Inexact Low-Rank Newton-ADI Method for Large-Scale Algebraic Riccati Equations

Transcription

1 Max Panck Institute Magdeburg Preprints Peter Benner 1 Matthias Heinkenschoss 2 Jens Saak 1 Heiko K. Weichet 1 Inexact Low-Rank Newton-ADI Method for Large-Scae Agebraic Riccati Equations MAX PLANCK INSTITUT FÜR DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG MPIMD/15-06 September 9, 2015

2 Affiiations: 1 Research Group Computationa Methods in Systems and Contro Theory (CSC), Max Panck Institute for Dynamics of Compex Technica Systems Magdeburg, Sandtorstr. 1, Magdeburg, Germany ({benner,saak,weichet}@mpi-magdeburg.mpg.de) 2 Department of Computationa and Appied Mathematics (CAAM), Rice University, MS-134, 6100 Main Street, Houston, TX , USA (heinken@rice.edu) The research of this author was supported in part by grants AFOSR FA and NSF DMS Corresponding author: Heiko K. Weichet 1 weichet@mpi-magdeburg.mpg.de Imprint: Max Panck Institute for Dynamics of Compex Technica Systems, Magdeburg Pubisher: Max Panck Institute for Dynamics of Compex Technica Systems Address: Max Panck Institute for Dynamics of Compex Technica Systems Sandtorstr Magdeburg

3 Abstract This paper improves the inexact Keinman-Newton method for soving agebraic Riccati equations by incorporating a ine search and by systematicay integrating the ow-rank structure resuting from ADI methods for the approximate soution of the Lyapunov equation that needs to be soved to compute the Keinman-Newton step. A convergence resut is presented that taiors the convergence proof for genera inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the Lyapunov equation residua, which in genera do not hod for ow-rank approaches. In the convergence proof of this paper, the ine search is needed to ensure that the Riccati residuas decrease monotonicay in norm. In the numerica experiments, the ine search can ead to substantia reduction in the overa number of ADI iterations and, therefore, overa computationa cost. 1 Introduction We present improvements of the inexact Keinman Newton method for the soution of arge-scae continuous-time agebraic Riccati equations (CARE) R(X) = C T C + A T X + XA XBB T X = 0 (1.1) with C R p n, A R n n, X = X T R n n, B R n r, and p + r n. The agorithmic improvements consist of incorporating a ine search and of systematicay integrating the ow-rank structure resuting from ADI methods for the soution of the Lyapunov equation where (A (k) ) T X (k+1) + X (k+1) A (k) = C T C X (k) BB T X (k), (1.2) A (k) = A BB T X (k), which has to be approximatey soved in the k-th iteration. The paper is motivated by the recent work of Feitzinger et a. [9] who propose and anayze inexact Keinman Newton methods without ine search, by Benner and Byers [2] who incorporate ine search into the exact Keinman Newton method, and by the recent work of Benner et a. [3, 5] on agorithmic improvements of ow-rank ADI methods. The convergence resut in [9] makes positive semi-definiteness assumptions on the difference between certain matrices and the residua of the Lyapunov equation that are in genera not vaid when the Lyapunov equation is soved with ow-rank methods ike, e.g., the owrank ADI iteration [7]. Our convergence resut foows the theory of genera inexact Newton methods, but uses the structure of Riccati equations. We add the inexact soution of the Lyapunov equation to [2] and incorporate the ow-rank structure. Our convergence proof makes use of the fact that the Riccati residuas decrease monotonicay in norm, which is ensured by the ine search. There is no proof that the inexact Keinman Newton, ow-rank ADI iteration converges gobay without ine search. On test exampes resuting from the finite eement approximation of LQR 1

4 probems governed by an advection diffusion equation, the incorporation of a ine search into the inexact Keinman Newton, ow-rank ADI iteration can ead to substantia reduction in the overa number of ADI iterations and, therefore, overa computationa cost. The paper is organized as foows. In the next section, we reca a basic existence and uniqueness resut for the unique symmetric positive semi-definite stabiizing soution of the CARE (1.1). Section 3 introduces the inexact Keinman Newton method with ine search and presents the basic convergence resut. The basic ingredients of ADI methods that are needed for this paper are reviewed in Section 4. Section 5 discusses the efficient computation of various quantities ike the Newton residua using the ow-rank structure. As a resut, the computationa cost of our overa agorithm is proportiona to the tota number of ADI iterations used; in comparison the cost of other components, such as execution of the ine search, are negigibe. Finay, we demonstrate the contributions of the various improvements on the overa performance gains in Section 6. As mentioned before, in our numerica tests, our improved inexact Keinman Newton method is seven to tweve times faster than the exact Keinman Newton method without ine search. Notation. Throughout the paper we consider the Hibert space of matrices in R n n endowed with the inner product M, N = tr ( M T N ) = n i,j=1 M ijn ij and the corresponding (Frobenius) norm M F = ( M, M ) 1/2 = ( n i,j=1 M ij 2 )1/2. Furthermore, given rea symmetric matrices M, N, we write M N if and ony if M N is positive semi-definite, and M N if and ony if M N is positive definite. The spectrum of a symmetric matrix M is denoted by σ(m). 2 The Riccati Equation We reca an existence and uniqueness resut for the continuous-time Riccati equation (1.1). Definition 1 Let A R n n, B R n r, and C R p n. The pair (A, B) is caed stabiizabe if there exists a feedback matrix K R n r such that A BK T is stabe, which means that A BK T has ony eigenvaues in the open eft haf compex pane C. The pair (C, A) is caed detectabe if (A T, C T ) is stabiizabe. Notice that (A, B) is stabiizabe if and ony if (A, BB T ) is stabiizabe and (C, A) is detectabe if and ony if (C T C, A) is detectabe. Furthermore, we aways use the word stabe as defined in [15], whereas, in other iterature, this is usuay caed asymptoticay stabe. Since, as in [15], asymptoticay stabe is the required property in a our appications we do not need to distinguish between stabe and asymptoticay stabe and, therefore, simpy use stabe everywhere. Assumption 2 The matrices A R n n, B R n r, and C R p n are given such that (A, B) is stabiizabe and (C, A) is detectabe. 2

