1 + δ) δ + [γ(1 + δ) δ] +4δ <γ(1 + δ) δ + (1 δ) +4δ = γ(1 + δ) δ +1 δ =+(γ 1)(1 + δ). (.9) follows then from this relation and (.8). In particular, when δ< 1 3 we have ρ<1. Using Theorem.1, we can estimate the convergence rates of the errors e x i and ey i :

7 Two new variants of nonlinear inexact Uzawa algorithms 339 Theorem. Under the assumptions (.) and (.5), Algorithm.1 converges for any δ< 1 3. Moreover, we have (.11) e x i A ( 1+4δ + ρ)ρ i 1 E 0, i =1,, and (.1) e y i C ρ i E 0, i =1,,. Here the rate ofconvergence ρ has the following bounds: { ρ 1 ρ γ 1 6 γ for 0 γ 4δ (.13) 1+δ, ρ 3 (δ + δ 4δ ) for 1+δ <γ<1. Proof. The estimate (.1) is a direct consequence of (.7). We next prove (.11). To do so, we start with the derivation of a useful expression for A 1/ f i. Clearly, we can write f i = Ae x i + Be y i, Bt x i+1 g = B t e x i+1, this with (.3) and (.4) leads to and A 1 e x i+1 = A 1 (e x i Ψ(f i ))=(A 1 fi A 1 Ψ(fi )) A 1 Be y i A 1 Be y i+1 = A 1 B(e y i + Ĉ 1 B t e x i+1) (.14) So we have = A 1 B [e y i + Ĉ 1 B t A 1 ( )] (A 1 fi A 1 Ψ(fi )) A 1 Be y i = A 1 B Ĉ 1 B t A 1 (A 1 fi A 1 Ψ(fi )) +(I A 1 B Ĉ 1 B t A 1 )A 1 Be y i. (.15) A 1 fi+1 = A 1 e x i+1 + A 1 Be y i+1 =(I + A 1 B Ĉ 1 B t A 1 )(A 1 fi A 1 Ψ(fi )) (A 1 B Ĉ 1 B t A 1 )A 1 Be y i. Because of (.), for any given natural number i one can construct a symmetric and positive definite n n matrix Q Ai such that Q Ai Ψ(f i )=f i

8 340 Q. Hu, J. Zou and I 1 1 Q Ai AQ Ai δ, see, for example, Lemma 9 of [3] for such a construction of Q Ai. Thus we have (.16) and (.17) Ψ(f i )=Q 1 Ai f i I A 1 Q 1 Ai A 1 δ. By means of (.14)-(.16), we can further write A 1 fi+1 =(I + A 1 B Ĉ 1 B t A 1 )(I A 1 Q 1 Ai A 1 )A 1 fi (A 1 B Ĉ 1 B t A 1 )A 1 Be y i, or equivalently write (replacing i by i 1) where A 1 fi =(I + Q ) Q 1i A 1 fi 1 Q A 1 Be y i 1 Q 1i = I A 1 Q 1 Ai A 1 and Q = A 1 B Ĉ 1 B t A 1. Applying the arithmetic-geometric mean inequality yields (.18) A 1 fi (1 + η) (I + Q ) Q 1i A 1 fi 1 +(1 + η 1 ) Q A 1 Be y i 1, where η is any positive number. One can easily see that the matrix Q has the same positive eigenvalues as the matrix Ĉ 1 C. Hence, the assumption (.6) implies that the eigenvalues of Q are in the interval [0, 1]. Namely, we have Q 1, I + Q, this, with (.18) and (.17), leads to A 1 fi (1 + η)4δ A 1 fi 1 +(1+η 1 ) A 1 Be y i 1 =4δ(1 + η) δa 1 fi 1 +(1+η 1 ) e y i 1 C.

9 Two new variants of nonlinear inexact Uzawa algorithms 341 Taking η =1/(4δ) above and using (.7) we obtain Now using the relation A 1 fi 1+4δ ρ i 1 E 0. A 1 e x i = A 1 fi A 1 Be y i, (.11) follows immediately from (.7) and the triangle inequality. It remains to show (.13). Let us first consider 0 <γ 4δ/(1 + δ)). We have (.19) ρ ρ = δ + γ 4δ + + γ +4δ ( δ + δ + δ) = γ 4(δ + + δ)+γ 4(δ + δ) = γ + γ ( 4(δ + δ)+γ + 4(δ + δ)). Using δ<1/3 and γ<1, we know 4(δ + δ)+γ + 4(δ + δ) < 3. This along with (.19) yields (.0) ρ ρ> γ + γ 6. Now consider the remaining case: 4δ/(1 + δ) <γ<1. We have ρ ρ = δ + γ (δ) + + γ +4δ { γ δ(1 γ } [γ(1 + δ) δ] + +4δ (.1) ( = γ ) (δ) + γ [γ(1 + δ) δ] δ + ( (δ) + γ +4δ + [γ(1 + δ) δ] +4δ). Let it is easy to see that f(γ) =(δ) + γ [γ(1 + δ) δ], f(γ) =γδ[4(1 + δ) γ( + δ)]

