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1 SIAM J. SCI. COMPUT. Vo.?, No.?, pp.?? c? Society for Industria and Appied Mathematics ABSOLUTE VALUE PRECONDITIONING FOR SYMMETRIC INDEFINITE LINEAR SYSTEMS EUGENE VECHARYNSKI AND ANDREW KNYAZEV Abstract. We introduce a nove strategy for constructing symmetric positive definite (SPD) preconditioners for inear systems with symmetric indefinite coefficient matrices. The strategy is motivated by the observation that the preconditioned minima residua method with the inverse of the absoute vaue of the coefficient matrix as a preconditioner converges to the exact soution of the system in at most two steps. Neither the exact absoute vaue of the coefficient matrix, nor its exact inverse are computationay feasibe to construct in genera. However, as the proof of concept, we provide two practica exampes of SPD preconditioners, which are based on the suggested approach, caed absoute vaue preconditioning. The first exampe is for stricty (bock) diagonay dominant coefficient matrices, where we propose using the inverse to the absoute vaue of the (bock) diagona as the preconditioner. Our second exampe is ess intuitive. We consider a mode probem with a shifted discrete negative Lapacian, and suggest a geometric mutigrid preconditioner, where the inverse to the absoute vaue of the coefficient matrix appears ony on the coarse grid, whie operations on finer grids are based on the Lapacian. Our numerica tests demonstrate practica effectiveness of the new mutigrid preconditioner for moderatey sma shifts. Key words. Preconditioning, inear system, preconditioned minima residua method, poar decomposition, matrix absoute vaue, geometric mutigrid AMS subject cassifications. 15A06, 65F08, 65F10, 65N22, 65N55 1. Introduction. We consider a system of inear equations (1.1) Ax = b, A = A R n n, b R n, where the coefficient matrix A is nonsinguar and symmetric indefinite, i.e., the spectrum of A contains both positive and negative eigenvaues. For a reativey sma probem size n, the exact (up to the effects of round-off errors) soution of (1.1) can be efficienty found using a direct method; see, e.g., the survey in [7]. However, in most cases, the computationa cost of such a method does not optimay scae with respect to n, i.e., the amount of invoved computations does not grow proportionay to the increasing number of matrix eements. Therefore, if the probem size is significanty arge, the appication of a direct method may become infeasibe. Additionay, it is often required to find ony an approximate soution of (1.1), as opposed to the exact soution targeted by a direct sover. The above arguments motivate an iterative technique, which can be expected to provide a (neary) optima compexity and, instead of the exact soution of (1.1), deivers a sequence of its approximations. In this manuscript, it is assumed that the coefficient matrix A is extremey arge and possiby sparse. We focus ony on iterative methods for soving inear system (1.1). Let us remark that, in the suggested framework, under soving a inear system we understand approximatey soving a inear system, i.e., finding a satisfactory approximation to the exact soution of (1.1). Linear systems with arge, sparse, symmetric, and indefinite coefficient matrices arise in a variety of appications. For exampe, in the form of sadde point probems, such systems resut from mixed finite eement discretizations of underying differentia Department of Computer Science and Engineering, University of Minnesota, 200 Union Street S.E., Minneapois, MN (eugenev@cs.umn.edu). Department of Mathematica and Statistica Sciences, University of Coorado Denver, P.O. Box , Campus Box 170, Denver, CO (andrew.knyazev@ucdenver.edu). 1

2 2 EUGENE VECHARYNSKI AND ANDREW KNYAZEV equations of fuid and soid mechanics; see, e.g., [3] and references therein. In acoustics, arge sparse symmetric indefinite inear systems are obtained after discretizing the Hemhotz equation for certain media types and boundary conditions. Often the need of soving indefinite probem (1.1) comes as an auxiiary task within other computationa routines, such as the inner step in the interior point methods in inear and noninear optimization, see, e.g., [3, 24], or soution of the correction equation in the Jacobi-Davidson method [29] for a symmetric eigenvaue probem. There is a number of iterative methods deveoped specificay to sove symmetric indefinite inear systems, ranging from modifications of the Richardson s iteration, e.g., [8, 22, 27], to the optima Kryov subspace minima residua method deivered through short-term recurrent schemes, such as, e.g., the MINRES agorithm [25]. It is known, however, that in practica probems the coefficient matrix A in (1.1) is very i-conditioned which, aong with the ocation of the spectrum to both sides of the origin, makes the straightforward appication of the existing techniques inefficient due to extremey sow convergence. In order to improve the convergence, it is possibe to introduce a preconditioner T R n n and formay repace (1.1) with (1.2) T Ax = T b. If T is propery chosen, then an iterative method for (1.2) can exhibit a better convergence behavior compared to the origina scheme appied to (1.1). The preconditioned matrix T A is never expicity computed. If T is not symmetric positive definite (SPD), then the matrix T A of preconditioned inear system (1.2), in genera, is not symmetric with respect to any inner product; see, e.g. [26, Theorem ]. Thus, the introduction of a non-spd preconditioner eads to the substitution of origina symmetric probem (1.1) by generay nonsymmetric probem (1.2). As a resut, speciaized methods for symmetric inear systems, suitabe for (1.1), may no onger be appicabe to (1.2), and, hence, shoud be repaced by iterative schemes for nonsymmetric inear systems; e.g., GMRES or GMRES(m) [28], Bi-CGSTAB [35], QMR [13], etc. Though known to be effective for a number of appications, the approach based on the choice of a non-spd preconditioner, which eads to soving nonsymmetric probem (1.2), has severa disadvantages. First, no short-term recurrent scheme that deivers an optima Kryov subspace method is typicay avaiabe for a nonsymmetric inear system; see [11]. In practice, this means that impementations of the reevant optima methods (e.g., GMRES) require an increasing amount of work and storage at every new step, and are often computationay expensive. Second, the convergence behavior of iterative methods for nonsymmetric inear systems is not competey understood. In particuar, the convergence may not be characterized in terms of reasonaby accessibe quantities, such as, e.g., the (estimated) spectrum of the preconditioned coefficient matrix; see the corresponding resuts for GMRES and GMRES(m) in [17, 37]. This makes it difficut to predict the computationa cost of the seected approach. If T is chosen to be SPD, i.e., T = T > 0, then the matrix T A of preconditioned inear system (1.2) is symmetric (indefinite), however, with respect to the T 1 inner product defined by (u, v) T 1 = (u, T 1 v) for any u, v R n. Here (, ) denotes the Eucidean inner product, i.e., (u, v) = v u, in which the matrices A and T are symmetric. Due to this symmetry preservation, inear system (1.2) can be soved using an optima Kryov subspace method which admits a short-term recurrent impementation. Moreover, the convergence of the method can be fuy estimated in terms of the spectrum of the preconditioned matrix T A.