5 If Assumption 2 hods, there exists a unique symmetric positive semi-definite soution X ( ) of the CARE (1.1) which is aso the unique stabiizing soution. This foows from Theorems and (see aso p. 244) in [15]. Furthermore, it can be shown that a symmetric positive semi-definite soutions of the CARE (1.1) are stabiizing. Theorem 3 If Assumption 2 hods, every symmetric soution X ( ) 0 of the CARE (1.1) is stabiizing. Proof. Let X = X T 0 sove the CARE (1.1). We show that A BB T X is stabe by contradiction. Assume that µ is an eigenvaue of A BB T X with Re (µ) 0 and et v C n \{0} be a corresponding eigenvector. The CARE (1.1) can be written as (A BB T X) T X + X(A BB T X) = C T C XBB T X. (2.1) Mutipy (2.1) with v H from the eft and v from the right. The eft-hand side of (2.1) yieds whereas the right-hand side of (2.1) yieds 2 Re (µ) v H Xv 0, since X = X T 0, v H C T Cv v H XBB T Xv 0, since C T C 0 and XBB T X 0. Hence, eft- and right-hand sides of (2.1) mutipied by v H from the eft and v from the right are equa to zero, that is v H Xv = 0 and v H C T Cv +v H XBB T Xv = 0, which yieds Cv = 0 and B T Xv = 0. Since (A BB T X)v = µv, it foows that v is an eigenvector of A with eigenvaue µ and Re (µ) 0. The Hautus-Popov Test for Detectabiity, e.g., [12, Sec. 80.3], states that (C, A) is detectabe if and ony if Ax = λx, x 0 and Re (λ) 0 impies Cx 0. Thus, the existence of v 0 and Re (µ) 0 with Av = µv contradicts the detectabiity of (C, A) by the Hautus-Popov test. 3 The Inexact Keinman Newton Method with Line Search This section introduces the inexact Keinman Newton method with ine search and gives a convergence resut. The fundamenta ideas are identica to what is we known for inexact Newton methods, see, e.g., Keey [13, Sec. 8.2], but are taiored to the structure of the Riccati equations. The presentation of the basic agorithm in Section 3.1 combines ideas from genera inexact Newton methods, from Keinman Newton with inexactness, see, e.g., Feitzinger et a. [9], and Keinman Newton with ine search, see, 3

6 e.g., Benner and Byers [2]. In Section 3.3, we wi show that the assumptions made in Feitzinger et a. [9], to ensure convergence of the inexact Keinman Newton method, are in genera not vaid if ow-rank Lyapunov sovers are used to compute the inexact Keinman Newton step, and we wi present an aternative convergence resut that foows more cosey that of genera inexact Newton methods. 3.1 Derivation of the Method We want to compute the symmetric, positive semi-definite, stabiizing soution X ( ) of the CARE (1.1). The operator R : R n n R n n defined in (1.1) is twice Fréchet differentiabe with derivatives given by R (X)N = (A BB T X) T N + N(A BB T X), R (X)(N 1, N 2 ) = N 1 BB T N 2 N 2 BB T N 1. (3.1a) (3.1b) Since R is quadratic in X, the 2nd order Fréchet derivative is independent of X and R(Y ) can be expressed via a Tayor series as R(Y ) = R(X) + R (X)(Y X) R (X)(Y X, Y X). (3.2) The CARE (1.1) can be soved using Newton s method, which in this context is referred to as the Keinman Newton method [14]. Given an approximate symmetric soution X (k) of (1.1), the new Keinman Newton iterate is given by R (X (k) )X (k+1) = R (X (k) )X (k) R(X (k) ). (3.3) Equation (3.3) is the Lyapunov equation (1.2). Instead of soving (3.3) for the new iterate, one coud sove R (X (k) )S (k) = R(X (k) ) for the step S (k) = X (k+1) X (k). Whie the atter equation may be favorabe in cases where the Lyapunov equation is soved using direct methods (see, e.g., [2, p. 101]), (3.3) is favorabe when the Lyapunov equation is soved inexacty using iterative methods. The right hand side in (3.3) is X (k) BB T X (k) C T C = GG T, where G = [C T X (k) B] R n (p+r). As we wi see ater, this ow-rank factorization (p + r n) is important when the Keinman Newton method is combined with ow-rank approximation methods. Expressions of R(X (k) ) which ead themseves to the appication of ow-rank approximation methods, and which are equa to R(X (k) ) in the exact Keinman Newton method, fai when used in an inexact Keinman Newton method as shown in Feitzinger et a. [9]. If Assumption 2 hods, then the specia structure of R aows one to prove goba convergence of the Keinman Newton method: If the initia iterate X (0) is symmetric and stabiizing, then the Keinman Newton method is we defined (i.e., (1.2) has a unique soution), the iterates generated by the Keinman Newton method converge with a q-quadratic convergence rate, and satisfy X (1) X (2)... X ( ) 0; see, e.g., Keinman [14] or Lancaster and Rodman [15, Sec. 9.2]. Even though the Keinman Newton method exhibits goba convergence, it was observed by Benner and Byers [2] that a ine search improves its efficiency. Especiay 4

7 in the first iteration of the Keinman Newton method, the residua may increase dramaticay if no ine search is used. For arge scae probems, the Newton equation (the Lyapunov equation) (3.3) is soved iterativey, and the residua error in the Lyapunov equation has to be controed appropriatey to ensure convergence. We integrate the inexact soution of (3.3) and a ine search into the Keinman Newton method. As we have mentioned before, the fundamenta ideas are identica to what is we known for inexact Newton methods, see, e.g., Keey [13, Sec. 8.2]. Given a symmetric X (k) R n n and α > 0, η k (0, 1), we compute a symmetric step S (k) R n n with R (X (k) )S (k) + R(X (k) R(X ) η (k) k ) (3.4) F F and then compute the next iterate X (k+1) = X (k) + λ k S (k), where the step size λ k > 0 is such that the sufficient decrease condition ( R X (k) + λ k S (k)) ( F (1 λ k α) R X (k)) F (3.5) is satisfied and the step size λ k is not unnecessariy sma. If we define the Newton step residua R (X (k) )S (k) + R(X (k) ) = L (k+1), (3.6) then (3.4) reads L (k+1) F R(X η (k) k ). (3.7) F Using the definition (1.1), (3.1a), and X (k+1) = X (k) + S (k), the equation (3.6) is equivaent to (A (k) ) T X(k+1) + X (k+1) A (k) = X (k) BB T X (k) C T C + L (k+1) (3.8a) and the new iterate is X (k+1) = (1 λ k )X (k) + λ k X(k+1). (3.8b) The Riccati residua at X (k+1) = X (k) + λ k S (k) can be expressed using (3.2) and (3.6) as R(X (k) + λ k S (k) ) = R(X (k) ) + λ k R (X (k) )S (k) + λ2 k 2 R (X (k) )(S (k), S (k) ) = (1 λ k )R(X (k) ) + λ k L (k+1) λ 2 ks (k) BB T S (k). (3.9) 5