10 34 Q. Hu, J. Zou and Thus by the monotonicity of f, f (γ) =δ > 0. (.) f(γ) >f ( ) 4δ = 16δ 1+δ (1 + δ). Moreover, using δ<1/3 and γ<1 we have (δ) + γ +4δ + [γ(1 + δ) δ] +4δ<3. This combining with (.1)-(.) leads to ρ ρ> 3 δ + 16δ 6(1 + δ) > 3 (δ + δ ), which completes the proof of (.13). Remark.1 The key points in the proof of Theorem.1 is the replacement of the error e x i by the residual f i and the use of the norm. The n n matrix Q Ai used in the proof can be viewed as a preconditioner of the matrix A (which is just the preconditioner  in [3] and the preconditioner Q A in [5]). However, this preconditioner changes with the iteration number i. This makes the norm used in [3] (involving ) and the norm used in the proof  of Theorem 3.1 of [5] (involving Q A A ) not applicable here for the analysis of the convergence of Algorithm.1, because the local contraction of errors with a changing norm can not guarantee the global convergence of the algorithm (refer to Theorem 4., [5]). Remark. The numerical results shown in [5] indicated that Algorithm.1 may still converge even if the condition (.6) is not satisfied. It was seen as a surprising phenomenon (see [5]). Now we would like to illustrate a bit about this phenomenon. It can be verified directly by (.10) that F < 1 is equivalent to Thus, F < 1 if and only if 1 δ β(1 + δ) < 1 δ. (.3) β< (1 δ) 1+δ, β σ(q 0). Under the assumption (.5), we can only obtain β max 1. Here β max denotes the maximal eigenvalue of the matrix Q 0. Solving the inequality

11 Two new variants of nonlinear inexact Uzawa algorithms < (1 δ) 1+δ, we get δ< 1 3, which means that the covergence condition δ< 1 3 is sharp theoretically when β max =1. However, if the preconditioner Ĉ is not properly scaled β max may be bounded far away from one (namely, close to zero). In this case (e.g., see the example considerd in [5]), the inequality (.3) is also valid even if δ is close to one. But the convergence can be very slow (cf. [5]) because the assumption (.5) holds only with the γ close to one (note that 0 < 1 γ β max ). These facts indicate that the afore-mentioned phenomenon is not too surprising. 3 New algorithm I: the nonlinear inexact Uzawa combined with the steepest descent method In general, the crucial assumption (.5) is not so easy to fulfill as it requires a good estimate on the maximum eigenvalue of the preconditioned Schur complement Ĉ 1 C in order to have the desired proper scaling of Ĉ. Without such a good estimate this assumption is not met or it is satisfied only with some γ very close 1, which may cause the divergence of the algorithm or results in very slow convergence of the algorithm. Obtaining such a good estimate of the eigenvalues is not very convenient or practical. In this and next sections, we propose two new algorithms to avoid such estimates. Our first idea is to introduce a relaxation parameter τ i in (.4) such that the norm τ i Ĉ 1 g i C 1 g i C is minimized, where g i = B t x i+1 g.ifg i /=0, the direct calculation gives τ = (g i, Ĉ 1 g i ) Ĉ 1 g i C = (g i, Ĉ 1 g i ) (A 1 BĈ 1 g i,bĉ 1 g i ). Since the action of A 1 is not allowed, A 1BĈ 1 g i will be then replaced by Ψ(BĈ 1 g i ), which results in the following new algorithm: Algorithm 3.1 (Nonlinear inexact Uzawa-steepest descent). Given x 0 R n and y 0 R m, the sequence {(x i,y i )} R n R m is defined for i =1,,, by Step 1. Compute f i = f (Ax i + By i ) and Ψ(f i ), update (3.1) x i+1 = x i + Ψ(f i ). Step. Compute g i = B t x i+1 g, d i = Ĉ 1 g i and (3.) τ i = { 1 (g i,d i ) (Ψ(Bd i ),Bd i ) for g i 0; 1 for g i =0.

