Absolute Value Preconditioning for Symmetric Indefinite Linear Systems

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Absoute Vaue Preconditioning for Symmetric Indefinite Linear Systems Vecharynski, E.; Knyazev, A.V. TR March 2013 Abstract We introduce a nove strategy for constructing symmetric positive definite (SPD) preconditioners for inear systems with symmetric indefinite matrices. The strategy, caed absoute vaue preconditioning, is motivated by the observation that the preconditioned minima residua method with the inverse of the absoute vaue of the matrix as a preconditioner converges to the exact soution of the system in at most two steps. Neither the exact absoute vaue of the matrix nor its exact inverse are computationay feasibe to construct in genera. However, we provide a practica exampe of an SPD preconditioner that is based on the suggested approach. In this exampe we consider a mode probem with a shifted discrete negative Lapacian, and suggest a geometric mutigrid (MG) preconditioner, where the inverse of the matrix absoute vaue 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 MG preconditioner, which eads to a robust iterative scheme with minimaist memory requirements. SIAM Journa of Scientifc Computing This work may not be copied or reproduced in whoe or in part for any commercia purpose. Permission to copy in whoe or in part without payment of fee is granted for nonprofit educationa and research purposes provided that a such whoe or partia copies incude the foowing: a notice that such copying is by permission of Mitsubishi Eectric Research Laboratories, Inc.; an acknowedgment of the authors and individua contributions to the work; and a appicabe portions of the copyright notice. Copying, reproduction, or repubishing for any other purpose sha require a icense with payment of fee to Mitsubishi Eectric Research Laboratories, Inc. A rights reserved. Copyright c Mitsubishi Eectric Research Laboratories, Inc., Broadway, Cambridge, Massachusetts 02139

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3 ABSOLUTE VALUE PRECONDITIONING FOR SYMMETRIC INDEFINITE LINEAR SYSTEMS EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV Abstract. We introduce a nove strategy for constructing symmetric positive definite (SPD) preconditioners for inear systems with symmetric indefinite matrices. The strategy, caed absoute vaue preconditioning, is motivated by the observation that the preconditioned minima residua method with the inverse of the absoute vaue of the matrix as a preconditioner converges to the exact soution of the system in at most two steps. Neither the exact absoute vaue of the matrix nor its exact inverse are computationay feasibe to construct in genera. However, we provide a practica exampe of an SPD preconditioner that is based on the suggested approach. In this exampe we consider a mode probem with a shifted discrete negative Lapacian, and suggest a geometric mutigrid (MG) preconditioner, where the inverse of the matrix absoute vaue 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 MG preconditioner, which eads to a robust iterative scheme with minimaist memory requirements. Key words. Preconditioning, inear system, preconditioned minima residua method, poar decomposition, matrix absoute vaue, mutigrid, poynomia fitering AMS subect cassifications. 15A06, 65F08, 65F10, 65N22, 65N55 1. Introduction. Large, sparse, symmetric, and indefinite systems 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 equations of fuid and soid mechanics; see, e.g., [3] and references therein. In acoustics, arge sparse symmetric indefinite systems are obtained after discretizing the Hemhotz equation for certain media types and boundary conditions. Often the need to sove symmetric indefinite probems comes as an auxiiary task within other computationa routines, such as the inner step in interior point methods in inear and noninear optimization [3, 26], or soution of the correction equation in the Jacobi-Davidson method [36] for a symmetric eigenvaue probem. We consider an iterative soution of a inear system Ax = b, where the matrix A is rea nonsinguar and symmetric indefinite, i.e., the spectrum of A contains both positive and negative eigenvaues. In order to improve the convergence, we introduce a preconditioner T and formay repace Ax = b by the preconditioned system T Ax = T b. If T is propery chosen, an iterative method for this system can exhibit a better convergence behavior compared to a scheme appied to Ax = b. Neither the preconditioner T nor the preconditioned matrix T A is normay expicity computed. If T is not symmetric positive definite (SPD), then T A, in genera, is not symmetric with respect to any inner product [29, Theorem ]. Thus, the introduction of a non-spd preconditioner repaces the origina symmetric probem Ax = b by a generay nonsymmetric T Ax = T b. Speciaized methods for symmetric inear sys- Preiminary posted at This materia is based upon work partiay supported by the Nationa Science Foundation under Grant No The work is partiay based on the PhD thesis of the first coauthor [42]. Computationa Research Division, Lawrence Berkeey Nationa Laboratory, Berkeey, CA (eugene.vecharynski[at]gmai.com) Department of Mathematica and Statistica Sciences; University of Coorado Denver, P.O. Box , Campus Box 170, Denver, CO , USA (andrew.knyazev[at]ucdenver.edu) Mitsubishi Eectric Research Laboratories; 201 Broadway Cambridge, MA and 1