8 Agorithm 1 Inexact Keinman-Newton Method with Line Search Input: A, B, C, stabiizing initia iterate X (0), to Newton > 0, η (0, 1) and α (0, 1 η). Output: Approximate soution of the CARE (1.1). 1: for k = 0, 1,... do 2: if R(X (k) ) to Newton then 3: Return X (k) as an approximate soution of the CARE (1.1). 4: end if 5: Set A (k) = ( A BB T X (k)), G = [ C T X (k) B ]. 6: Seect η k (0, η]. 7: Compute an approximate soution X (k+1) of the Lyapunov equation such that (A (k) ) T X(k+1) + X (k+1) A (k) = GG T + L (k+1) and L (k+1) F η k R(X (k) ) F. 8: Set S (k) = X (k+1) X (k). 9: Compute λ k > 0 such that R(X (k) + λ k S (k) ) F (1 λ k α) R(X (k) ) F. 10: Set X (k+1) = X (k) + λ k S (k). 11: end for Therefore, if η k η < 1 and α (0, 1 η), then (3.7) and (3.9) impy for a λ with R(X (k) + λs (k) ) F (1 λ) R(X (k) ) F + λ L (k+1) F + λ 2 S (k) BB T S (k) F (1 λ + λ η) R(X (k) ) F + λ 2 S(k) BB T S (k) F R(X (k) ) F (1 αλ) R(X (k) ) F R(X (k) ) F R(X (k) ) F 0 < λ (1 α η). (3.10) S (k) BB T S (k) F In particuar, the sufficient decrease condition (3.5) is satisfied for a λ k with (3.10). In the actua computation of the step size λ k we use (3.9) which impies that f(λ) = R(X (k) + λs (k) ) 2 F (3.11) = (1 λ) 2 α (k) + λ 2 β (k) + λ 4 δ (k) + 2λ(1 λ)γ (k) 2λ 2 (1 λ)ε (k) 2λ 3 ζ (k) is a quartic poynomia with α (k) = R(X (k) ) 2 F, δ (k) = S (k) BB T S (k) 2 F, β (k) = L (k+1) 2 F, ε (k) = R(X (k) ), S (k) BB T S (k), γ (k) = R(X (k) ), L (k+1), ζ (k) = L (k+1), S (k) BB T S (k). (3.12) 6

9 The derivative is f (λ) = R(X (k) + λs (k) ), R(X (k) ) + L (k+1) 2λS (k) BB T S (k). In particuar, using the Cauchy Schwarz inequaity and (3.7), we find f (0) < 0, which again confirms that S (k) is a descent direction. Remark 4 If the current iterate X (k) is symmetric positive semi-definite, if the soution X (k+1) of (3.8a) is symmetric positive semi-definite, and if λ k (0, 1], then X (k+1) = X (k) + λ k ( X (k+1) X (k) ) is aso symmetric positive semi-definite. The basic inexact Keinman Newton method with ine search is summarized in Agorithm Line Search There are many possibiities to compute a step size λ k that satisfies the sufficient decrease condition (3.5). We review two. In both cases, the representation (3.11) of the Riccati residua as a quartic poynomia can be used for the efficient impementation of the respective procedure Armijo Rue Given β (0, 1), the Armijo rue in its simpest form seects λ k = β, where is the smaest integer such that the sufficient decrease condition (3.5) is satisfied. See Keey [13, Sec. 8.2] for more detais. Since the sufficient decrease condition (3.5) is satisfied for a step sizes with (3.10) and is the smaest integer such that λ k = β satisfies (3.5), the step size λ k generated by the Armijo rue satisfies R(X (k) ) F λ k > β(1 α η). (3.13) S (k) BB T S (k) F Using the structure of the CARE (1.1) we can bound the right hand side in (3.13). Theorem 5 Assume that A (k) is stabe and et r k denote the rank of S (k). If the forcing parameter η k that contros the size of the Lyapunov residua, see (3.7), satisfies η k η < 1, then the step size generated by the Armijo rue obeys λ k > β(1 α η) 1 r k (1 + η) 2 BB T F R(X (k) (3.14) ) F exp(a 0 (k) t) 2 2 dt. Proof. First we bound the step S (k), the soution of (3.6). Since A (k) is stabe, the step is given by S (k) = ( ) exp((a (k) ) T t) L (k+1) R(X (k) ) exp((a (k) )t) dt. Therefore, 0 S (k) 2 L (k+1) R(X (k) ) 2 exp(a (k) t) 2 2 dt and S (k) F r k L (k+1) R(X (k) ) F exp(a (k) t) 2 2 dt, 0 0 7

10 since for a square matrix M with rank r, M 2 M F r M 2. The bound on S (k), (3.6), (3.7), and η k η < 1 impy S (k) BB T S (k) F BB T F S (k) 2 F r k (1 + η) 2 BB T F R(X (k) ) 2 F 0 exp(a (k) t) 2 2 dt. Inserting this into (3.13) gives the desired ower bound (3.14). Remark 6 If A (k) is a norma matrix with spectra abscissa α(a (k) ) := max{re (µ) : µ σ(a (k) )}, then exp(a (k) t) 2 exp(tα(a (k) )). If α(a (k) ) < 0, then 0 exp(a (k) t) 2 2 dt 1/(2 α(a (k) ) ). Otherwise we can use the ɛ-pseudospectrum σ ɛ (A (k) ) of A (k) and the ɛ-pseudospectra abscissa α ɛ (A (k) ) := sup{re (µ) : µ σ ɛ (A (k) )}. If σ ɛ (A (k) ) has a boundary with finite arc ength L ɛ,k, then exp(a (k) t) 2 L ɛ,k exp(t α ɛ (A (k) )) 2πɛ ɛ > 0, t 0 [22, p. 139], and, if α ɛ (A (k) ) < 0, then Exact Line Search 0 exp(a (k) t) 2 2 dt L 2 ɛ,k 8π 2 ɛ 2 α ɛ (A (k) ). Equation (3.11) shows that R(X (k) +λs (k) ) is quadratic in λ. Hence, min λ>0 R(X (k) + λs (k) ) 2 F corresponds to the minimization of the quartic poynomia (3.11). For the Keinman Newton method with exact Lyapunov equation soves, L (k+1) = 0, the exact ine search is anayzed by Benner and Byers [2]. In particuar, they show that there is a oca minimum λ k (0, 2], and that if A (k) is stabe and X (k+1) is computed with a step ength λ k (0, 2], then A (k+1) is aso stabe. However, both resuts are no onger true, in genera, for the inexact case. 3.3 Convergence Feitzinger et a. [9] extend the convergence resuts for the Keinman Newton method with step size λ k = 1 to the inexact case, provided the Lyapunov residua L (k+1) satisfies certain positive semi-definiteness assumptions. The first resut estabishes the we-posedness of the inexact Keinman Newton method. 8