12 344 Q. Hu, J. Zou Update (3.3) y i+1 = y i + τ i d i. Remark 3.1 It is clear that when both f i and g i vanish, then the vectors x i and y i are the exact solution of the equation (1.1), and thus Algorithm 3.1 terminates. For the sake of convenience, we introduce two parameters δ and δ 0 which are the minimal positive numbers satisfying (3.4) (3.5) A 1 f i Ψ(f i ) A δ A 1 f i A A 1 Bd i Ψ(Bd i ) A δ 0 A 1 Bd i A. Remark 3. Note that the assumptions (3.4) and (3.5) are in fact consequences of the assumption (.). So we can simply assume the condition (.) instead of (3.4)-(3.5) and all the results in this and next sections hold. We choose to formulate the above two conditions here so that we can see how the convergence rate of Algorithm 3.1 depends more explicitly on the accuracies of the two nonlinear inner iterations in both (3.1) and (3.3). The evaluations of A 1 f i and A 1 Bd i by the nonlinear iteration Ψ can be different. (3.6) Let κ = cond(ĉ 1 C), β = 1 4κ(1 δ 0) (1 + κ) (1 δ 0 ), then we have Theorem 3.1 Assume that the approximation parameters δ and δ 0 satisfy δ< 1 3 and δ 0 < 1, then Algorithm 3.1 converges. Moreover, we have (3.7) e x i A ( 1+4δ +ˆρ)ˆρ i 1 E 0, i =1,, and (3.8) e y i C ˆρ i E 0, i =1,,. Here the rate ofconvergence ˆρ (< 1) can be estimated by (3.9) ˆρ = { δ + δ + δ for 0 < 1+β 4δ 1+δ ; (1 β)(1 + δ) for 4δ 1+δ < 1+β < 1.

13 Two new variants of nonlinear inexact Uzawa algorithms 345 Remark 3.3 From Theorem 3.1 we know that the convergence of Algorithm 3.1 is independent of the spectrum of the preconditioned Schur complement Ĉ 1 C, and the convergence rate of this new algorithm depends only on the condition number κ, not necessary to have a proper scaling of the preconditioner Ĉ and a good apriori estimate on the maximum and minimum eigenvalues of Ĉ 1 C. This seems to be an important advantage of the new algorithm over the existing nonlinear inexact Uzawa algorithms. The assumptions of Theorem 3.1 are natural and can be satisfied by the approximate inverse associated with the PCG iteration and the approximate inverse defined by one sweep of a multigrid algorithm with a suitable smoother (cf. [3,5,4] and [3]). For the proof of Theorem 3.1 we need the following lemma: Lemma 3.1 Assume that the condition (3.5) holds with δ 0 < 1. Then for any given natural number i there is a symmetric and positive definite m m matrix Q Bi such that (i) Bi g i = τ i Ĉ 1 g i. (ii) All eigenvalues ofthe matrix Q 1 1 β BiC are in the interval [, 1]. Q 1 ProofofTheorem 3.1. With Lemma 3.1, we can view the matrix Q Bi as the preconditioner Ĉ for the Schur complement C, which satisfies the condition (.5) with γ = 1 β. Thus Theorem 3.1 is a direct consequence of Theorems.1-.. Here we have to make use of the important fact that the norms adopted in Theorems.1-. are independent of the preconditioner Ĉ, in comparison with [5]. ProofofLemma 3.1. It is easy to see that (3.10) (Ψ(BĈ 1 g i ),BĈ 1 g i )= Ĉ 1 g i C +(Ψ(BĈ 1 g i ) A 1 (BĈ 1 g i ),BĈ 1 g i ). Using the Cauchy s inequality and (3.5), yields (Ψ(BĈ 1 g i ) A 1 (BĈ 1 g i ),BĈ 1 g i ) Ψ(BĈ 1 g i ) A 1 (BĈ 1 g i ) A BĈ 1 g i ) A 1 δ 0 A 1 (BĈ 1 g i ) A BĈ 1 g i ) A 1 = δ 0 Ĉ 1 g i C. Substituting the above inequality into (3.10) leads to (1 δ 0 ) Ĉ 1 g i C (Ψ(BĈ 1 g i ),BĈ 1 g i ) (1 + δ 0 ) Ĉ 1 g i C.

14 346 Q. Hu, J. Zou Thus, (3.11) (g i, Ĉ 1 g i ) (1 + δ 0 ) Ĉ 1 g i C τ i (g i, Ĉ 1 g i ) (1 δ 0 ) Ĉ 1 g i C, where the parameter τ i is defined by For τ>0, let Clearly, τ i = (g i, Ĉ 1 g i ). Ψ(BĈ 1 g i ) A F (τ) = τĉ 1 g i C 1 g i C. (3.1) F (τ) = C 1 g i C τ(g i, Ĉ 1 g i )+τ Ĉ 1 g i C. It is easy to verify that F (τ) is an increasing function for and is a decreasing function for τ (g i, Ĉ 1 g i ), Ĉ 1 g i C τ (g i, Ĉ 1 g i ). Ĉ 1 g i C Using this property and (3.11), we know (3.13) where F ( τ i ) max { F ( τ0i 1+δ 0 τ 0i = (g i, Ĉ 1 g i ). Ĉ 1 g i C ) ( )} τ0i,f 1 δ 0 On the other hand, it follows from (3.1) that ( ) τ0i F = C 1 g i C (1 + δ 0) 1 (g i, Ĉ 1 g i ) 1+δ 0 (1 + δ 0 ) { Ĉ 1 g i C } = 1 (1 + δ 0) 1 (g i, Ĉ 1 g i ) (1 + δ 0 ) C 1 g Ĉ 1 g i C C 1 g i i C. C