4 2 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV tems are no onger appicabe to the preconditioned probem, and must be repaced by iterative schemes for nonsymmetric inear systems; e.g., GMRES or GMRES(m) [35], Bi-CGSTAB [41], and QMR [17]. The approach based on the choice of a non-spd preconditioner, which eads to soving a nonsymmetric probem, has severa disadvantages. First, no short-term recurrent scheme that deivers an optima Kryov subspace method is typicay avaiabe for a nonsymmetric inear system [15]. In practice, this means that impementations of the optima methods (e.g., GMRES) require an increasing amount of work and storage at every new step, and hence 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 the spectrum of the preconditioned matrix; see the corresponding resuts for GMRES and GMRES(m) in [20, 43]. This makes it difficut to predict computationa costs. If T is chosen to be SPD, i.e., T = T > 0, then the matrix T A of the preconditioned inear system is symmetric with respect to the T 1 inner product defined by (u, v) T 1 = (u, T 1 v) for any pair of vectors u and v. Here (, ) denotes the Eucidean inner product (u, v) = v u, in which the matrices A and T are symmetric. Due to this symmetry preservation, system T Ax = T b can be soved using an optima Kryov subspace method that admits a short-term recurrent impementation, such as preconditioned MINRES (PMINRES) [14, 28]. Moreover, the convergence of the method can be fuy estimated in terms of the spectrum of T A. In ight of the above discussion, the choice of an SPD preconditioner for a symmetric indefinite inear system can be regarded as natura and favorabe, especiay if corresponding non-spd preconditioning strategies fai to provide convergence in a sma number of iterations. We advocate the use of SPD preconditioning. The question of constructing SPD preconditioners for symmetric indefinite systems has been widey studied in many appications. For sadde point probems, the bock-diagona SPD preconditioning has been addressed, e.g., in [16, 37, 44]. In [2], it was proposed to use an inverse of the negative Lapacian as an SPD preconditioner for indefinite Hemhotz probems. This approach was further extended in [25] by introducing a shift into the preconditioner. Another strategy was suggested in [18], primariy in the context of inear systems arising in optimization. It is based on the so-caed Bunch-Parett factorization [9]. We introduce here a different idea of constructing SPD preconditioners that resembe the inverse of the absoute vaue of the coefficient matrix. Throughout, the absoute vaue of A is defined as a matrix function A = V Λ V, where A = V ΛV is the eigenvaue decomposition of A. We are motivated by the observation that PMIN- RES with A 1 as a preconditioner converges to the exact soution in at most two steps. We refer to the new approach as the absoute vaue (AV) preconditioning and ca the corresponding preconditioners the AV preconditioners. The direct approach for constructing an AV preconditioner is to approximatey sove A z = r. However, A is generay not avaiabe, which makes the appication of standard techniques, such as, e.g., incompete factorizations, approximate inverses, probematic. The vector A 1 r can aso be found using matrix function computations, normay fufied by a Kryov subspace method [19, 23] or a poynomia approximation [30, 31]. Our numerica experience shows that the convergence, with respect to the outer iterations, of a inear sover can be significanty improved with this approach, but the computationa costs of approximating f(a)r = A 1 r may be