3 ABSOLUTE VALUE PRECONDITIONING 3 In the ight of the discussion above, the choice of a propery defined SPD preconditioner for a symmetric indefinite inear system can be regarded as natura and favorabe. We advocate the use of SPD preconditioning for probem (1.1) and propose a nove strategy for constructing SPD preconditioners. Given an SPD preconditioner T, we consider soving symmetric indefinite inear system (1.1) with the preconditioned minima residua method, impemented in the form of the PMINRES agorithm [16, 25]. In the absence of round-off errors, at step i, the method constructs an approximation x (i) to the soution of (1.1) of the form (1.3) x (i) x (0) + K i (T A, T r (0)), such that the residua vector r (i) = b Ax (i) satisfies the optimaity condition (1.4) r (i) T = min u AK i(t A,T r (0) ) r(0) u T. Here, K i (T A, T r (0)) = span {T r (0), (T A)T r (0),..., (T A) i 1 T r (0)} is the Kryov subspace generated by the matrix T A and the vector T r (0), and AK i (T A, T r (0)) = span {(AT )r (0),..., (AT ) i r (0)} is the corresponding Kryov residua subspace; v 2 T = (v, v) T for any v R n, and x (0) is the initia guess. The PMINRES impementation of the optima preconditioned minima residua method (1.3) (1.4) for (1.1) is based on a short-term recurrence. It can be viewed as the MINRES agorithm appied to (1.2) with occurring Eucidean inner products repaced by T 1 based inner products, written in a way to avoid computations invoving T 1. The conventiona convergence rate bound for method (1.3) (1.4) can be found, e.g., in [16], and reies soey on the distribution of eigenvaues of the preconditioned matrix T A. In this work, we describe a nove approach for constructing SPD preconditioners for symmetric indefinite inear systems. First, in Section 2, we present an idea SPD preconditioner for probem (1.1), which we define as the inverse of the absoute vaue of the coefficient matrix. Next, motivated by this idea preconditioner, we introduce a notion of an absoute vaue preconditioner, and refer to the new preconditioning strategy as the absoute vaue preconditioning. The rest of the paper deas with the question if absoute vaue preconditioners can be efficienty constructed in practice. In Section 3, we provide simpe exampes of such preconditioners for inear systems with stricty (bock) diagonay dominant coefficient matrices. In Section 4, we suggest a different exampe, a geometric mutigrid (MG) absoute vaue preconditioner for a mode probem resuting from a discretization of a shifted negative Lapace operator on a unit square. The efficiency of this preconditioner for moderatey sma shifts is demonstrated in our numerica tests, aso in Section 4. In particuar, we compare the proposed absoute vaue preconditioner to severa other SPD preconditioners, which resut from the known ideas of using the inverted Lapacian and the so-caed Bunch- Parett factorization for preconditioning symmetric indefinite inear systems. Throughout this paper, we assume the exact arithmetic. The choice of the rea vector spaces has been made to simpify the presentation. The generaization of the resuts to the compex case is straightforward. We note that the resuts presented in this work are partiay based on the PhD thesis of the first co-author [36], defended at the University of Coorado Denver, under the supervision of the second co-author.

4 4 EUGENE VECHARYNSKI AND ANDREW KNYAZEV 2. Absoute vaue preconditioning for symmetric indefinite inear systems. Let A R n n be a symmetric matrix with an eigendecomposition A = V ΛV, where V is an orthogona matrix of eigenvectors, and Λ = diag{λ j }, j = 1,..., n, is a diagona matrix of eigenvaues of A. We consider the factorization of the form (2.1) A = A sign(a) = sign(a) A, where A = V Λ V is an absoute vaue of A (matrix absoute vaue), and sign(a) = V sign(λ)v is a sign of A (matrix sign); Λ = diag{ λ j }, sign(λ) = diag{sign(λ j )}. Factorization (2.1) is, in fact, a poar decomposition, see, e.g., [19], of the symmetric matrix A, with the symmetric positive (semi) definite factor A and the orthogona factor sign(a). If, additionay, A is nonsinguar, then A is SPD. The foowing theorem regards the inverted absoute vaue of the coefficient matrix, i.e., A 1, as an SPD preconditioner for symmetric indefinite inear system (1.1). Theorem 2.1. Preconditioned minima residua method (1.3) (1.4) with the preconditioner T = A 1 converges to the exact soution of (1.1) in at most two steps. Proof. Minimization property (1.4) at a step i of a preconditioned minima residua method can be equivaenty written as (2.2) r (i) T = min p P i, p(0)=1 p(at )r(0) T, where P i is a set of a poynomias of degree at most i. Then, according to decomposition (2.1), the choice T = A 1 resuts in the matrix AT = sign(a) with ony two distinct eigenvaues: 1 and 1. Hence, the minima poynomia of AT is of the second degree. Thus, by (2.2), r (i) T = 0 for at most i = 2. Theorem 2.1 impies that T = A 1 is an idea SPD preconditioner for symmetric indefinite inear system (1.1). Indeed, a preconditioner T can be considered idea if it deivers the preconditioned matrix T A with the corresponding minima poynomia of the east possibe degree; e.g., see reated discussions in [14, 23]. The degree of the minima poynomia gives the number of iterations typicay required by a Kryov subspace method to guarantee the convergence to the exact soution of (1.1). Since the symmetric matrix A has both positive and negative eigenvaues, then so does T A, provided that T is SPD. This impies that the degree of the minima poynomia of T A is at east 2. Therefore, an idea SPD preconditioner for symmetric indefinite inear system (1.1), in genera, ensures the convergence to the exact soution in two steps. The one step convergence can occur for specia choices of the initia guess. We aso note that Theorem 2.1 hods not ony for preconditioned minima residua method (1.3) (1.4), but for a other methods, where convergence to the exact soution is determined by the degree of the minima poynomia of the preconditioned matrix. In practica situations, the computation of an idea SPD preconditioner T = A 1 is prohibitivey costy. However, we show that it is possibe to construct reasonaby inexpensive SPD preconditioners, which resembe, but are not equa to, A 1, and are abe to significanty acceerate the convergence of an underying iterative method. We refer to such a preconditioning strategy as the absoute vaue preconditioning and define absoute vaue preconditioners as foowing. Definition 2.2. We ca an SPD preconditioner T for a symmetric indefinite inear system (1.1) an absoute vaue preconditioner, if it satisfies the reation (2.3) δ 0 (v, T 1 v) (v, A v) δ 1 (v, T 1 v), v R n