11 Theorem 7 ([9, Thm. 4.3]) Let X (k) be symmetric and positive semi-definite such that A BB T X (k) is stabe and hods. Then L (k+1) C T C (3.15) (i) the iterate X (k+1) = X (k+1) of the inexact Keinman Newton method with stepsize λ k = 1 is we defined, symmetric and positive semi-definite, (ii) and the matrix A BB T X (k+1) is stabe. We wi use the ow-rank ADI method, see, e.g., [16], to approximatey sove the Lyapunov equation, see Section 4. This means that in our agorithm L (k+1), X (k), and other matrices are ow-rank. In particuar, we wi see in Section 4 (see equation (4.4)), L (k+1) = W (k+1) (W (k+1) ) T = F GG T F T, where F is a matrix with spectrum inside the unit ba and G = [C T X (k) B]. Lemma 8 If M, N are symmetric positive semi-definite matrices with M N, then ker (M) ker (N) and range (N) range (M). Proof. Assume there exists v ker (M) with v / ker (N), then v T Mv v T Nv = v T Nv < 0, which contradicts M N. Hence, range (M) = ker (M) ker (N) = range (N) and, consequenty, range (N) range (M). The definition of G and appication of the previous emma give that C T C L (k+1) = F GG T F T impies range ( F C ) T range (F G) range ( C ) T range (G). However, the invariance property range (F G) range ( C ) T, or even range ( F C ) T range ( C ) T, is typicay not satisfied. Reca that C T R n p whie F G R n (p+r) for k > 1. Therefore, in genera C T C L (k+1). Under an additiona semidefiniteness condition on the Lyapunov residua, Feitzinger et a. [9] prove quadratic convergence of the inexact Keinman Newton method. Theorem 9 ([9, Thm. 4.4]) Let Assumption 2 be satisfied and et X (0), symmetric and positive semi-definite, be such that A BB T X (0) is stabe. Let (3.15) hod for a k N, and et X (k) be the iterates of the inexact Keinman Newton method with step size λ k = 1. If 0 L (k+1) (X (k+1) X (k) )BB T (X (k+1) X (k) ) (3.16) hod for a k N, then the iterates of inexact Keinman Newton (3.8) with step size λ k = 1 satisfy (i) im k X (k) = X and 0 X X (k+1) X (k) X (1), (ii) (A BB T X ) is stabe and X is the maxima soution of R(X) = 0, 9

12 (iii) X (k+1) X F c X (k) X 2 F, k N. The condition (3.16) impies the monotonicity 0 X (k+1) X (k) X (1), which impies convergence of the sequence of the iterates. See the proof of [9, Thm. 4.4]. It is aso interesting to note that under the condition (3.16), the inexact Keinman Newton method convergences q quadraticay, independent of how the forcing parameter η k in (3.4) is chosen. Unfortunatey, the semidefiniteness condition (3.16) impies range (F G) range ( (X (k+1) X (k) )B ), which is generay not satisfied. Therefore, the convergence anaysis in [9] is not appicabe if the ow-rank ADI method, or any other ow-rank sover, is used to approximatey sove the Lyapunov equation. Our convergence proof foows that of inexact Newton methods, see, e.g., Keey [13, Sec. 8.2]. First, we prove R(X (k) ) F 0 and then we use the structure of the Riccati equations to argue convergence of {X (k) }. In particuar, Benner and Byers [2, Lem. 6] prove that if (A, B) is controabe and {R(X (k) )} is bounded, then {X (k) } is aso bounded. Since controabiity of (A, B) impies stabiizabiity of (A, B), the assumption of controabiity is stronger than Assumption 2. Guo and Laub [10] removed the controabiity assumption and showed that if (A, B) is stabiizabe, {R(X (k) )} is bounded, and the matrices A (k) are stabe, then {X (k) } is aso bounded. The papers [14] on exact Keinman Newton, [2] on Keinman Newton with ine search and [9] on inexact Keinman Newton contain proofs that the matrices A (k) corresponding to the iterates X (k) are stabe, provided that A (0) is stabe. This impies the unique soution of the Lyapunov equation (1.2) and, therefore, the we-posedness of the respective method. Since the definiteness assumption in [9, Thm. 4.3] typicay does not hod in the ow-rank case, there is no resut yet on the we-posedness of the inexact Keinman Newton method and we have to assume existence of X(k+1) such that (3.8a) and (3.7) are satisfied. Theorem 10 Let Assumption 2 be satisfied and assume that for a k there exists a symmetric positive semi-definite X (k+1) such that (3.8a) and (3.7) hod. (i) If the step sizes are bounded away from zero, λ k λ min > 0 for a k, then R(X (k) ) F 0. (ii) If, in addition to (i), the matrices A (k) are stabe for k K 0, and X (k) 0 for a k K 0, then X (k) X ( ), where X ( ) 0 is the unique stabiizing soution of the CARE. Proof. (i) The first part is a standard ine search argument. The sufficient decrease condition (3.5) impies that for any integer K, R(X (0) ) F R(X (0) ) F R(X (K+1) ) F K K = R(X (k) ) F R(X (k+1) ) F λ k α R(X (k) ) F 0. k=0 Taking the imit K and using λ k λ min > 0 impies R(X (k) ) F 0. k=0 10

13 (ii) If the matrices A (k) are stabe for k K 0 and {R(X (k) )} is bounded, [10, Lem. 2.3] guarantees that {X (k) } is bounded. Hence, {X (k) } has a converging subsequence. For any converging subsequence im j X (kj) 0 and 0 = im j R(X (kj) ) F = R(im j X (kj) ) F. Since the symmetric positive semi-definite soution of the CARE (1.1) is unique and stabiizing, every converging subsequence of {X (k) } has the same imit X ( ). Therefore, the entire sequence converges. Remark If the step size λ k (0, 1], then X (k) 0 for a k, see Remark Lower bounds for the step size computed by the Armijo rue are estabished in Theorem 5. In particuar if } { R(X (k) ) F exp(a (k) t) 2 2 dt, (3.17) 0 k N is bounded, the step size is bounded away from zero. Since R(X (k) ) F < R(X (0) ) F, the sequence (3.17) is bounded, if exp(a (k) t) dt is bounded, which is a condition on the uniform stabiity of the matrices A (k), k N. As it is we known for inexact Newton methods (see, e.g., Keey [13, Sec. 8.2]), the specific choice of the forcing parameter η k in (3.4) determines the rate if convergence. In particuar, if η k 0 the inexact Keinman Newton method converges superineary (under the assumptions of Theorem 10) and if η k = O( R(X (kj) ) F ), the convergence is quadratic. 4 ADI Method To compute the new iterate X (k+1) within the Keinman Newton method one has to sove the Lyapunov equation (1.2). A powerfu approach to sove such arge-scae Lyapunov equations with ow-rank right-hand sides is the aternating directions impicit (ADI) method, see, e.g., [7, 16]. In this section, we review the basic ingredients of the ADI method combined with recent agorithmic improvements from [3, 5, 4]. To simpify the notation, we drop the index k and write (1.2) in a more genera form as F X + XF T = GG T (4.1) with G := [ C T X (k) B ] R n (p+r) and F = (A BB T X (k) ) T R n n. We assume that F is stabe. The origina ow-rank ADI method computes a ow-rank soution factor Ẑ Cn ((p+r)) such that ẐẐH X R n n is the approximated soution of the Lyapunov equation (4.1); see, e.g., [7]. For given ADI shifts {q 1,..., q } C, the ow-rank ADI method successivey computes V 1 = (F + q 1 I) 1 G C n (p+r), (4.2a) V = V 1 (q + q 1 )(F + q I) 1 V 1 C n (p+r), 2. (4.2b) 11