5 ABSOLUTE VALUE PRECONDITIONING 3 too high, i.e., much higher than the cost of matrix-vector mutipication with A. Introduction of the genera concept of the AV preconditioning is the main theoretica contribution of the present work. As a proof of concept exampe of the AV preconditioning, we use a geometric mutigrid (MG) framework. To investigate appicabiity and practica effectiveness of the proposed idea, we choose a mode probem resuting from discretization of a shifted Lapacian (Hemhotz operator) on a unit square with Dirichet boundary conditions. The obtained inear system is rea symmetric indefinite. We construct an MG AV preconditioner that, used in the PMINRES iteration, deivers an efficient computationa scheme. Let us remark that the same mode probem has been considered in [4], where the authors utiize the coarse grid approximation to reduce the indefinite probem to the SPD system. Satisfactory resuts have been reported for sma shifts, i.e., for sighty indefinite systems. However, the imitation of the approach ies in the requirement on the size of the coarse space, which shoud be chosen sufficienty arge. As we show beow, the MG AV preconditioner presented in this paper aows keeping the coarsest probem reasonaby sma, even if the shift is arge. Numerica soution of Hemhotz probems is an obect of active research; see, e.g., [1, 6, 12, 13, 21, 27, 40]. A typica Hemhotz probem is approximated by a compex symmetric (non-hermitian) system. The rea symmetric case of the Hemhotz equation, considered in this paper, is ess common. However, methods for compex probems are evidenty appicabe to our particuar rea case, which aows us to make numerica comparisons with known Hemhotz sovers. We test severa of sovers, based on the inverted Lapacian and the standard MG preconditioning, to compare with the proposed AV preconditioning. In fact, the inverted (shifted) Lapacian preconditioning [2, 25] for rea Hemhotz probems can be viewed as a specia case of our AV preconditioning. In contrast to preconditioners in [18] reying on the Bunch-Parett factorization, we show that the AV preconditioners can be constructed without any decompositions of the matrix, which is crucia for very arge or matrix-free probems. This paper is organized as foows. In Section 2, we present and ustify the genera notion of an AV preconditioner. The rest of the paper deas with the question of whether AV preconditioners can be efficienty constructed in practice. In Section 3, we give a positive answer by constructing an exampe of a geometric MG AV preconditioner for the mode probem. The efficiency of this preconditioner is demonstrated in our numerica tests in Section 4. We concude in Section AV preconditioning for symmetric indefinite systems. Given an SPD preconditioner T, we consider soving a inear system with the preconditioned minima residua method, impemented in the form of the preconditioned MINRES (PMINRES) agorithm [14, 28]. In the absence of round-off errors, at step i, the method constructs an approximation x (i) to the soution of Ax = b of the form (2.1) 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 (2.2) 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), the T -norm is defined by

6 4 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV v 2 T = (v, v) T for any v, and x (0) is the initia guess. Scheme (2.1) (2.2) represents an optima Kryov subspace method and the PMINRES impementation is based on a short-term recurrence. The conventiona convergence rate bound for (2.1) (2.2) can be found, e.g., in [14], and reies soey on the distribution of eigenvaues of T A. The foowing trivia, but important, theorem regards A 1 as an SPD preconditioner for a symmetric indefinite system. Theorem 2.1. The preconditioned minima residua method (2.1) (2.2) with preconditioner T = A 1 converges to the soution of Ax = b in at most two steps. Theorem 2.1 impies that T = A 1 is an idea SPD preconditioner. Note that the theorem hods not ony for the preconditioned minima residua method (2.1) (2.2), but for a methods where convergence is determined by the degree of the minima poynomia of T A. 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 inexpensive SPD preconditioners that resembe A 1 and can significanty acceerate the convergence of an iterative method. Definition 2.2. We ca an SPD preconditioner T for a symmetric indefinite inear system Ax = b an AV preconditioner if it satisfies (2.3) δ 0 (v, T 1 v) (v, A v) δ 1 (v, T 1 v), v with constants δ 1 δ 0 > 0, such that the ratio δ 1 /δ 0 1 is reasonaby sma. Let us remark that Definition 2.2 of the AV preconditioner is informa because 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 approximates A 1, up to a positive scaing. If A represents a hierarchy of mesh probems then it is desirabe that δ 1 /δ 0 is independent of the probem size. In this case, if A is SPD, Definition 2.2 of the AV preconditioner is consistent with the we known concept of spectray equivaent preconditioning for SPD systems; see [10]. The foowing theorem provides bounds for eigenvaues of the preconditioned matrix T A in terms of the spectrum of T A. We note that T and A, and thus T A and T A, do not in genera commute. Therefore, our spectra anaysis cannot be based on a traditiona matrix anaysis too, a basis of eigenvectors. Theorem 2.3. Given a nonsinguar symmetric indefinite A R n n and an SPD T R n n, 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 +1 λ µ p +1, = 1,..., p; µ p λ µ, = p + 1,..., n. Proof. We start 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.