5 ABSOLUTE VALUE PRECONDITIONING 5 with constants δ 1 δ 0 > 0, such that the ratio δ 1 /δ 0 1 is reasonaby sma. If inear system (1.1) represents a hierarchy of mesh probems, then the ratio must be independent of the probem size n, i.e., matrices A and T 1 are spectray equivaent [9]. Let us remark that Definition 2.2 of the absoute vaue preconditioner is informa, since no precise assumption is made of how sma the ratio δ 1 /δ 0 shoud be. It is cear from (2.3) that δ 1 /δ 0 measures how we the preconditioner T resembes an idea SPD preconditioner A 1 up to a positive scaing. If A is SPD, then A = A, and, for mesh probems, Definition 2.2 is consistent with the we known concept of spectray equivaent preconditioning for inear systems with SPD coefficient matrices; see [9]. The foowing theorem provides bounds for eigenvaues of the preconditioned matrix T A in terms of the spectrum of T A. Theorem 2.3. Let us be given a nonsinguar symmetric indefinite A R n n and an SPD T R n n, and et µ 1 µ 2... µ n be the eigenvaues of T A. Then eigenvaues λ 1... λ p < 0 < λ p+1... λ n of T A are ocated in intervas (2.4) µ n j+1 λ j µ p j+1, j = 1,..., p; µ j p λ j µ j, j = p + 1,..., n. Proof. We start the proof by observing that the absoute vaue of the Rayeigh quotient of the generaized eigenvaue probem Av = λ A v is bounded by 1, i.e., (2.5) (v, Av) (v, A v), v R n. Now, we reca that the spectra of matrices T A and T A are given by the generaized eigenvaue probems A v = µt 1 v and Av = λt 1 v, respectivey, and introduce the corresponding Rayeigh quotients (2.6) ψ(v) (v, A v) (v, T 1 v), φ(v) (v, Av) (v, T 1 v), v Rn. Further, et us fix any index j {1, 2,..., n}, and denote by S an arbitrary subspace of R n, such that dim(s) = j. Since inequaity (2.5) aso hods on S, using (2.6), we can write (2.7) ψ(v) φ(v) ψ(v), v S. Moreover, since taking maxima in vectors v S, and then minima in subspaces S S j = {S R n : dim(s) = j}, of a sides of (2.7) preserves the inequaity, we have (2.8) min max S S j v S ( ψ(v)) min max S S j v S φ(v) min S S j max v S ψ(v). By the Courant-Fischer theorem (see, e.g., [19, 26]) for the Rayeigh quotients ±ψ(v) and φ(v), defined in (2.6), we concude from (2.8) that µ n j+1 λ j µ j. Recaing that j has been arbitrariy chosen, we obtain the foowing bounds on the eigenvaues of the matrix T A: (2.9) µ n j+1 λ j < 0, j = 1,..., p; 0 < λ j µ j, j = p + 1,..., n.

6 6 EUGENE VECHARYNSKI AND ANDREW KNYAZEV Next, in order to derive nontrivia upper and ower bounds for the p negative and n p positive eigenvaues λ j in (2.9), we use the fact that eigenvaues ξ j and ζ j of the generaized eigenvaue probems A 1 v = ξt v and A 1 v = ζt v are the reciprocas of the eigenvaues of the probems A v = µt 1 v and Av = λt 1 v, respectivey, i.e., (2.10) 0 < ξ 1 = 1 µ n ξ 2 = 1 µ n 1... ξ n = 1 µ 1, and (2.11) ζ 1 = 1 λ p... ζ p = 1 λ 1 < 0 < ζ p+1 = 1 λ n... ζ n = 1 λ p+1. Simiar to (2.5), (v, A 1 v) (v, A 1 v), v R n. Thus, we can use the same arguments as those foowing (2.5) to show that reations (2.7) and (2.8), with a fixed j {1, 2,..., n}, aso hod for (2.12) ψ(v) (v, A 1 v), φ(v) (v, A 1 v) (v, T v) (v, T v), v Rn, where ψ(v) and φ(v) are now the Rayeigh quotients of the generaized eigenvaue probems A 1 v = ξt v and A 1 v = ζt v, respectivey. The Courant-Fischer theorem for ±ψ(v) and φ(v), defined in (2.12), aows us to concude from (2.8) that ξ n j+1 ζ j ξ j. Due to an arbitrary choice of j in the inequaity above, by (2.10) (2.11), we get the foowing bounds on the eigenvaues of the matrix T A: (2.13) 1/µ p j+1 1/λ j < 0, j = 1,..., p; 0 < 1/λ j 1/µ j p, j = p + 1,..., n. Combining (2.9) and (2.13), we obtain (2.4). This competes the proof. Theorem 2.3 suggests two usefu impications, which are given by the corresponding theorems beow. In particuar, the foowing resut describes the ocation of spectrum of the preconditioned matrix T A in terms of δ 0 and δ 1 in (2.3). Theorem 2.4. Let us be given a nonsinguar symmetric indefinite A R n n, an SPD T R n n, and constants δ 1 δ 0 > 0, such that (2.3) hods. Then (2.14) Λ(T A) [ δ 1, δ 0 ] [δ 0, δ 1 ], where Λ(T A) is the spectrum of T A. Proof. Foows directy from (2.3) and (2.4) with j = 1, p, p + 1, n. The next theorem shows that the presence of reasonaby popuated custers of eigenvaues in the spectrum of T A guarantees the occurrence of corresponding custers in the spectrum of the preconditioned matrix T A. Theorem 2.5. Let us be given a nonsinguar symmetric indefinite A R n n and an SPD T R n n, and et µ µ µ +k 1 be a sequence of k eigenvaues of T A, where 1 < + k 1 n and τ = µ µ +k 1. Then, if k p + 2,