14 Agorithm 2 rea ow-rank ADI method [5] Input: F, G, to ADI, shifts q C. Output: Z such that ZZ T X soves Eq. (4.1). 1: Set = 1, Z = [ ], W 0 = G. 2: whie W 1 T W 1 F > to ADI do 3: Sove V = (F + q I) 1 W 1. 4: if Im (q ) = 0 then 5: 6: W = W 1 2q V Ṽ = 2q V 7: ese 8: γ = 2 Re (q ), δ = Re (q ) / Im (q ) 9: W +1 [ = W 1 + γ 2(Re (V ) + δ Im (V )) ] 10: Ṽ = γ (Re (V ) + δ Im (V )) γ (δ 2 + 1) Im (V ) 11: = : end [ if ] 13: Z = Z Ṽ 14: = : end whie In the -th iteration, the approximate ow-rank soution factor is [ Ẑ = 2 Re (q1 ) V 1,..., ] 2 Re (q ) V C n ( (p+r)). (4.3) We use two important modifications of the origina ADI method, which are due to Benner et a. [3, 5, 4]. The first reorganizes the computation of the V s to obtain a ow-rank representation of the Lyapunov residua in the ADI iterations. The second expoits the fact that the ADI shifts need to occur either as a rea number q R or as a pair of compex conjugate numbers q C, q +1 = q, to write a matrices in the ADI iterations as rea matrices. We summarize the main ideas. In [4, Sec. 4.2], Benner et a. introduced a nove ow-rank residua formuation for ADI. For 2 the identity (4.2b) can be written as V = (I (q + q 1 )(F + q I) 1 ) V 1 = (F q 1 I)(F + q I) 1 V 1 ( = (F q j 1 I)(F + q j I) 1) (F + q 1 I) 1 G. j=2 Because (F ± qi) and (F + ˆqI) 1 commute for a q, ˆq C\σ(F ), these products can be regrouped to yied V = (F + q I) 1( 1 (F q j I)(F + q j I) 1) G =: (F + q I) 1 Ŵ 1. j=1 12

15 By definition of Ŵ (and setting Ŵ0 = G), Moreover, Ŵ = (F q I) V = (F q )(F + q I) 1 Ŵ 1 = (I 2 Re (q ) (F + q I) 1 )Ŵ 1 = Ŵ 1 2 Re (q ) V C n (p+r). Ŵ = (F q I) V = (F q j I)(F + q j I) 1 G = F G j=1 with F = F(F, q 1,..., q ) := j=1 (F q ji)(f + q j I) 1 an anaytic matrix function depending on F and the ADI shifts q 1,..., q. Using this formuation, which is mathematicay equivaent to the origina agorithm in [7, 16], Benner et a. [4, Sec. 4.2] show that the Lyapunov residua after ADI step, can be written as L = F ẐẐH + ẐẐH F T + GG T = ŴŴ H = FGG T FH R n n. Using the ow-rank structure L = ŴŴ H, Ŵ C n (p+r), of the Lyapunov residua together with the commony known resut that the eigenvaues σ(ŵŵ H) \ {0} = σ(ŵ HŴ)\{0}, see, e.g., [11, Theorem 1.32], eads to an efficient way to compute and accumuate the Lyapunov residua and its spectra or Frobenius norm to contro the accuracy of the ADI iteration [4]. The previous versions of the ow-rank ADI method compute compex ow-rank factors V, Ŵ C n (p+r), Ẑ C n ((p+r)). To avoid compex arithmetic and storage of compex matrices as much as possibe, Benner et a. [3, 5] introduced a reformuated ow-rank ADI iteration, where they expoit the fact that the ADI shifts need to occur either as a rea number q R or as a pair of compex conjugate numbers q C, q +1 = q. The resuting ow-rank ADI iteration works with rea ow-rank factors V, W R n (p+r), Z R n ((p+r)). The rea ow-rank ADI method is shown in Agorithm 2. The approximate soution of the Lyapunov equation (4.1) is X Z Z T R n n. The corresponding Lyapunov residua has a rea ow-rank representation L = W W T = FGG T F T R n n, (4.4) where F(F, q 1,..., q ) R n n is an anaytic matrix function depending on F and the ADI shifts q 1,..., q. Notice that F F iff {q i } i=1 = {q i} i=1, i.e., the ADI shifts are cosed under compex conjugation. See, e.g., [7, 5] for detais. 5 Low-Rank Residua Newton-ADI Method Using Agorithm 2 as the inner oop to sove the Lyapunov equations in Line 5 of Agorithm 1, we arrive at an agorithm for the Keinman-Newton method, where the 13

16 ow-rank structure can be used to efficienty compute residuas and the quartic function (3.11) that arises in the ine search computation. As we have seen in (4.4) in the previous section (we now keep track of the Keinman- Newton iteration counter k), the Lyapunov residua is L (k+1) = W (k+1) (W (k+1) ) T, where is the iteration counter in the inner ADI iteration and W (k+1) R n (p+r). Since L (k+1) 2 F is the sum of the squares of the eigenvaues of L(k+1) and σ(w (k+1) (W (k+1) ) T ) \ {0} = σ((w (k+1) ) T W (k+1) ) \ {0}, the norm L (k+1) 2 F can be efficienty computed by soving a sma (p + r) (p + r) eigenvaue probem. 5.1 Norm of the Difference of Outer Products Let W R n m and K R n p with m+p n be generic matrices. We frequenty need to compute Frobenius or 2-norms of the difference W W T KK T. This can be done efficienty using the indefinite ow-rank factorization W W T KK T = UDU T R n n, where U = [ W K ] [ ] Im 0 and D =. 0 I p For the spectrum we have σ(udu T ) \ {0} = σ(u T UD) \ {0} (see, e.g., [11, Theorem 1.32]). Since U T UD is a sma (m+p) (m+p) matrix, its spectrum can be computed efficienty and we can use W W T KK T 2 = max{ λ : λ σ(w W T KK T )} = max{ λ : λ σ(u T UD)}, W W T KK T 2 F = λ 2 i = λ 2 i. λ i σ(w W T KK T ) λ i σ(u T UD) Notice that since U T UD is not symmetric, max{ λ : λ σ(u T UD)} = U T UD 2 and λ i σ(u T UD) λ2 i U T UD 2 F. 5.2 Low-Rank Riccati Residua and Feedback Accumuation Reca that X (k+1) = X (k) + S (k). Consider S (k) B = X (k+1) B X (k) B =: K (k+1) K (k) =: K (k+1) R n r, (5.1) X (k+1) which defines the change of the feedback K corresponding to the tria soution of (3.8a). The key ingredient to use the ine search idea efficienty for arge-scae probems are the ow-rank formuations of the Lyapunov and Riccati residuas. Reca from (4.4) that L (k+1) = W (k+1) (W (k+1) ) T (5.2a) 14