7 ABSOLUTE VALUE PRECONDITIONING 5 Let us fix any index {1, 2,..., n}, and denote by S an arbitrary subspace of R n such that dim(s) =. Since inequaity (2.5) aso hods on S, using (2.6) we write (2.7) ψ(v) φ(v) ψ(v), v S. Moreover, taking the maxima in vectors v S, and after that the minima in subspaces S S = {S R n : dim(s) = }, of a parts of (2.7) preserves the inequaities, so (2.8) min max S S v S ( ψ(v)) min max S S v S φ(v) min S S max v S ψ(v). By the Courant-Fischer theorem (see, e.g., [24, 29]) for the Rayeigh quotients ±ψ(v) and φ(v) defined in (2.6), we concude from (2.8) that µ n +1 λ µ. Recaing that has been arbitrariy chosen, we obtain the foowing bounds on the eigenvaues of T A: (2.9) µ n +1 λ < 0, = 1,..., p; 0 < λ µ, = p + 1,..., n. Next, in order to derive nontrivia upper and ower bounds for the p negative and n p positive eigenvaues λ in (2.9), we use the fact that eigenvaues ξ and ζ 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 {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) in (2.12) aows us to concude from (2.8) that ξ n +1 ζ ξ. Given the arbitrary choice of in the above inequaity, by (2.10) (2.11) we get the foowing bounds on the eigenvaues of T A: (2.13) 1/µ p +1 1/λ < 0, = 1,..., p; 0 < 1/λ 1/µ p, = p + 1,..., n.

8 6 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV Combining (2.9) and (2.13), we obtain (2.4). Theorem 2.3 suggests two usefu impications given by the corresponding coroaries beow. In particuar, the foowing resut describes Λ(T A), i.e., the spectrum of the preconditioned matrix T A, in terms of δ 0 and δ 1 in (2.3). Coroary 2.4. Given a nonsinguar symmetric indefinite A R n n, an SPD T R n n, and constants δ 1 δ 0 > 0 satisfying (2.3), we have (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 = 1, p, p + 1, n. The next coroary 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. Coroary 2.5. Given a nonsinguar symmetric indefinite A R n n and an SPD T R n n, 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, 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). Coroary 2.4 impies that the ratio δ 1 /δ 0 1 of the constants from (2.3) measures the quaity of the AV preconditioner T. Indeed, the convergence speed of the preconditioned minima residua method is determined by the spectrum of T A, primariy by the intervas of the right-hand side of incusion (2.14). Additionay, Coroary 2.5 prompts that a good AV 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 PMINRES. In the next section, we construct an exampe of the AV preconditioner for a particuar mode probem. We appy the MG techniques. 3. MG AV preconditioning for a mode probem. Let us consider the foowing rea boundary vaue probem, (3.1) u(x, y) c 2 u(x, y) = f(x, y), (x, y) Ω = (0, 1) (0, 1), u Γ = 0, where = 2 / x / y 2 is the Lapace operator and Γ denotes the boundary of Ω. Probem (3.1) is a particuar instance of the Hemhotz equation with Dirichet boundary conditions, where c > 0 is a wave number. After introducing a uniform grid of size h in both directions and using the standard 5-point finite-difference stenci to discretize continuous probem (3.1), one obtains the corresponding discrete probem (3.2) (L c 2 I)x = b, where A L c 2 I represents a discrete negative Lapacian L (ater caed Lapacian ), satisfying the Dirichet boundary condition shifted by a scaar c 2. The common rue of thumb, see, e.g., [13, 22], for discretizing (3.1) is (3.3) ch π/5.