7 ABSOLUTE VALUE PRECONDITIONING 7 the k p positive eigenvaues λ +p λ +p+1... λ +k 1 of T A are such that λ +p λ +k 1 τ. Aso, if k (n p) + 2, the k (n p) negative eigenvaues λ n k λ p λ p +1 of T A are such that λ n k +2 λ p +1 τ. Proof. Foows directy from bounds (2.4). Theorem 2.4 suggests that the ratio δ 1 /δ 0 1 of the constants from (2.3) measures the quaity of the absoute vaue preconditioner T in terms of the convergence speed of the preconditioned minima residua method, which is determined by the spectrum of T A. Additionay, Theorem 2.5 prompts that a good absoute vaue preconditioner shoud ensure custers of eigenvaues in the spectrum of T A. This impies the custering of eigenvaues of the preconditioned matrix T A, which has a favorabe effect on the convergence behavior of a poynomia iterative method, such as the preconditioned minima residua method, for inear system (1.1). At the same time, we reca that the costs of the construction and appication of T shoud preferaby be simiar to the costs of the matrix-vector mutipication by the coefficient matrix A. Beow we discuss severa possibe approaches to construct an absoute vaue preconditioner T for inear system (1.1) with a genera symmetric indefinite coefficient matrix. Without oss of generaity, we assume that T can be accessed ony through a matrix-vector mutipication. In other words, given a vector r R n, we define a preconditioner as a mapping r w = T r, where the matrix T may not be expicity constructed and stored. Iterative schemes are normay designed to hande such matrix-free preconditioning. One of the approaches to construct an absoute vaue preconditioner is to approximatey sove for z the equation A z = r. However, the coefficient matrix A is generay not avaiabe. Thus, the probem of approximatey soving the inear system A z = r can be repaced by the probem of finding a vector w which approximates the action of the matrix function f(a) = A 1 on the vector r, i.e., w f(a)r = A 1 r, moreover the construction of w requires ony knowedge of A, not A. The atter constitutes a we estabished task in matrix function computations, which is standardy fufied by a Kryov subspace method, e.g., [15, 18]. Our numerica experience shows that the convergence, with respect to the (outer) iteration count, of a inear sover can be significanty improved with this approach, but the computationa costs of approximating f(a)r = A 1 r, e.g., by the Lanczos method in [4], are too high for their appication in the context of absoute vaue preconditioners. We aso note that a Kryov subspace method for constructing w = T r A 1 r may resut in the preconditioner T, which is symmetric positive semidefinite and variabe, whereas the assumption for T to be SPD and non-variabe is typicay crucia for the conventiona convergence anayses of iterative inear sovers. In particuar, the use of variabe preconditioners eads to the oss of the goba optimaity of a Kryov subspace method for (1.1); see, e.g., the corresponding convergence bounds for the conjugate gradients method in [20]. Another approach to construct an absoute vaue preconditioner is to use a preconditioning technique, which aims to deiver T satisfying (2.3) with the ratio δ 1 /δ 0 reasonaby sma (and independent of n for mesh probems), however, which possiby does not correspond to a sover for equation A z = r, i.e., the norm T A 1 may not necessariy be sma. In Section 4, we demonstrate, for a mode probem, that such a construction of an efficient absoute vaue preconditioner is indeed possibe, e.g., if based on MG techniques. The discussion above concerns the construction of absoute vaue preconditioners for a genera symmetric indefinite inear system (1.1). In the next section, we consider

8 8 EUGENE VECHARYNSKI AND ANDREW KNYAZEV a specia case, where the coefficient matrix A has a (bock) diagona approximation C, so that C 1 is avaiabe at a sma computationa cost. A reasonabe absoute vaue preconditioner can then be expected to be given by T = C Absoute vaue preconditioning for stricty (bock) diagonay dominant matrices. For certain types of symmetric indefinite matrices, absoute vaue preconditioners can be easiy constructed. For exampe, if A = {a ij } is stricty diagonay dominant, i.e., (3.1) a ii > n a ij, i = 1,..., n, then the preconditioner T may be chosen as a diagona matrix with eements a ii 1. Assumptions simiar to (3.1) appear, e.g., in eectronic structure cacuations using pain-wave discretization; see [31]. In the foowing theorems we do not estabish (2.3), but rather directy prove a statement simiar to (2.14). Theorem 3.1. Let A = {a ij } R n n be a stricty diagonay dominant symmetric indefinite matrix, such that (3.2) δ a ii n a ij, i = 1,..., n, for a fixed δ [0, 1). Let T = diag { a 11 1, a 22 1,..., a nn 1}. Then (3.3) Λ(T A) {y R : y + 1 δ} {y R : y 1 δ}. Proof. We observe that the matrix T A = {a ij / a ii }; i, j = 1,..., n. Since A is symmetric indefinite and T is SPD, the spectrum of T A resides to both sides of the origin on the rea axis. In particuar, the Gershgorin circe theorem, see, e.g., [19, Theorem 6.1.1], suggests that a eigenvaues of T A are ocated in the union of n n intervas G i, where i=1 (3.4) G i From (3.2), we notice that Thus, y R : y sign(a ii ) 1 a ii 1 a ii n a ij, i = 1,..., n. n a ij δ, i = 1,..., n. G i {y R : y sign(a ii ) δ} = {y R : y + 1 δ} {y R : y 1 δ}, i = 1,..., n,

9 ABSOLUTE VALUE PRECONDITIONING 9 which competes the proof. Theorem 3.1 shows that for a stricty diagonay dominant symmetric indefinite coefficient matrix A, with δ sufficienty cose to zero in (3.3), a reasonabe absoute vaue preconditioner is given by T = diag { a 11 1, a 22 1,..., a nn 1}. The spectrum of T A, in this case, is ocated around 1 and 1. In fact, a simiar resut hods for stricty bock diagonay dominant matrices as we now show. A bock partitioned matrix A = {A ij } R n n, where A ij R ni nj, i, j = 1,..., s, is stricty bock diagonay dominant (reative to the partitioning {A ij }) if the diagona bocks A ii are nonsinguar, and if (3.5) ( A 1 ii ) 1 > s A ij, i = 1,..., s. The matrix norms in (3.5) are defined as spectra norms, i.e., A ij = σ max (A ij ) and A 1 ii = σ 1 min (A ii), where σ max (A ij ) and σ min (A ij ) are the argest and the smaest singuar vaues of A ij, respectivey. Inequaity (3.5) can be viewed as a direct generaization of definition (3.1) of the strict diagona dominance to the case of bock partitioned matrices. Severa other generaizations can be found, e.g., in [12, 32]. In particuar, the foowing extension of the Gershgorin circe theorem hods. Theorem 3.2 ([12, 32]). Let A = {A ij } R n n be a bock partitioned matrix with A ij R ni nj ; i, j = 1,..., s. Let us define (3.6) G i Λ(A ii ) s A ij y C \ Λ(A ii ) : ( (yi i A ii ) 1 ) 1 for i = 1,..., s, where Λ(A ii ) C is the spectrum of the submatrix A ii, I i is the n i -by-n i identity matrix, and denotes the spectra norm 1. Then the spectrum of s A is encosed in the union of sets (3.6), i.e., Λ(A) G i. We further use Theorem 3.2 to show that for a stricty bock diagonay dominant symmetric indefinite A = {A ij } in (1.1), the preconditioner T may be chosen as a bock diagona matrix with bocks A ii 1 on the diagona. Theorem 3.3. Let A = {A ij } R n n, i, j = 1,..., s, be a stricty bock diagonay dominant symmetric indefinite matrix, such that (3.7) δ ( A 1 ii ) 1 i=1 s A ij, i = 1,..., s, for a fixed δ [0, 1). Let T = diag { A 11 1, A 22 1,..., A ss 1}. Then (3.8) Λ(T A) {y R : y + 1 δ} {y R : y 1 δ}. 1 Matrix norms in (3.5) and (3.6) can be defined more generay as conventiona operator norms, induced by the vector norms on the corresponding n i - (and n j -) dimensiona spaces; see [12, 32]. The choice of the spectra norm has been made ony for the purposes of the current paper.