17 and assume that with R(X (k) ) = W (k) (W (k) ) T K (k) ( K (k) ) T = U (k) D(U (k) ) T D = [ ] Ir+p 0 0 I r and U (k) = [ W (k) K (k)] R n (2r+p). (5.2b) (5.2c) For k = 0 and X (0) = 0, (5.2) hods with W (0) = C T and K (0) = 0. We ca a factorization of the form (5.2b) an indefinite ow-rank factorization (compare Section 5.1). If one uses (5.2) and the feedback change (5.1), than (3.9) impies R(X (k+1) ) = R(X (k) + λ k S (k) ) = (1 λ k )U (k) D(U (k) ) T + λ k W (k+1) (W (k+1) ) T λ 2 k K ( (k+1) K (k+1)) T ( ) = (1 λ k ) W (k) (W (k) ) T K (k) ( K (k) ) T + λ k W (k+1) (W (k+1) ) T λ 2 k K ( (k+1) K (k+1)) T [ [ = (1 λk )W (k) λw (k+1)] [ ] (1 λk ) K (k) λ k K (k+1)] [ ] I(s+1)(p+r) 0 0 I (s+1)r [ [ (1 λk )W (k) λ k W (k+1)] [ ] T (1 λk ) K (k) λ k K (k+1)], (5.3) where s {0, 1,..., } is the number of iterations immediatey before the k-th iteration in which the step size was ess than one. See beow for more detais. If λ k = 1, then X (k+1) = X (k+1), K (k+1) = K (k+1) and (5.3) simpifies to R(X (k+1) ) = R( X (k+1) ) = W (k+1) (W (k+1) ) T K (k+1) ( K (k+1) ) T =: U (k+1) D(U (k+1) ) T (5.4) with U (k+1) = [ W (k+1) K (k+1)], which is of the form (5.2b). If λ k (0, 1), we can redefine [ W (k+1) (1 λk )W (k) λ k W (k+1)] R n (s+1)(p+r), [ K (k+1) (1 λk ) K (k) λ k K (k+1)] R n (s+1)r, [ ] I(s+1)(p+r) 0 D. 0 I (s+1)r After this redefinition, (5.4) hods. Notice that if λ k < 1, the sizes of W (k+1) and K (k+1) grow. As mentioned before, their sizes depend on the number s {0, 1,..., } 15

18 of iterations immediatey before the k-th iteration in which the step size was ess than one, i.e., on s {0, 1,..., } with λ k s 1 = 1, λ k s < 1, λ k < 1. The representation (5.4) can be used to compute the Riccati residua R(X (k) + λ k S (k) ) F in dependence of λ k efficienty (see Section 5.1). It is important to mention that we need to keep U (k) R n ((s+1)(2r+p)) to perform the ine search; it is not sufficient to just keep R(X (k) ) F. The tria iterate X (k+1) is computed by Agorithm 2 iterativey and, consequenty, the tria feedback K (k+1) = X (k+1) B R n r can aready be computed during the execution of Agorithm 2. Let be the iteration counter in Agorithm 2. We have K (k+1) = = X (k+1) B = K (k+1) 1 ] [Ṽ1... Ṽ + Ṽ(Ṽ T B), (k+1) If we define K 0 = K (k), then Ṽ1 T. Ṽ T K(k+1) 0 = 0. B = j=1 Ṽ j (Ṽ T j B) (k+1) K = K (k+1) K (k) = K (k+1) 1 + Ṽ(Ṽ T B) K (k) (k+1) = K 1 + Ṽ(Ṽ T B). Thus, the feedback change can be assembed efficienty during the ADI iteration. The ow-rank Riccati residua factor for the k+1-st Riccati step after ADI steps can be written as U (k+1) (k+1) = [W K ] R n (2r+p). The Riccati residua norm R(X (k+1) ) F can be computed easiy during the ADI iteration by computing the eigenvaues of the sma matrix (U (k+1) ) T U (k+1) D, see Section Low-Rank Line Search Impementation To compute the step size as discussed in Section 3.2 for arge-scae probems, we need to compute the quartic poynomia (3.11). We can compute the coefficients defined in (3.12) efficienty. The coefficient α (k) = R(X (k) 2 F can be computed using (5.2b) (see Section 5.1). Simiary, β (k) = L (k+1) 2 F = W (k+1) (W (k+1) ) T 2 F can be computed efficienty as shown at the beginning of this section. Instead of using eigenvaues of M, N R n n, we can use the property tr (MN) = tr (NM) and, for symmetric matrices M, tr ( M 2) = i,j (M ij) 2, and compute β (k) = W (k+1) (W (k+1) ) T 2 F = tr (W ) (k+1) (W (k+1) ) T W (k+1) (W (k+1) ) T ( = tr (W (k+1) ) T W (k+1) (W (k+1) ) T W (k+1)) = (W (k+1) ) T W (k+1) 2 F. Simiary, with K (k+1) = S (k) B R n r, δ (k) = K (k+1) ( K (k+1) ) T 2 F = ( K (k+1) ) T K (k+1) 2 F. 16

19 Appication of trace identities gives γ (k) = R(X (k) ), L (k+1) = tr (U ) (k) D(U (k) ) T W (k+1) (W (k+1) ) T ( ) = tr (U (k) ) T W (k+1) (W (k+1) ) T U (k) D ([ ] (W = tr (k) ) T W (k+1) [(W ( K (k) ) T W (k+1) (k+1) ) T W (k) (W (k+1) ) T K (k)]) = [ ] [ ] (W (k) ) T W (k+1) (W (k) ) T (W (k+1) ) ( K (k) ) T W (k+1) ( K (k) ) T W (k+1) i,j ij ij ( ((W (k) ) T W (k+1)) 2 ( ( K ij) (k) ) T W (k+1)) 2 ij) = i,j i,j and, anaogousy, ε (k) = R(X (k) ), S (k) BB T S (k) = tr (U (k) D(U (k) ) T K (k+1) ( K ) (k+1) ) T ( ((W (k) ) T K (k+1)) 2 ) 2. ij) ( (( K (k) ) T K (k+1)) ij = i,j i,j Finay, ζ (k) = L (k+1), S (k) BB T S (k) = tr (W (k+1) (W (k+1) ) T K (k+1) ( K ) (k+1) ) T = ( ((W (k+1) ) T K (k+1)) ) 2. ij i,j After choosing λ k appropriatey, the next iterate X (k+1) (3.8b) and the feedback K (k+1) can be computed. Using a ow-rank ADI method (see Section 4), X(k+1) = Z (k+1) ( Z(k+1)) T as ow-rank approximation of the soution of (3.8a) and the ow-rank approximations of the previous iterate X (k) = Z (k) (Z (k) ) T are used. X (k+1) = X (k) + λ k S (k) = (1 λ k )X (k) + λ k X(k+1) ( = (1 λ k )Z (k) Z (k)) T ( ) + λk Z(k+1) Z(k+1) T [ = 1 λk Z (k) ] [ λ Z(k+1) k 1 λk Z (k) ] λ Z(k+1) T k. (5.5) K (k+1) = X (k+1) B = (1 λ k )X (k) B + λ k X(k+1) B = (1 λ k )K (k) + λ k K(k+1). (5.6) Notice that the size of Z (k) and Z (k+1) depends on the number of ADI steps that are needed to sove (3.8a). Athough (5.5) might be very arge, it is important to mention that it ony needs to be computed at the end of the Newton iteration, because 17