15 Two new variants of nonlinear inexact Uzawa algorithms 347 Let λ min and λ max denote respectively the minimal and maximal eigenvalues of the matrix Ĉ 1 CĈ 1. Then using the well-known matrix-eigenvalue inequality we obtain (g i, Ĉ 1 g i ) Ĉ 1 g i C C 1 g i C = (Ĉ 1 g i, Ĉ 1 g i ) (Ĉ 1 CĈ 1 (Ĉ 1 g i ), Ĉ 1 g i )((Ĉ 1 CĈ 1 ) 1 (Ĉ 1 g i ), Ĉ 1 g i ) 4λ min λ max (λ min + λ max ) = 4κ (1 + κ). Hence ( ) { τ0i F 1 (1 + δ } 0) 1 4κ (3.14) 1+δ 0 (1 + δ 0 ) (1 + κ) C 1 g i C. Similarly we can show (note that δ 0 < 1 ) ( ) { τ0i F 1 (1 δ } 0) 1 4κ (3.15) 1 δ 0 (1 δ 0 ) (1 + κ) C 1 g i C. Now by the direct calculation, we have namely, (1 + δ 0 ) 1 (1 + δ 0 ) (1 δ 0) 1 (1 δ 0 ) ( 1 = 1 )( ) 1 δ 0 1+δ 0 1 δ 0 1+δ 0 ( 1 = 1 ) δ 0 1 δ 0 1+δ 0 1 δ0 > 0, (1 δ 0 ) 1 (1 δ 0 ) < (1 + δ 0) 1 (1 + δ 0 ), which, together with (3.13)-(3.15), yields { F ( τ i ) 1 (1 δ } 0) 1 4κ (1 δ 0 ) (1 + κ) C 1 g i C = β C 1 g i C. Summarizing the above, we have proved τ i Ĉ 1 g i C 1 g i C β C 1 g i C.

16 348 Q. Hu, J. Zou It is clear that β<1when δ 0 < 1. Hence by Lemma 9 of [3] we know that there is a symmetric and positive definite m m matrix ˆQ Bi such that ˆQ 1 Bi g i = τ i Ĉ 1 g i and I ˆQ 1 Bi C ˆQ 1 Bi β. Now Q Bi =ˆQ Bi is the desired matrix, since the matrix Q 1 BiC has the same eigenvalues as the matrix Q 1 1 Bi CQ Bi. 4 New algorithm II: the nonlinear inexact Uzawa combined with the conjugate gradient method In the last section, we proposed a nonlinear inexact Uzawa combined with the steepest descent method (Algorithm 3.1), which converges when a general preconditioner is used for the Schur complement system and a general nonlinear iteration is used for solving Ax = b involved in the inner iteration. However, the steepest descent method converges with a reasonable rate only when a good preconditioner is available for the Schur complement system. This is the case when the saddle-point problem arises, for example, from the Stokes problem, mixed finite element formulations of second order elliptic problems [], and domain decomposition methods with Lagrange multipliers [17]. Without such a good preconditioner the method may converge with a slow rate. This fact can be observed from the numerical experiments shown in Sect. 5. In this section, we propose another new algorithm which combines the nonlinear inexact Uzawa algorithm with the conjugate gradient (CG) method, in an effort to accelerate the nonlinear inexact Uzawa algorithm when a good preconditioner is not available for the Schur complement system. This is the case when the saddle-point problems arise from the Lagrange multiplier formulations for optimization problems [16] and the parameter identification [1] [19]. In the following, we will first derive a new nonlinear inexact Uzawa algorithm and combine it with the CG method, then analyse the convergence of the new algorithm and compare its convergence rate with that of Algorithm 3.1, which uses the steepest descent method. For a given sequence {f i } R n, let {Q Ai } be the sequence of symmetric and positive definite n n matrices satisfying (.16) and (.17), and set H i = B t Q 1 Ai B. Clearly, H i is also a symmetric and positive definite m m