9 ABSOLUTE VALUE PRECONDITIONING 7 Beow, we ca (3.2) the mode probem. We assume that the shift c 2 is different from any eigenvaue of the Lapacian and is greater than the smaest but ess than the argest eigenvaue. Thus, the matrix L c 2 I is nonsinguar symmetric indefinite. In the foowing subsection, we appy the idea of the AV preconditioning to construct an MG AV preconditioner for system (3.2). Whie our main focus throughout the paper is on the 2D probem (3.1), in order to simpify presentation of theoretica anaysis, we aso refer to the 1D anaogue (3.4) u (x) c 2 u(x) = f(x), u(0) = u(1) = 0. The concusions drawn from (3.4), however, remain quaitativey the same for the 2D probem of interest, which we test numericay Two-grid AV preconditioner. Aong with the fine grid of mesh size h underying probem (3.2), et us consider a coarse grid of mesh size H > h. We denote the discretization of the Lapacian on this grid by L H, and I H represents the identity operator of the corresponding dimension. We assume that the exact fine-eve absoute vaue L c 2 I and its inverse are not computabe, whereas the inverse of the coarse-eve operator LH c 2 I H can be efficienty constructed. 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. Whie L c 2 I is not avaiabe, et us assume that we have its SPD approximation B, i.e., B L c 2 I and B = B > 0. The operator B can be given in the expicit matrix form or through the action on a vector. We suggest the foowing genera scheme as a two-grid AV preconditioner for mode probem (3.2). Agorithm 3.1 (The two-grid AV preconditioner). Output: w. 1. Presmoothing. Appy ν smoothing steps, ν 1: Input: r, B L c 2 I. (3.5) w (i+1) = w (i) + M 1 (r Bw (i) ), i = 0,..., ν 1, w (0) = 0, where M defines a smoother. Set w pre = w (ν). 2. Coarse grid correction. Restrict (R) r Bw pre to the coarse grid, appy LH c 2 I H 1, and proongate (P ) to the fine grid. This deivers the coarse grid correction, which is added to w pre : (3.6) (3.7) w H = L H c 2 I H 1 R (r Bw pre ), w cgc = w pre + P w H. 3. Postsmoothing. Appy ν smoothing steps: (3.8) w (i+1) = w (i) + M (r Bw (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 (3.6) we assume that LH c 2 I H is nonsinguar, i.e., c 2 is different from any eigenvaue of L H. The presmoother is defined by the nonsinguar M, whie the postsmoother is deivered by M. Note that the (inverted) absoute vaue appears ony on the coarse grid, whie the fine grid computations are based on the approximation B. It is immediatey seen that if B = L c 2 I, Agorithm 3.1 represents a forma two-grid cyce [8, 39] for system (3.9) L c 2 I z = r.

10 8 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV Note that the introduced scheme is rather genera in that different choices of approximations B and smoothers M ead to different preconditioners. We address these choices in more detai in the foowing subsections. It can be verified that the AV preconditioner given by Agorithm 3.1 impicity constructs a mapping r w = T tg r, where the operator is (3.10) T tg = ( I M B ) ν P L H c 2 I H 1 R ( I BM 1 ) ν + F, with F = B 1 (I M B) ν B ( 1 I BM 1) ν. The fact that the constructed preconditioner T = T tg is SPD foows directy from the observation that the first term in (3.10) is SPD provided that P = αr for some nonzero scaar α, whie the second term F is SPD if the spectra radii of I M 1 B and I M B are ess than 1. The atter condition requires the pre- and postsmoothing iterations (3.5) and (3.8) to represent convergent methods for By = r. Note that the above argument essentiay repeats the one used to ustify symmetry and positive definiteness of a preconditioner based on the standard two-grid cyce for an SPD system; see, e.g., [5, 38]. In this paper we consider two different choices of the approximation B. The first choice is given by B = L, i.e., it is suggested to approximate the absoute vaue L c 2 I by the Lapacian L. The second choice is deivered by B = p m (L c 2 I), where p m is a poynomia of degree at most m such that p m (L c 2 I) L c 2 I Agorithm 3.1 with B = L. If B = L, Agorithm 3.1 can be regarded as a step of a standard two-grid method [8, 39] appied to the Poisson equation (3.11) Ly = r, modified by repacing the operator L H by L H c 2 I H on the coarse grid. The question remains if the agorithm deivers a form of an approximate sove for absoute vaue probem (3.9), and hence is suitabe for AV preconditioning of (3.2). To be abe to answer this question, we anayze the propagation of the initia error e AV 0 = L c 2 I 1 r of (3.9) under the action of the agorithm. We start by reating errors of (3.9) and (3.11). Lemma 3.1. Given a vector w, consider errors e AV (w) = L c 2 I 1 r w and e P (w) = L 1 r w for (3.9) and (3.11), respectivey. Then (3.12) e AV (w) = e P (w) + (c 2 I W p )L 1 L c 2 I 1 r, where W p = 2V p Λ p V p, V p is the matrix of eigenvectors of L c 2 I corresponding to the p negative eigenvaues λ 1... λ p < 0, and Λ p = diag { λ 1,..., λ p }. Proof. Observe that for any w, e AV (w) = L c 2 I 1 r w = L c 2 I 1 r + (e P (w) L 1 r) = e P (w) + L c 2 I 1 L 1 (L L c 2 I )r. Denoting A = L c 2 I, we use the expression A = A 2V p Λ p V p to get (3.12) Agorithm 3.1 transforms the initia error e P 0 = L 1 r of equation (3.11) into (3.13) e P = S ν 2 KS ν 1 e P 0, where S 1 = I M 1 L and S 2 = I M L are pre- and postsmoothing operators, K = I P L H c 2 I H 1 RL corresponds to the coarse grid correction step, and e P = L 1 r w post. Denoting the error of absoute vaue system (3.9) after appying

11 ABSOLUTE VALUE PRECONDITIONING 9 Agorithm 3.1 by e AV = L c 2 I 1 r w post and observing that e P 0 = L c 2 I L 1 e AV 0, by (3.12) (3.13) we obtain (3.14) e AV = ( S ν 2 KS ν 1 L c 2 I + c 2 I W p ) L 1 e AV 0. The ast expression gives an expicit form of the desired error propagation operator, which we denote by G: (3.15) G = ( S ν 2 KS ν 1 L c 2 I + c 2 I W p ) L 1. Beow, as a smoother, we use a simpe Richardson s iteration, i.e., S 1 = S 2 = I τl, where τ is an iteration parameter. The restriction R is given by the fu weighting and the proongation P by the standard piecewise inear interpoation; see [8, 39]. At this point, in order to simpify further presentation, et us refer to the onedimensiona anaogue (3.4) of mode probem (3.1). In this case, the matrix L is tridiagona: L = tridiag { 1/h 2, 2/h 2, 1/h 2}. We assume that n, the number of interior grid nodes, is odd: h = 1/(n + 1). The coarse grid is then obtained by dropping the odd-numbered nodes. We denote the size of the coarse grid probem by N = (n + 1)/2 1; H = 1/(N + 1) = 2h. The tridiagona matrix L H denotes the discretization of the 1D Lapacian on the coarse eve. Reca that the eigenvaues of L are = 4 h sin 2 πh 2 2 with corresponding eigenvectors v = 2h [sin πh] n =1. Simiary, the eigenvaues of L H are θ H = 4 H sin 2 πh 2 2, and the coarse grid eigenvectors are denoted by v H = 2H [sin πh] N =1. It is cear that operators L c 2 I and L H c 2 I H have the same sets of eigenvectors as L and L H with eigenvaues t = c 2 and t H = θ H c 2, respectivey. Let e AV 0 = n =1 α v be the expansion of the initia error in the eigenbasis of L. Since e AV = Ge AV 0 = n =1 α (Gv ), we are interested in the action of the error propagation operator (3.15) on the eigenmodes v. The action of the operators R and P on v and v H, respectivey, is we known; see, e.g., [8, pp ]. Thus, it is easy to obtain the foowing expression for Kv : (3.16) Kv = ( ) 1 c 4 t H v + s 2 c 2 t H v n+1, = 1,..., N, v(, ) = N + 1, 1 c 4 t H n+1 v + s 2 c 2 t H n+1, = N + 2,..., n. n+1 vh Here, c = cos πh 2 and s = sin πh 2. Since v are the eigenvectors of S 1 = S 2 = I τl, L c 2 I, L 1 and W p, (3.15) eads to expicit expressions for Gv. Theorem 3.2. Let c 2 < θ N+1 = 2/h 2. Then the error propagation operator G in (3.15) acts on the eigenvectors v of 1D Lapacian as foows: (3.17) Gv = g (11) v + g (12) v n+1, = 1,..., N, g v, = N + 1, g (21) v + g (22) v n+1, = N + 2,..., n,