10 10 EUGENE VECHARYNSKI AND ANDREW KNYAZEV Proof. The proof is simiar to that of Theorem 3.1. Since A is symmetric indefinite and T is SPD, the spectrum of T A is rea. Given the bock diagona } structure of T, the preconditioned matrix T A has the bock form T A = { A ii 1 A ij ; i, j = 1,..., s. Since each submatrix A ii of A is symmetric, the i-th diagona bock ( A ii 1 A ii ) of T A is equa to sign(a ii ) R ni ni, and Λ(sign(A ii )) = { 1, 1}. By Theorem 3.2, s Λ(T A) G i, where i=1 G i { 1, 1} y R \ { 1, 1} : ( (yi i sign(a ii )) 1 ) 1 s A ii 1 A ij. For any y R \ { 1, 1}, the matrix yi i sign(a ii ) is symmetric and nonsinguar, with eigenvaues y ± 1. Hence, by definition of the spectra norm, we have (3.9) (yi i sign(a ii )) 1 ) 1 = min { y + 1, y 1 }, i = 1,..., s. Next, the submutipicativity of a matrix norm gives s (3.10) A ii 1 A ij A ii 1 s A ij, i = 1,..., s. The definition of the spectra norm impies that A ii 1 = A 1 ii. Thus, from (3.7) and (3.10), we obtain (3.11) s A ii 1 A ij δ, i = 1,..., s. Combining (3.9) and (3.11), we get G i { 1, 1} {y R \ { 1, 1} : min { y + 1, y 1 } δ} = {y R : y + 1 δ} {y R : y 1 δ}, i = 1,..., s. This competes the proof. Theorem 3.3 shows that for symmetric indefinite coefficient matrices, which are we approximated by their bock diagona submatrices with reativey sma bock sizes, absoute vaue preconditioners can be efficienty constructed. Finay, et us again remark that the (bock) diagona preconditioners, which resut from Theorem 3.1 and Theorem 3.3, have been introduced without expicit vaidation of Definition 2.2 of an absoute vaue preconditioner, i.e., without a forma cacuation of constants δ 0 and δ 1 in (2.3). If δ in (3.2) (3.3) or (3.7) (3.8) is sufficienty sma, we sti refer to these preconditioners as absoute vaue preconditioners. In the next (and ast) section, we present a different exampe of the absoute vaue preconditioning.

11 ABSOLUTE VALUE PRECONDITIONING MG absoute vaue preconditioning for a mode probem. Let us consider the foowing rea boundary vaue probem, (4.1) u(x, y) c 2 u(x, y) = f(x, y), (x, y) Ω = (0, 1) (0, 1), u Γ = 0, where = 2 x is the Lapace operator (Lapacian), and Γ denotes the boundary of Ω. Probem (4.1) is a particuar instance of the Hemhotz equation with y2 Dirichet boundary conditions, where c R is a wavenumber. After introducing a uniform grid of the size h in both directions and using the standard 5-point finite-difference stenci to discretize continuous probem (4.1), one obtains the corresponding discrete probem (1.1) of the form (4.2) (L c 2 I)x = b, where the coefficient matrix A = L c 2 I represents a discrete negative Lapace operator L, satisfying the Dirichet boundary condition at the grid points on the boundary, shifted by a scaar c 2 times the identity matrix I. The right-hand side b in (4.2) is the vector of function vaues of f(x, y) cacuated at the grid points (numbered in the exicographica order). The soution of system (4.2) provides an approximation to the soution of the boundary vaue probem (4.1), evauated at the grid points. Beow, we ca (4.2) the mode probem. Assuming that c 2 is different from any eigenvaue of the SPD negative Lapacian L and is greater than its smaest, however ess than its argest, eigenvaue, i.e., λ min (L) < c 2 < λ max (L), where λ min (L) = 2π 2 +O(h 2 ) and λ max (L) = 8h 2 +O(1), we concude that A = L c 2 I is nonsinguar symmetric indefinite. Thus, as an SPD preconditioner T for inear system (4.2) one can choose an absoute vaue preconditioner; see Definition 2.2 with A = L c 2 I. There is a variety of existing preconditioning techniques for inear systems, which are based on the knowedge of the coefficient matrix A; see, e.g., the survey in [2]. Many popuar approaches rey on the sparsity of A, and aim to construct sparse preconditioners, which (partiay) preserve the structure of the coefficient matrix. Since we propose to use the absoute vaue of A to derive a preconditioner, it is interesting to check if A can be as sparse as A, at east after dropping sma eements. Figure 4.1 demonstrates the spy pots of the absoute vaue of the shifted Lapacian from our mode probem (4.2) for different shift vaues, after dropping the entries of magnitude ess than 1% of the argest eement of L c 2 I. We observe that athough the fi increases for a arger shift vaue, the matrix L c 2 I, up to the dropping, remains sparse and structured; moreover, the symmetry and positive definiteness can be preserved for sufficienty sma drop toerances. This brings a hope that methods for constructing sparse preconditioners, e.g., based on incompete factorizations or sparse inverse approximations of A, can be extended to constructing absoute vaue preconditioners, even though A is not expicity known. Beow, however, we do not foow this option, but use the MG approach instead The MG absoute vaue preconditioner. In this section, we combine principes underying (geometric) MG methods, e.g., [6, 33], with the idea of the absoute vaue preconditioning to construct a preconditioner for symmetric indefinite system (4.2) with a sighty or moderatey indefinite operator L c 2 I, i.e., for the mode probem with a reativey sma shift vaue c 2.