17 Two new variants of nonlinear inexact Uzawa algorithms 349 matrix. For a given sequence {g i } R m, consider the linear system (with the index i fixed) (4.1) H i z i = g i. We intend to solve (4.1) for each fixed index i by the PCG iteration with the preconditioner Ĉ. This process is to find a sequence of approximate solutions {ẑ j i } as follows: initialize : ẑi 0 =0, ˆr0 i = g i, ˆp 0 i = Ĉ 1 g i, ĥ 0 i =0; iterate for j =1,, : ẑ j i =ẑj 1 i +ˆτ j i ˆpj 1 i, ˆτ j i = (ˆrj 1 i,ˆp j 1 i ) ĥ j i = ĥj 1 i +ˆτ j i H i ˆp j 1 i, ˆr j i = g i ĥj i, ˆp j i = Ĉ 1ˆr j i ˆθ j i ˆpj 1 i, ˆθj, (H i ˆp j 1 i,ˆp j 1 i ) i = (Ĉ 1ˆr j i,h i ˆp j 1 i ). (H i ˆp j 1 i,ˆp j 1 i ) Note that one has the relation ĥj i = H i ẑ j i between the variables ĥj i and ẑj i above, and each action of H i involves the action of Q 1 Ai. In general it is difficult to obtain the explicit form of the matrix Q Ai. Thus, it is important and practical to use the inexact solver Ψ(B ˆp j 1 i ) to replace Q 1 Ai B ˆpj 1 i. The resulting iteration can be formulated as follows: initialize : zi 0 =0, r0 i = g i, p 0 i = Ĉ 1 g i, h 0 i =0; iterate for j =1,, : z j i = zj 1 i + τ j i pj 1 i, τ j i = (r j 1 i,p j 1 i ) + τ j i Bt Ψ(Bp j 1 i ), h j i = hj 1 i r j i = g i h j i, p j i = Ĉ 1 r j i θj i pj 1 i, (Ψ(Bp j 1 i ),Bp j 1 i ), θ j i = (BĈ 1 r j i,ψ(bpj 1 i )). (Ψ(Bp j 1 i ),Bp j 1 i ) For convenience, we use Φ k (g i ) to denote the kth iterate zi k generated from the above process. Φ k defines a (nonlinear) mapping from R m to R m, and Φ k (g i ) can be regarded as an approximate solution of the system (4.1). Note that we may not have Ψ(Bb) =Q 1 Ai Bbfor a vector b Rm in general, so Φ k (g i ) is not the true approximation generated by the k steps of PCG iteration for (4.1). We assume that there is a positive number β k < 1 such that (4.) Φ k (g i ) z i Hi = Φ k (g i ) Hi 1 g i Hi β k Hi 1 g i Hi.

18 350 Q. Hu, J. Zou Then by Lemma 9 of [3], there is a symmetric and positive definite m m matrix ˆQ 1 ki, such that ˆQ ki g i = Φ k (g i ) and I ˆQ 1 ki H i ˆQ 1 ki β k, which implies 1 (4.3) (H i w, w) ( 1+β ˆQ ki w, w) 1 (H i w, w) w R m. k 1 β k On the other hand, it follows from (.17) that (1 δ)(a 1 w, w) (Q 1 Ai w, w) (1 + δ)(a 1 w, w) w R m. This, together with (4.3), leads to 1 δ (4.4) (Cw,w) ( 1+β ˆQ ki w, w) 1+δ (Cw,w). k 1 β k Define Q ki = 1 δ ˆQ ki. It is easy to see that (4.5) and (4.6) Q 1 ki g i = 1 δ Φ k(g i ), (1 δ)(1 β k ) (Q ki w, w) (Cw,w) (1 + δ) (Q ki w, w) w R m by using (4.4) and the fact that β k < 1. Based on (4.5) and Algorithm.1, we propose the following new algorithm: Algorithm 4.1 (Nonlinear inexact Uzawa PCG method). Given x 0 R n and y 0 R m, the sequence {(x i,y i )} R n R m is defined for i =1,,, by Step 1. Compute f i = f (Ax i + By i ) and Ψ(f i ), update (4.7) x i+1 = x i + Q 1 Ai f i = x i + Ψ(f i ). Step. Compute g i = B t x i+1 g, and d ki = Φ k (g i ), update (4.8) y i+1 = y i + Q 1 ki g i = y i + 1 δ d ki. The inequality (4.6) indicates that the matrix Q ki satisfies the inequality (.5) with γ k = 1+3δ+β k(1 δ) (1+δ). Thus by Theorem. we obtain