12 10 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV where (3.18) (3.19) (3.20) (3.21) (3.22) g (11) = (1 τ ) (1 2ν c 4 t H g (12) g (21) ) t = (1 τ ) ν s 2 c 2 t t H (1 τθ n+1 ) ν, g = (1 τ ) 2ν t + c2, = (1 τ ) (1 2ν c 4 t H n+1 g (22) ) + c2 β, t + c2, = (1 τ ) ν s 2 c 2 t t H n+1 (1 τθ n+1 ) ν ; and β = { 2(c 2 ), < c 2, 0, > c 2. Theorem 3.2 impies that for reativey sma shifts, Agorithm 3.1 with B = L and a proper choice of τ and ν reduces the error of (3.9) in the directions of amost a eigenvectors v. In a few directions, however, the error may be ampified. These directions are given by the smooth eigenmodes associated with that are cose to c 2 on the right, as we as with that are distant from c 2 on the eft. The number of the atter, if any, is sma if ch is sufficienty sma, and becomes arger as ch increases. Indeed, et τ = h 2 /3, so that 1 τ < 1 for a and 1 τ < 1/3 for > N. This choice of the parameter provides the east uniform bound for 1 τ that correspond to the osciatory eigenmodes [34, p.415]. It is then readiy seen that (3.19) and (3.22) can be made arbitrariy sma within a reasonaby sma number ν of smoothing steps. Simiary, (3.20) and (3.21) can be made arbitrariy cose to c 2 / < 1. If c 2 << θ N+1, then c 2 / in (3.20) and (3.21) is cose to zero. Thus, Theorem 3.2 shows that for reativey sma shifts, smoothing provides sma vaues of (3.19) (3.22) and, hence, damps of the osciatory part of the error. Note that the damping occurs even though the smoothing is performed with respect to (3.11), not (3.9). Now et us consider (3.18). Theorem 3.2 shows that if c 2 is cose to an eigenvaue θ H of the coarse-eve Lapacian, i.e., if t H 0, then the corresponding reduction coefficient (3.18) can be arge. This means that Agorithm 3.1 with B = L has a potentia difficuty of ampifying the error in the directions of a few smooth eigenvectors. Simiar effect is known to appear for standard MG methods appied to Hemhotz type probems; see [7, 13]. Beow, we anayze (3.18) in more detai. Let > c 2. Then, using the reation θ H = c 2, we can write (3.18) as ) ( ) g (11) = (1 τ ) (1 2ν c c 2 /θ H 1 c2 + c2. Here, it is easy to see that as c 2 /θ H 0, g (11) (1 τ ) 2ν s 2 < 1/2, meaning that the smooth eigenmodes corresponding to away from c 2 on the right are we damped. If < c 2, then (3.18) takes the form g (11) = (1 τ ) 2ν ( c 2 / c 2 c4 c 2 / c 2 ) (c ) (2 c2 ).

13 ABSOLUTE VALUE PRECONDITIONING 11 Since c 2 (1/2, 1), for any c2 / > 1, we can obtain the bound c 2 / 2 c 2 / 1 c2 / c 2 c4 c 2 / c 2 Additionay, 3 2ν < (1 τ ) 2ν < 1. Thus, < g (11) < 3(c2 / ) 1 4(c 2 / ) 2, c2 / 3/4 c 2 / 1/2. where = 0 if 1 < c 2 / 2, and = 2 c 2 / if c 2 / > 2. The inequaity impies that g (11) < 1 for 1 < c 2 / 3, i.e., the agorithm reduces the error in the directions of severa smooth eigenvectors associated with to the eft of c 2. At the same time, we note that as c 2 /, g (11), i.e., the smooth eigenmodes corresponding to that are distant from c 2 on the eft can be ampified. Ceary, if ch is sufficienty sma then the number of such error components is not arge (or none), and grows as ch increases. The above anaysis shows that Agorithm 3.1 with B = L indeed represents a sove for (3.9), where the soution is approximated everywhere, possiby except for a subspace of a sma dimension. In the context of preconditioning, this transates into the fact that the preconditioned matrix has spectrum custered around 1 and 1 with a few outiers generated by the ampification of the smooth eigenmodes. If the shift is sufficienty sma, the number of such outiers is not arge, which ony sighty deays the convergence of the outer PMINRES iterations and does not significanty affect the efficiency of the overa scheme Agorithm 3.1 with B = p m (L c 2 I). The anaysis of the previous subsection suggests that the quaity of Agorithm 3.1 with B = L may