12 12 EUGENE VECHARYNSKI AND ANDREW KNYAZEV Shift vaue c 2 = 100 (6 negative eigenvaues) Shift vaue c 2 = 400 (28 negative eigenvaues) Number of nonzeros=1065 (2.1%) Number of nonzeros=6825 (13.5%) Fig Sparsity patterns of L c 2 I after dropping the entries of magnitude ess than 1% of the argest eement; n = 225. Aong with the (fine) grid of the mesh size h, underying discrete probem (4.2), et us consider a (coarse) grid of a mesh size H > h. We denote the discretization of the negative Lapacian on this grid by L H, I H represents the identity operator of the corresponding dimension. Further, we assume that the fine-eve absoute vaue L c 2 I and its inverse are not computabe, whie the inverse of the coarse-eve operator LH c 2 I H can be efficienty constructed, e.g., by the fu eigendecomposition. In the two-grid framework, we use the subscript H to refer to the quantities defined on the coarse grid. No subscript is used for denoting the fine-grid quantities. We suggest the foowing scheme as an exampe of the two-grid absoute vaue preconditioner for mode probem (4.2). Agorithm 4.1 (The two-grid absoute vaue preconditioner). Input r, output w. 1. Presmoothing. Appy ν smoothing steps: (4.3) w (i+1) = w (i) + M 1 (r Lw (i) ), i = 0,..., ν 1, w (0) = 0, where the (nonsinguar) matrix M defines the choice of a smoother, ν 1. Set w pre = w (ν). 2. Coarse grid correction. Restrict the vector r Lw pre to the coarse grid, mutipy it by the inverted coarse-eve absoute vaue L H c 2 I H, and then proongate the resut back to the fine grid. This deivers the coarse grid correction, which is added to w pre to obtain the corrected vector w cgc : (4.4) (4.5) w H = LH c 2 I H 1 R (r Lw pre ), w cgc = w pre + P w H, where P and R are proongation and restriction operators, respectivey. 3. Postsmoothing. Appy ν smoothing steps: (4.6) w (i+1) = w (i) + M (r Lw (i) ), i = 0,..., ν 1, w (0) = w cgc, where M and ν are the same as in step 1. Return w = w post = w (ν). In (4.4) we assume that the coarse-grid operator L H c 2 I H is invertibe, i.e., c 2 is different from any eigenvaue of L H. The number of smoothing steps in (4.3)

13 ABSOLUTE VALUE PRECONDITIONING 13 and (4.6) is the same; the presmoother is defined by the nonsinguar matrix M, whie the postsmoother is deivered by M. Agorithm 4.1 can be viewed as a modified two-grid cyce for inear system L c 2 I z = r. Here, the computationay expensive or unavaiabe absoute vaue of the fine-grid operator, L c 2 I, is repaced by the easiy accessibe negative Lapacian L = L at smoothing steps (4.3) and (4.6), as we as in the restricted residua in (4.4). The operator L c 2 I appears ony at the coarse grid (4.4). If used repeatedy, Agorithm 4.1 does not sove any inear system. However, as seen ater, its use as a preconditioner (aong with the respective MG extension described beow) significanty acceerates the convergence of an iterative scheme appied to mode probem (4.2) with a reativey sma shift vaue; moreover, the convergence is independent of the mesh size. Two-grid Agorithm 4.1 impicity constructs a mapping r w = T tg r, where the operator T = T tg has the foowing structure: (4.7) T tg = ( I M L ) ν P LH c 2 I H 1 R ( I LM 1 ) ν + F, with F = L 1 (I M L) ν L ( 1 I LM 1) ν. The fact that the constructed preconditioner T = T tg is SPD foows directy from the observation that the first term in (4.7) is symmetric positive semidefinite, provided that P = αr for some nonzero scaar α, whie the second term F is symmetric and positive definite, if the spectra radii of I M 1 L and I M L are ess than 1. The atter condition, in fact, requires the pre- and postsmoothing iterations (4.3) and (4.6) to represent convergent methods for system (4.2) with c = 0 and b = r (i.e., for the discrete Poisson s equation) on their own. We note that the above argument for the operator T = T tg to be SPD essentiay repeats the corresponding pattern to justify symmetry and positive definiteness of a two-grid preconditioner appied within an iterative scheme, e.g., the preconditioned conjugate gradient method (PCG), to sove a system of inear equations with an SPD coefficient matrix; see, e.g., [5, 30]. Now et us consider a hierarchy of m + 1 grids numbered by = m, m 1,..., 0 with the corresponding mesh sizes {h } in the decreasing order (h m = h corresponds to the finest, and h 0 to the coarsest, grid). For each eve we define the discretization L c 2 I of the differentia operator in (4.1), where L is the discrete negative Lapacian on grid, and I is the identity of the same size. In order to extend the two-grid absoute vaue preconditioner given by Agorithm 4.1 to the mutigrid, instead of inverting the absoute vaue LH c 2 I H in step 2 (formua (4.4)), we recursivey appy the agorithm to the restricted vector R(r Lw pre ). This pattern is then foowed, in the V-cyce fashion, on a eves, with the exact inversion of the absoute vaue of the shifted discrete negative Lapace operator on the coarsest grid. If started from the finest grid = m, the foowing scheme gives the mutieve extension of the two-grid absoute vaue preconditioner defined by Agorithm 4.1. We note that the subscript is introduced to match the occurring quantities to the corresponding grid. Agorithm 4.2 (AVP-MG(r ): the MG absoute vaue preconditioner). Input r, output w. 1. Presmoothing. Appy ν smoothing steps: (4.8) w (i+1) = w (i) + M 1 (r L w (i) ), i = 0,..., ν 1, w (0) = 0,