19 Two new variants of nonlinear inexact Uzawa algorithms 351 Theorem 4.1 Assume that (.) holds for δ< 1 3, and (4.) holds for β k < 1. Then Algorithm 4.1 converges. Moreover, we have (4.9) e x i A ( 1+4δ +ˆρ k )(ˆρ k ) i 1 E 0, i =1,, and (4.10) e y i C (ˆρ k ) i E 0, i =1,,, and the convergence rate ˆρ k (< 1) can be estimated by (4.11) ˆρ k { δ + δ + δ for 0 β k 5δ 1 1 δ, (1 β 5δ 1 k)(1 δ) for 1 δ <β k < 1. Remark 4.1 One can easily see from (4.5)-(4.6) that Theorem 4.1 also holds if the factor (1 δ)/ in (4.8) is replaced by any number α (0, 1/3). In fact, the numerical results in Sect. 5 indicate that Algorithm 4.1 still converges when the factor (1 δ)/ is replaced by 1. So Algorithm 3.1 is a special case of Algorithm 4.1 with k =1. As noted in Remark 3.3, the first assumption of Theorem 4.1 holds when the mapping Ψ is properly defined. However, it is difficult to verify the condition (4.) in general. The following proposition gives a condition which guarantees (4.) when all H i s are taken to be the same: Proposition 4.1 Assume that there is a symmetric and positive definite n n matrix Q A, such that Q 1 A φ = Ψ(φ) for any φ Rn. Define H = B t Q 1 A B. Then the following inequality holds for some β k < 1 : (4.1) Φ k (g i ) H 1 g i H β k H 1 g i H. Proof. Under the assumption of Proposition 4.1, Φ k (g i ) is just the approximate solution generated by the k steps of PCG iteration for the linear system (4.13) Hz i = g i. Thus, the parameter β k satisfying the relation (4.1) can be estimated by (cf. []) { β k = min ( κ 1 κ +1 ) k }, κ 1 κ < 1 +1 with κ =cond(ĉ 1 H).

20 35 Q. Hu, J. Zou Remark 4. The assumption in Proposition 4.1 is satisfied in some applications. For example, it is true when the mapping Ψ is defined by the preconditioned Richardson iteration, which includes the standard multigrid sweeps. To see this, for any b R n, recall the preconditioned Richardson iteration for solving the system (4.14) Ax= b with the preconditioner  is x i+1 = x i +  1 (b Ax), i =1,,. If the eigenvalues of  1 A is in the range 0 <λ<, the iteration converges. Taking x 0 =0, we can easily see x i+1 = x (I  1 A) i+1 x i+1 = ( 1) j 1 C j i+1 ( 1 A) j x j=1 i+1 = ( 1) j 1 C j i+1 ( 1 A) j 1  1 b. j=1 This suggests us to define Q 1 A i+1 Q 1 A = as ( 1) j 1 C j i+1 ( 1 A) j 1  1 = j=1 ( I (I  1 A) i+1) A 1. We can easily verify that the matrix Q A is symmetric and positive definite, and satisfies Q 1 A b = x i+1 = Ψ(b) b R n. Note that if A is the stiffness matrix which arises from the finite element discretization of self-adjoint second order elliptic problems, then one can choose  1 to be the operator corresponding to one V-cycle sweep of multigrid method [5] or the hierarchical basis multigrid method [3]. In the following we make some comparison between the convergence rates of Algorithm 3.1 and Algorithm 4.1. Proposition 4. Assume that the parameters δ, κ and β k in (.), (3.6) and (4.) satisfy δ< 1 5, κ = cond(ĉ 1 C) >, lim k β k =0

21 Two new variants of nonlinear inexact Uzawa algorithms 353 respectively. Then Algorithm 4.1 converges faster than Algorithm 3.1 when k is sufficiently large. But this may not be true for general k. Proof. We first show that the parameter β in (3.6) is larger than 1/3 when κ>. It is easy to see that and so β 1 δ 0 (1 δ 0 ) 1, 1 4κ (1 + κ) = κ 1 κ +1 > 1 3 when κ>. Now let ˆρ and ˆρ k be the convergence rates of Algorithm 3.1 and Algorithm 4.1 (see Theorems 3.1 and 4.1). Since β> 1 3 and δ<1 5, it is easy to see 4δ 1+δ < 3 < 1+β Then by (3.9) and (4.11) we have and < 1 and 5δ 1 1 δ < 0 β k. ˆρ =1 1 3 (1 β)(1 + δ) > 1 (1 β) 4 10 ˆρ k =1 1 4 (1 β k)(1 δ) < (1 β k). Therefore one has ˆρ k < ˆρ if 1 5 (1 β k) > 3 (1 β), 10 that is, (4.15) β k < 3 β 1. It is clear that when β> 1 3 the inequality (4.15) can be satisfied for large k. It remains to show the second part of Proposition 4.. Form Theorem.1 and the proof of Lemma 3.1, we know that the convergence rate of Algorithm 4.1 is determined by the parameter ˆβ k satisfying Φ k (g i ) C 1 g i C ˆβ k C 1 g i C, g i R m,