20 the previous iterate X (k) enters the right-hand side of (3.8a) ony as product with the input matrix B from the right. This means one ony needs the inexpensivey accumuated feedback K (k+1) = X (k+1) B in Eq. (5.6) to proceed with the Newton iteration. Furthermore, typicay, a ine search wi ony be necessary in the first few Newton steps, so that (5.5) might never be used after the first few iterations and instead simpy X (k+1) = X (k+1) = Z (k+1) ( Z (k+1) ) T is used. 5.4 Compete Impementation We concude this section with a summary of the resuting agorithm and some comments on the ine search and the convergence of the inexact Keinman-Newton method with ine search. We perform a ine search if after reaching the condition (3.7) at ADI step it hods that R( cases: X (k+1) ) > (1 α) R(X (k) ). We aso perform a ine search in the foowing a) Before reaching the condition (3.7), the actua step 2 yieds L F > L 1 F, i.e., the norm of the Lyapunov residua exceeds the norm of the initia Lyapunov residua. b) The number of ADI steps exceeds the maxima number of aowed ADI steps without reaching the condition (3.7). If the conditions in a) or b) are observed, it indicates that the ADI method does not converge, e.g., because the matrix A (k) is not stabe. Athough condition (3.7) is vioated, we perform a ine search, since the cost of its execution is sma, and accept X (k+1) = X (k) + λ k S (k) if the sufficient decrease condition (3.5) is fufied. If the ine search method determines a λ k that is too sma, we switch to an exact Keinman-Newton method, i.e., we use Agorithm 1 with ADI Agorithm 2 with toerance to ADI = 10 1 to Newt as the inner sover. Since we cannot guarantee stabiity of A (k), it is not guaranteed that Agorithm 2 converges. If we observe that Agorithm 2 does not converge, we restart the entire process using the exact Keinman-Newton method as described above. During the exact Keinman-Newton scheme, the agorithm switches back to the inexact scheme as soon as the Riccati residua shows the expected convergence behavior. We note that in the numerica exampe studies in the next section, the ADI Agorithm aways reached the required toerance, i.e., condition (3.7) was aways achieved, and the ine search was aways successfu. The inexact Keinman-Newton method with ine search and a rea ow-rank ADI method as inner sover is summarized in Agorithm 3. The residua R( X (k+1) ) = ) T is accumuated during the ADI iteration. In practice, the factor U of the indefinite ow-rank decomposition of the Riccati residua in ines 20, 28, and 32 is never assembed expicity since norm computation and ine search directy use W and K. U (k+1) D(U (k+1) 18

21 Agorithm 3 inexact Keinman-Newton-ADI method with ine search Input: A, B, C, initia feedback K (0), to Newt, η (0, 1), and α (0, 1 η). Output: K (k+1) (optiona: Z (k+1) such that Z (k+1) (Z (k+1) ) T is a stabiizing approximate soution of the CARE (1.1)). 1: Set k = 0, res (0) Newt = CT C + K ( (0) K (0)) T ( ). 2: whie res (k) Newt > to Newt res (0) Newt do 3: Set A (k) = ( A T K (k) B ) T, G = [ C T K (k)]. 4: Compute ADI shifts {q } n max,adi =1 C and choose η k (0, η]. 5: Set = ( 1, W 0 = G, K 0 = K ) (k) (optiona Z = [ ]). 6: whie W W T F > η k res (k) Newt do 7: V = ( A (k) + q I ) 1 W 1 8: if Im (q ) = 0 then 9: W = W 1 2q V 10: Ṽ = 2q V 11: K = K 1 + (Ṽ B) Ṽ 12: ese 13: γ = 2 Re (q ), δ = Re (q ) / Im (q ) 14: W +1 [ = W 1 + γ 2 (Re (V ) + δ Im (V )) 15: Ṽ = γ (Re (V ) + δ Im (V )) γ ] (δ 2 + 1) Im (V ) 16: = : K = K 2 + (Ṽ B) Ṽ T 18: end if [ ] 19: (optiona Z = Z Ṽ ) 20: U = [W K ] 21: = : end whie 23: if U DU T F > (1 α)res (k) Newt then 24: Choose λ k (0, 1) using Armijo or exact ine search. 25: K 1 = λ k K 1 26: W (k+1) = [ 1 λ k W (k) λ k W (k+1)] 27: K (k+1) = [ 1 λ k K (k) K 1 ] 28: U (k+1) = [ W (k+1) K (k+1)] 29: (optiona Z (k+1) = [ 1 λ k Z (k) λ k Z ] ) 30: ese 31: W (k+1) = W, K (k+1) = K 1 32: U (k+1) = U 33: (optiona Z (k+1) = Z) 34: end if 35: K (k+1) = K (k) + K 1 36: res (k+1) Newt = U (k+1) D ( U (k+1)) T F 37: k = k : end whie 19

22 6 Numerica experiments Consider the infinite dimensiona optima contro probem minimize 1 ( ) 2 γ x(ξ, t)dξ + u 2 (t) dt, 2 0 Ω O subject to x x (ξ, t) = x(ξ, t) + 20 (ξ, t) x(ξ, t) + f(ξ)u(t), t ξ 2 ξ Ω, t > 0, x(ξ, t) = 0, ξ Ω, t > 0, with Ω = (0, 1) d, d 2, 3, Ω O Ω, γ > 0, and { 100 ξ ΩC, f(ξ) := 0 ese, where Ω C = (0.1, 0.3) (0.4, 0.6) if d = 2 and Ω C = (0.1, 0.3) (0.4, 0.6) (0.1, 0.3) if d = 3. For d = 2, this exampe was aso used by Feitzinger et a. [9] and by Morris and Navasca [20]. 1 We use piecewise inear finite eements to discretize the optima contro probem. More specificay, we use P1 finite eements on a uniform trianguation. If d = 2, Ω = (0, 1) 2 is divided into squares of size h h and each square is divided into two trianges. If d = 3, Ω = (0, 1) 3 is divided into cubes of size h h h and each cube is divided into six tetrahedra. We use mesh sizes h such that the mesh is aigned with the boundaries of Ω O and of Ω C. This eads to the inear quadratic contro probem Minimize y(t) T y(t) + u 2 (t) dt, (6.1a) subject to Eẋ(t) = Ax(t) + Bu(t), t > 0, (6.1b) y(t) = γcx(t), t > 0, (6.1c) where E R n n is the symmetric positive definite mass matrix, A R n n, B R n 1, C R 1 n, x(t) R n, u(t) R r, and y(t) R, where n is proportiona to 1/h 2. For the output, we consider the cases Ω O = Ω and Ω O = Ω C. If Ω O = Ω, then the finite eement discretization resuts in the output matrix C = e T E, where e is the vector of a ones, and if Ω O = Ω C, then the finite eement discretization resuts in the output matrix C = B T /100. The matrix E 1 A is stabe and, therefore, for this LQR probem (6.1) the Assumption 2 is satisfied. It is a basic resut, see, e.g., [18], that u (t) = Kx (t) with 1 Both papers [9] and [20] use a centra finite difference method on a uniform grid with mesh size h = 1/(n + 1). Feitzinger et a. [9] use the output matrix C = [0.1,..., 0.1] and γ = 1, and Morris and Navasca [20] use the output matrix C = B T and γ = 1. These output matrices correspond to scaed versions of the outputs resuting from Ω O = Ω and Ω O = Ω C, respectivey, in the PDE mode. In fact, if Ω O = Ω = (0, 1) 2, the finite difference spatia discretization of the objective function is γ Ω x(ξ, t)dξ γ h2 n i,j=1 x ij(t) = γcx(t) where x ij (t) x((ih, jh), t), i, j = 1,..., n, and C = [h 2,..., h 2 ] R 1 n2. This output matrix C scaed by 1/(10h 2 ) corresponds to the output matrix in [9]. If Ω O = Ω C (0, 1) 2, then the finite difference spatia discretization of the objective function eads to the output matrix (h 2 /100)B T, which is the output matrix in [20] scaed by 100/h 2. 20