14 14 EUGENE VECHARYNSKI AND ANDREW KNYAZEV where the (nonsinguar) matrix M defines the choice of a smoother on eve, ν 1. Set w pre = w (ν). 2. Coarse grid correction. Restrict the vector r L w pre to the grid 1. If = 1, then mutipy the restricted vector by the inverted coarse-eve absoute vaue L0 c 2 I 0, (4.9) w 0 = L 0 c 2 I 0 1 R0 (r 1 L 1 w pre 1 ), if = 1. Otherwise, recursivey appy AVP-MG to the restricted vector, (4.10) w 1 = AVP-MG (R 1 (r L w pre )), if > 1. Proongate the resut back to the fine grid. This deivers the coarse grid correction, which is added to w pre to obtain the corrected vector w cgc : (4.11) w cgc = w pre + P w 1, where w 1 is given by (4.9) (4.10). The operators R 1 and P define the restriction from the eve to 1 and the proongation from the eve 1 to, respectivey. 3. Postsmoothing. Appy ν smoothing steps: (4.12) w (i+1) = w (i) + M (r L w (i) ), i = 0,..., ν 1, w (0) = w cgc, where M and ν are the same as in step 1. Return w = w post = w (ν). The described MG absoute vaue preconditioner impicity constructs a mapping r w = T mg r, where the operator T = T mg has the foowing structure: (4.13) T mg = ( I M L ) ν P T (m 1) mg R ( I LM 1) ν + F, with F as in (4.7) and T (m 1) mg (4.14) T () T (0) defined according to the recursion beow, mg = ( I M ) ν L P T mg ( 1) ( R 1 I L M 1 ) ν + F, mg = L0 c 2 I 0 1, = 1,..., m 1, where F = L 1 ( I M ) ν ( L L 1 I L M 1 ) ν. In (4.13), we skip the subscript in the notation for the quantities associated with the finest eve = m. The structure of the mutieve preconditioner T = T mg in (4.13) is simiar to that of the two-grid preconditioner T = T tg in (4.7), with L H c 2 I H 1 (m 1) repaced by the recursivey defined operator T mg in (4.14). If the assumptions on the fine-grid operators M, M, R and P, sufficient to ensure that the two-grid preconditioner in (4.7) is SPD, remain vaid throughout the coarser eves, i.e., P = αr 1, and the spectra radii of I M 1 L and I M L are ess than 1, = 1,..., m 1, then the symmetry and positive definiteness of the MG preconditioner T = T mg in (4.13) can be obtained from the same property of the two-grid operator through reations (4.14). We remark that preconditioner (4.13) (4.14) is non-variabe, i.e., it can preserve the goba optimaity of an underying sover. We do not theoreticay prove that (2.3), with A = L c 2 I, hods independenty of h, thus formay estabishing that Agorithm 4.2 represents an absoute vaue preconditioner. Instead, we verify (2.3) numericay in Section 4.3.

15 ABSOLUTE VALUE PRECONDITIONING Aternative SPD preconditioners. We consider two known strategies for constructing SPD preconditioners for mode probem (4.2) here. The first one is based on the inverted Lapacian, whie the second one reies on the Bunch-Parett factorization of the coefficient matrix. In Section 4.3, we compare preconditioners, resuting from these approaches, to the absoute vaue preconditioner of Agorithm The inverted Lapacian preconditioner. The absoute vaue preconditioner given by Agorithm 4.2 can be viewed as a modified MG V-cyce for equation L c 2 I z = r, where the absoute vaue of the shifted discrete negative Lapace operator on finer eves is repaced by the computationay inexpensive discrete negative Lapacian. Aternativey, Agorithm 4.2 can be interpreted as a modified MG V-cyce for the discrete Poisson s equation. Specificay, if coarse-eve step (4.9) in Agorithm 4.2 is repaced by (4.15) w 0 = (L 0 ) 1 R 0 (r 1 L 1 w pre 1 ), then one gets the standard V-cyce for the discrete Poisson s equation, which deivers an approximation of L 1. The idea of using an (approximate) inverse of the discrete negative Lapacian L as a preconditioner for the corresponding Hemhotz probem is we known. Introduced in Turke et a. [1], it remains an object of active research; see, e.g., [10, 21, 34]. Since inear system (4.2) formay corresponds to the discrete Hemhotz equation, the strategy based on (approximatey) inverting L represents one of the few estabished approaches to construct SPD preconditioners for (4.2). We use this approach in the numerica experiments of Section Preconditioning based on the Bunch-Parett factorization. This preconditioning is suggested in [14], primariy, in the context of inear systems arising in optimization. It is based on decomposing the matrix A in (1.1) into the product (4.16) A = Q U DUQ, where Q is a permutation, U is upper trianguar, and D is (symmetric) bock diagona with bock sizes 1 or 2. Factorization (4.16) is often referred to as the Bunch-Parett factorization, and is known to exist for any symmetric A; see [7]. Given decomposition (4.16) of the coefficient matrix and the spectra factorization D = W ΘW of the bock diagona term, a perfect SPD preconditioner for probem (1.1) is defined in [14] as (4.17) T = ( Q U DUQ ) 1, D = W Θ W, where the orthogona W and the diagona Θ = diag {θ j } are the matrices of eigenvectors and eigenvaues of D, respectivey; Θ = diag { θ j }. One can check, using (4.16), that the preconditioned matrix T A, with a nonsinguar symmetric indefinite A and SPD T as in (4.17), has exacty two distinct eigenvaues: 1 and 1. Foowing the argument in the proof of Theorem 2.1, this observation aows us to concude that choice (4.17) of the preconditioner T enforces minima residua method (1.3) (1.4) to converge to the exact soution of (1.1) in at most two steps. Thus, aong with the preconditioner T = A 1 introduced in Section 2, expression (4.17) represents an aternative instance of an idea SPD preconditioner for a symmetric indefinite inear system. Both of these approaches guarantee the ocation of a eigenvaues of the preconditioned matrix at 1 and 1, but the corresponding

16 16 EUGENE VECHARYNSKI AND ANDREW KNYAZEV eigenspaces are not the same, i.e., the resuting matrices T A are different for each of the two options. If the coefficient matrix A in (1.1) is very arge, then the computationa cost of Bunch-Parett factorization (4.16) for constructing exact idea preconditioner (4.17) is prohibitive. The strategy chosen in [14] reies on the structure of the underying probem, associating A with its approximation, such that the atter admits an efficient Bunch-Parett factorization. The obtained factors are then used to construct a preconditioner, which can be viewed as an approximation of (4.17). We propose fitting the construction of preconditioners based on Bunch-Parett factorization, i.e., those resembing (4.17), into the same MG framework as in Section 4.1. In particuar, for mode probem (4.2), this can be done by foowing the steps of Agorithm 4.2, with the ony difference that, instead of inverting the matrix absoute vaue on the coarsest grid, one computes the Bunch-Parett factorization of the coarse-eve operator, i.e., L 0 c 2 I 0 = Q 0U 0 D 0 U 0 Q 0, and repaces (4.9) by (4.18) w 0 = ( Q 0U 0 D 0 U 0 Q 0 ) 1 R0 (r 1 L 1 w pre 1 ), D0 = W 0 Θ 0 W 0, where D 0 = W 0 Θ 0 W0 is the eigendecomposition of the bock diagona D 0. Simiar to (4.13) (4.14), it can be verified that the resuting preconditioner is SPD under mid assumptions on the restriction, proongation, and smoothers. In Section 4.3, we compare this preconditioner to the MG absoute vaue and the inverted Lapacian preconditioners described above Numerica experiments. In this section, we numericay examine the MG absoute vaue preconditioner given by Agorithm 4.2. As an underying iterative sover we use the matab impementation of the PMINRES agorithm. Let us highight that the experiments beow aim mainy at testing the preconditioning approach presented in this paper. In our numerica exampes, we consider inear system (4.2) with reativey sma shifts c 2, i.e., the coefficient matrix L c 2 I is ony sighty or moderatey indefinite. As a proof of concept, we provide an actua exampe of an absoute vaue preconditioner for such a probem, however, at the current stage of the research, we do not attempt to dea with inear systems, which are highy indefinite, i.e., where c 2 is reasonaby arge. Indeed, preconditioner in Agorithm 4.2 is designed to hande ony moderatey sma shifts, since the shift appears ony at the coarse grid correction and does not affect the choice of the coarse grid. We eave the question of constructing absoute vaue preconditioners for highy indefinite symmetric inear systems, which resut from known physica appications, for further investigation beyond the scope of this paper. Figure 4.2 iustrates the PMINRES performance appied to mode probem (4.2) with different SPD preconditioners. In particuar, we compare the MG absoute vaue preconditioner in Agorithm 4.2 ( AVP-MG ) to the MG inverted Lapacian preconditioner discussed in Section ( Lapace-MG ) and the MG preconditioner based on the Bunch-Parett factorization suggested in Section ( BP-MG ). Runs corresponding to the unpreconditioned iterative scheme (MINRES) and to PMINRES with the exacty inverted (using matab backsash operator) negative Lapacian as a preconditioner ( Lapace ) are aso presented in the figure. In these tests, inear system (4.2) corresponds to probem (4.1) discretized on the grid of size h = 2 7 (the fine probem size n = (2 7 1) ); c 2 = 100, 200, 300 and 400. The right-hand side vectors b, as we as initia guesses x 0, are randomy chosen (same for each shift vaue).