22 354 Q. Hu, J. Zou more accurately, it is determined by the value Φ k (g i ) C 1 g i C, i.e., the error at the k-th iteration. Since the results generated by Algorithm 4.1 with k =1are the same as the ones generated by Algorithm 3.1, for our purpose it suffices to construct a counterexample to show that the following inequality does not hold: (4.16) Φ (g i ) C 1 g i C < Φ 1 (g i ) C 1 g i C. In fact, we can show that (4.16) is not true when definition of Algorithm 4.1, we have Ĉ = C = I. Bythe Φ (g i )=z i = z 1 i + τ i p 1 i, thus Φ (g i ) C 1 g i C = z i g i (4.17) = z 1 i g i +τ i (z 1 i g i,p 1 i )+(τ i ) p 1 i. By the basic property of the CG method, we know (ri 1,p0 i )=0, which implies (r 1 i,p 1 i )=(r 1 i,r 1 i θ 1 i p 0 i ) > 0, and so τi > 0 by the definition of τi. Now the desired conclusion follows if we can find a matrix H such that (zi 1 g i,p 1 i ) > 0. Indeed, if this is true, then it follows from (4.17) that Φ (g i ) C 1 g i C > z 1 i g i = Φ 1 (g i ) C 1 g i C, which negates (4.16). It remains to find such a matrix H. By the definition of Algorithm 4.1, z 1 i = τ 1 i g i and Therefore we have p 1 i = r 1 i θ 1 i p 0 i = g i τ 1 i Hg i θ 1 i g i. (4.18) (zi 1 g i,p 1 i )=(τi 1 1)(g i, (1 θi 1 )g i τi 1 Hg i ) { } =(τi 1 1) (1 θi 1 ) g i τi 1 g i H = θi 1 (1 τi 1 ) g i, where we have used the fact that τi 1, it is easy to verify that θ 1 i = g i / g i H. By the definitions of (4.19) θi 1 (1 τi 1 )= ( g i Hg i g i 4 H )( g i g i H ) g i 6. H

23 Two new variants of nonlinear inexact Uzawa algorithms 355 By the Cauchy-Schwarz inequality, g i Hg i g i 4 H > 0 provided that H ci (c is a constant). Without loss of generality, we assume that the first component of g i is not zero, then define It is clear that H = diag( ). 4 g i g i H > 0, so we have from (4.19) that θi 1(1 τ i 1 ) > 0. This, together with (4.18), leads to (z 1 i g i,p 1 i ) > 0. 5 Numerical experiments In this section, we apply our newly proposed Algorithm 3.1 and Algorithm 4.1, and some other existing algorithms, to solve the two-dimensional Stokes problem and another system of purely algebraic equations. Let Ω be the unit square domain in R, and L 0 (Ω) be the set of all square integrable functions over Ω with zero averages. Let H 1 (Ω) be the usual Sobolev space of order one on Ω and H0 1(Ω) be a space consisting of those functions in H1 (Ω) whose traces vanish on the boundary of Ω. Our first example is the generalized Stokes problem whose variational formulation reads as follows: { (µ u, v) (p, v) =(f,v), v H 1 0 (Ω), (5.1) (q, u) =g, q L 0 (Ω), where f L (Ω) and µ L (Ω) with µ(x) c >0 almost everywhere in Ω. We use one of the well-known conforming Taylor-Hood elements, which has been widely used in engineering, to solve the system (5.1). For any positive integer N, we divide Ω into N N subsquares, and set h =1/N. Such a triangulation of Ω, with its elements being all squares, is denoted as T h. Let X h H0 1(Ω) and M h H 1 (Ω) L 0 (Ω) be the usual continuous Q and Q 1 finite element spaces defined on T h respectively (cf. [7,13]). The total number of unknowns for this finite element is n + m = (N

24 356 Q. Hu, J. Zou Table 1. Number of iterations (Algorithm 3.1) δ=0.1 δ=0. N PCG CG ) +[(N +1) 1]. The finite element approximation of the above Stokes system can be formulated as follows: (5.) (µ(x) u h, v) (p h, v) =(f,v), v Xh, (5.3) (q, u h )=0, q M h. It is known that the inf-sup condition is satisfied by the pair (Xh,M h) (see [7]), thus the Schur complement matrix B t A 1 B associated with the system (5.)-(5.3) has a condition number independent of h. As in [6] and [], we take the variable coefficient µ to be µ =1+x 1 x +x 1 x /, and know that the corresponding matrix A is block diagonal with two copies of the stiffness matrix associated with the bilinear form (µ(x), ) on the diagonal. Therefore, if µ =1, it can be solved by the fast Poisson solver. It is natural to choose this fast solver  as the preconditioner of A. It is clear that (5.4) 0.5(Âz, z) (Az, z).5(âz, z), thus the matrix  1 A is well-conditioned, and so is the matrix B t  1 B. In this case, we can choose for Algorithm 3.1 Ĉ = I and Ψ defined by the CG method (without preconditioning) or the PCG method with the preconditioner  for A. For all the numerical results shown below, the inner nonlinear iteration is terminated when Ψ satisfies φ AΨ(φ) / φ δ for a given vector φ R m. And the outer iteration of Algorithm 3.1 terminates when (5.5) ε = Mv i b / Mv 0 b with v i =(x i y i ) t,v 0 =(x 0 y 0 ) t and b =(fg) t. The number of iterations of Algorithm 3.1 is listed in Table 1, where we can see the rate of convergence independent of mesh size h. In comparison with Algorithm 3.1, we tried to solve the Stokes system also by Algorithm 4.1 with the same preconditioners as above. The number of iterations for different k is listed in Table 1.