23 K = B T XE minimizes the cost functiona (6.1a) with X as stabiizing soution of the generaized CARE γ 2 C T C + A T XE + E T XA E T XBB T XE = 0. (6.2) The extension of our resuts for the soution of CARE (1.1) to the generaized CARE (6.2) with nonsinguar E is straightforward. A γ 1 increases the effect that R(X (1) ) F R(X (0) ) F. The ADI shifts are computed foowing the V -shifts idea in [6]. In a computations the mesh size is h = 1/30. This eads to matrix sizes n = 841 in the 2D case and n = 24, 389 in the 3D case. We appy the Keinman-Newton-ADI method either exacty or inexacty. In the atter case we either use the forcing parameter η k in (3.4) given by η k = 1/(k 3 +1) or by η k = min{0.1, 0.9 R(X (k) ) F }. The first choice eads to superinear convergence, whie the second resuts in quadratic convergence (under the assumptions of Theorem 10). In a cases the Keinman-Newton-ADI method is stopped when the normaized residua R(X (k) ) / C T C drops beow to Newt = In the exact Keinman-Newton- ADI method, the ADI toerance is set to to ADI = to Newt /10. We appy a methods without ine search ( no LS ), i.e., set λ k = 1 in a iterations, and with ine search. If the sufficient decrease condition (3.5) is not satisfied for λ k = 1, then we compute a step size using a simpe impementation of the Armijo rue with β = 0.5, cf. Section The performances of the various Keinman-Newton-ADI methods are summarized in Tabes 1 to 7. In a tabes, # Newt. is the tota number of (inexact) Newton steps executed before the stopping criterion R(X (k) ) / C T C < to Newt = is satisfied, # ADI is the tota number of ADI iterations executed, and # LS is the tota number of times the step size λ k was chosen to be ess than one. The entry no LS indicates that the agorithm was run without ine search, i.e., that λ k = 1, k. In a variations of the Keinman-Newton-ADI method, the execution times are essentiay proportiona to the tota number of ADI steps performed. Due to the ow-rank structure, the execution times for other agorithm components, such as ine search, are negigibe compared to the execution of one ADI iteration. In a exampes shown, the exact and inexact versions of the Keinman-Newton methods converged, and the inexact versions of the Keinman-Newton method significanty outperform the exact version. We note that athough in a exampes the inexact Keinman-Newton method without ine search converged, there is no convergence proof to guarantee this (uness the conditions on the Lyapunov residua in Feitzinger et a. [9] can be satisfied, which is not the case when ow-rank ADI methods are used). The ine search performed differenty for the outputs Ω O = Ω C (C = B T /100) (see Tabes 1, 3) and Ω O = Ω (C = e T E) (see Tabes 5, 7). In the exampe Ω O = Ω C (C = B T /100), the ine search is active, i.e., λ k 1, in at most the first two iterations and it is ony active if γ 1. In this exampe, using the ine search aways resuted in fewer Newton iterations and ed to fewer ADI iterations overa. In the exampe Ω O = Ω (C = e T E), the ine search is active, i.e., λ k 1, in more iterations. The ine search is active in the first iterations and if λ k = 1 in one iteration 21

24 k, it is equa to one on a subsequent iterations. In the 2D case with γ = 1 (Tabe 6a), the ine search eads to significanty more Newton and ADI iterations. In this case, R(X (k) + S (k) ) F R(X (k) ) F for the first iterations, and a sma step size λ k is needed to satisfy the sufficient decrease condition (3.5). This eads to sma steps initiay and a substantia increase in Newton iterations. It may be possibe to improve the performance of the inexact Keinman-Newton method with ine search by refining the forcing parameter η k, i.e., the choice of Lyapunov residua toerances. This is part of future research. In a other cases, using the ine search eads to fewer Newton iterations and mosty fewer ADI iterations. Notice that the ine search enforces the monotonicity R(X (k+1) ) < R(X (k) ) F which can resut in a significanty smaer right hand side in (3.4), i.e., a smaer Lyapunov equation sover toerance, compared to when no ine search is used. Therefore, using a ine search can require more ADI iterations per Newton iteration; compare Tabe 6b ( superinear ), Tabe 8b ( inexact ), and Tabe 8c ( quadratic ). 7 Concusions We have presented an efficient impementation of the inexact Keinman-Newton method with a ow-rank ADI subprobem sover. On the theoretica side, we presented a convergence proof which is based on convergence proofs for genera inexact Newton methods. Because of the ow-rank case and ack of positive semi-definiteness conditions, ike the one in Theorem 7 [9, Thm. 4.3], it is not possibe to ensure that a iterates are stabiizing if the initia iterate is stabiizing. This was not an issue in our numerica exampe. In our convergence proof, the ine search is needed to ensure that the Riccati residuas decrease monotonicay in norm. Athough in our numerica exampes, the inexact Keinman-Newton method with a ow-rank ADI subprobem sover aways converged when used without ine search, there is no guarantee for this and we have observed other exampes where the inexact Keinman-Newton method without ine search faied. The numerica exampe showed that the ine search can ead to substantia reduction in the overa number of ADI iterations and, therefore, overa computationa cost, but there is one case where the ine search resuts in substantiay more Keinman-Newton iterations and in a substantiay higher number of tota ADI iterations. Possibe improvements by changing the forcing parameter, i.e., the choice of Lyapunov residua toerances, is part of future research. We have begun numerica experiments with the computation of feedback contros for incompressibe Navier-Stokes fows, simiar to [1], where stabiity of iterates can be an issue. A detaied report of these tests, and comparisons with other arge-scae Riccati sovers, ike [8, 21, 19, 17], is part of future research. F 22

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