17 ABSOLUTE VALUE PRECONDITIONING 17 Shift vaue c 2 = 100 (6 negative eigenvaues) Shift vaue c 2 = 200 (13 negative eigenvaues) Eucidean norm of residua No Prec. Lapace Lapace MG BP MG AVP MG Iteration number Eucidean norm of residua No Prec. Lapace Lapace MG BP MG AVP MG Iteration number Shift vaue c 2 = 300 (19 negative eigenvaues) Shift vaue c 2 = 400 (26 negative eigenvaues) Eucidean norm of residua No Prec. Lapace Lapace MG BP MG AVP MG Iteration number Eucidean norm of residua No Prec. Lapace Lapace MG BP MG AVP MG Iteration number Fig Comparison of severa SPD preconditioners for PMINRES appied to the mode probem of the size n = (2 7 1) In the experiments in Figure 4.2, as we as in the tests beow, we define the components for a three of our MG agorithms, i.e., for the absoute vaue preconditioner (Agorithm 4.2), the inverted Lapacian preconditioner (Agorithm 4.2 with (4.9) repaced by (4.15)), and the preconditioner based on the Bunch-Parett factorization (Agorithm 4.2 with (4.9) repaced by (4.18)), as foowing: ω-damped Jacobi iteration as a (pre- and post-) smoother with the damping parameter ω = 4/5, standard coarsening scheme (i.e., h 1 = 2h ), fu weighting for the restriction, and piecewise mutiinear interpoation for the proongation; see, e.g., [6, 33]. Uness otherwise is expicity stated, the coarsest grid is of the size 2 4 (the coarse probem size n 0 = 225). The number of (pre- and post-) smoothing steps is set to one, i.e., ν = 1. We note that for sufficienty rough coarse grids, the cost of the coarse grid computations for a of the considered MG agorithms, i.e., computations in (4.9), (4.15), and (4.18), is negigibe reative to the cost of smoothing and other operations at the fine grid. This means that the costs of the corresponding V-cyces are simiar for a our MG preconditioners tested. Thus, our numerica comparison, based on the number of iterations, is representative. As observed from Figure 4.2, PMINRES with MG absoute vaue preconditioner noticeaby outperforms the method with other SPD preconditioners, which are based on the inverted Lapacian and the Bunch-Parett factorization. We see that, for the mode probem with smaer shifts, the preconditioning approach based on the Bunch-Parett factorization resuts in a reasonabe SPD preconditioner. However, in

18 18 EUGENE VECHARYNSKI AND ANDREW KNYAZEV most cases, it is inferior to a other preconditioners considered in Figure 4.2. This behavior may be reated to the fact that the inverted negative Lapacian T = L 1 and the absoute vaue T = L c 2 I 1 preconditioners share the same eigenvectors with the coefficient matrix A = L c 2 I, whie the perfect preconditioner (4.17), which is based on the Bunch-Parett factorization, does not. Figure 4.2 reveas an interesting phenomenon the convergence history of the PMINRES method preconditioned by Agorithm 4.2 can be ceary separated into the initia reativey sow convergence phase and the second much faster convergence phase. Moreover, the duration of the first, sower, phase approximatey corresponds to the number of negative eigenvaues of A = L c 2 I. PMINRES with MG absoute vaue preconditioner, given by Agorithm 4.2 with (4.9), significanty acceerates in the second phase, whie MINRES with the traditiona MG preconditioner for the Lapacian, given by Agorithm 4.2 with (4.15), does not. Athough we do not report the corresponding tests here, it is aso observed that the increase in the number of smoothing steps (ν = 2) in Agorithm 4.2 sighty improves the quaity of the resuting absoute vaue preconditioner, however, correspondingy affects its computationa cost; see [36]. The same observation appies to the MG preconditioner based on the Bunch-Parett factorization. Number of iterations Performance of the MG absoute vaue preconditioners Coarse probem size 225 Coarse probem size Shift vaue c 2 Fig Performance of the MG absoute vaue preconditioners for the mode probem with different shift vaues. The probem size n = (2 7 1) The number of negative eigenvaues varies from 0 to 75. Figure 4.2 demonstrates that the quaity of the MG absoute vaue preconditioner deteriorates with the increase of the shift vaue. Figure 4.3, which pots the number of PMINRES iterations performed to decrease the 2-norm of the initia error by 10 8 for a given vaue of c 2, resembes the speed of this deterioration. We see that, for arger shift vaues, it may be desirabe to have a grid of a higher resoution on the coarsest eve in Agorithm 4.2. Let us remind the reader that Agorithm 4.2 with formua (4.9) repaced by (4.15) is a standard geometric MG for the Lapacian. Such a sover is known to have optima costs, i.e., ineary proportiona to n. We have aready discussed that Agorithm 4.2 with (4.9) has essentiay the same cost as Agorithm 4.2 with (4.15). Therefore, if, in addition, the number of iterations in the iterative sover preconditioned with Agorithm 4.2 does not depend on the probem size, the overa scheme is optima.