25 Two new variants of nonlinear inexact Uzawa algorithms 357 Table 1. Number of iterations (Algorithm 4.1) PCG inner iteration, δ =0.1 N k= k=5 k=10 k= We can see from Table 1 that Algorithm 3.1 outperforms Algorithm 4.1 in this example, and Algorithm 3.1 shows a more stable and consistent convergence history than Algorithm 4.1. The results are reasonable as Algorithm 3.1 should perform better than Algorithm 4.1 when a good preconditioner Ĉ is available for the Schur complement C. Indeed, the preconditioner h Ĉ = h I is spectrally equivalent to C (and the two algorithms are invariant under constant scalings). It can be verified by (5.4) that Ĉ =Isatisfies the assumption (.5) (cf. [6]). Thus we can also apply Algorithm.1 to solve the system (5.)- (5.3). However, the resulting convergence is very slow for this example because the parameter γ defined by (.5) is close to one. For example, 19 iterations are needed to reach the same tolerance as (5.5) when N =8and Ψ is defined by the CG iteration with δ =0.1. We can also use the diagonal preconditioner with diagonal block  and Ĉ = h I (spectrally equivalent to C) to solve the system (5.)-(5.3) by the minimum residual method. Unfortunately, its convergence seems slow. For example, 65 iterations are needed to reach the same tolerance when N =16. Our second example is a system of purely algebraic equations. We define the matrices A =(a ij ) n n and B =(b ij ) n m (n m) in (1.1) as follows: a ij = { i +1,i= j, 1, i j =1, 0, otherwise; b ij = { 15j, i = j + n m, 0, otherwise. The preconditioners  =(a ij) n n and Ĉ =(c ij) n m are defined by { { i, i = j, i a ij = c 0, otherwise; ij = +3, i = j, 0, otherwise. The right-hand side vectors f and g are defined such that the exact solutions x and y are the vectors with all components being 1. We solve the corresponding system (1.1) by Algorithm 3.1 with the preconditioner Ĉ and Ψ defined by N c iterations of the CG method (without preconditioner) and the PCG method with the preconditioner Â. Algorithm 3.1 terminates when the error ε , the resulting number of iterations is given in Table. To compare Algorithm 3.1 with Algorithm 4.1 for this example, we solved the considered system again by Alg The numerical experiments

26 358 Q. Hu, J. Zou Table. Numbers of iteration (Algorithm 3.1) n = 00,m= 150 n = 400,m= 300 n = 800,m= 600 PCG(N c =) CG(N c =6) Table 3. Number of iterations PCG(N c=) inner iteration, Ĉ = I (n,m) Algorithm 3.1 Algorithm 4.1 k=1 k= k=5 k=10 k=0 (00,150) (400,300) (800,600) Table 4. Number of iterations CG(N c =6) inner iteration, Ĉ = I (n,m) Algorithm 3.1 Algorithm 4.1 k=1 k= k=5 k=10 k=0 (00,150) (400,300) (800,600) indicate that both algorithms have almost the same number of iterations with 1 δ the preconditioner Ĉ and Ψ defined as above, even when the factor in (4.8) is replaced by 1. So the detailed number of iterations is omitted here. To further compare the performance of both algorithms in the case with a very bad preconditioner Ĉ, we take Ĉ = I, the worst preconditioner for C, namely without any preconditioning. Tables 3 and 4 below listed the number of iterations in this case. Tables 3 and 4 above indicate that Algorithm 4.1 is more efficient in the case with a very bad preconditioner Ĉ for the Schur complement C. We also tried to use Algorithm.1 to solve the considered system, but the numerical experiments indicate that this algorithm is divergent for this example. In fact, the assumption (.5) is not satisfied with the given preconditioner Ĉ. Also note that it is not easy to achieve a good estimate on the minimal generalized eigenvalue of the preconditioner Ĉ with respect to the matrix B t A 1 B in this case (except that the action of A 1 is allowed), then there seems no direct way to solve the considered system using Algorithm.1. Acknowledgements. The authors wish to thank the anonymous referees for many constructive comments which lead to the great improvement of the paper.

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