deteriorate as ch increases. This resut is not surprising, since for arger ch the reation L L c 2 I becomes no onger meaningfu. Beow we introduce a different approach for approximating the fine grid absoute vaue. In particuar, we consider constructing poynomia approximations B = p m (L c 2 I), where p m (λ) is a poynomia of degree at most m > 0, such that p m (L c 2 I) L c 2 I. Let us first refer to the idea particuar case, where p m (L c 2 I) = L c 2 I. This can happen, e.g., if p m (λ) is an interpoating poynomia of f(λ) = λ on the spectrum of L c 2 I, m = n 1. In such a situation, Agorithm 3.1 with B = p m (L c 2 I) resuts in the foowing transformation of the initia error: (3.23) e AV = S ν 2 K S ν 1 e AV 0, where S 1 = I M 1 L c 2 I and S 2 = I M L c 2 I are pre- and postsmoothing operators, and K = I P L H c 2 I H 1 R L c 2 I corresponds to the coarse grid correction step. The associated error propagation operator is further denoted by Ḡ, (3.24) Ḡ = S ν 2 K S ν 1. For the purpose of carity, we again consider the 1D counterpart (3.4) of the mode probem. As a smoother, we choose Richardson s iteration with respect to absoute vaue system (3.9), i.e., S 1 = S 2 = I τ L c 2 I. It is important to note here that the eigenvaues t of the absoute vaue operator are, in genera, no onger ascendingy ordered with respect to as is the case for s and t s. Moreover, in contrast to

14 12 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV L and L c 2 I, the top part of the spectrum of L c 2 I may be associated with both smooth and osciatory eigenmodes. In particuar, this means that Richardson s iteration may fai to propery eiminate the osciatory components of the error, which is an undesirabe outcome of the smoothing procedure. To avoid this, we require that t 1 < t N+1. It is easy to verify that the atter condition is fufied if (3.25) ch < 1. Note that (3.25) automaticay hods if discretization rue (3.3) is enforced. Repeating the above argument for the 2D case aso eads to (3.25). Let the restriction and proongation operators R and P be the same as in the previous subsection. Simiar to (3.16), we obtain an expicit expression for the action of the coarse grid correction operator K on eigenvectors v : ( ) 1 c 4 t t H v + s 2 c 2 t t H v n+1, = 1,..., N, (3.26) Kv = v(, ) = N + 1, 1 c 4 t t H n+1 v + s 2 c 2 t t H n+1, = N + 2,..., n. n+1 vh The foowing theorem is the anaogue of Theorem 3.2. Theorem 3.3. The error propagation operator Ḡ in (3.24) acts on the eigenvectors v of the 1D Lapacian as foows: ḡ (11) v + ḡ (12) v n+1, = 1,..., N, (3.27) Ḡv = ḡ v, = N + 1, ḡ (21) v + ḡ (22) v n+1, = N + 2,..., n, where (3.28) (3.29) (3.30) (3.31) (3.32) ) = (1 τ t ) (1 2ν c 4 t t H, ḡ (11) ḡ (12) ḡ (21) = (1 τ t ) ν s 2 c 2 t t H (1 τ t n+1 ) ν, ḡ = (1 τ t ) 2ν, ) = (1 τ t ) (1 2ν c 4 t t H n+1, ḡ (22) = (1 τ t ) ν s 2 c 2 t t H n+1 (1 τ t n+1 ) ν. We concude from Theorem 3.3 that in the idea case where p m (L c 2 I) = L c 2 I, Agorithm 3.1 with B = p m (L c 2 I) and a proper choice of τ and ν reduces the error of system (3.9) in the directions of a eigenvectors v, possiby except for a few that correspond to cose to the shift c 2. Unike in the case of Agorithm 3.1 with B = L, as ch grows, no ampified error components appear in the directions of eigenvectors associated with distant from c 2 on the eft. This suggests that Agorithm 3.1 with B = p m (L c 2 I) = L c 2 I provides a more accurate sove for (3.9) with arger ch.

15 ABSOLUTE VALUE PRECONDITIONING 13 To see this, et us first assume that τ = h 2 /(3 c 2 h 2 ). Since (3.25) impies that t = t for > N, this choice is known to give the smaest uniform bound on 1 τ t corresponding to the osciatory eigenmodes v, which is 1 τ t < 1/(3 ch) < 1/2 with the ast inequaity resuting from (3.25). Hence, coefficients (3.29) (3.32) can be reduced within a reasonaby sma number ν of smoothing steps. Next, we note that (3.28), which is not substantiay affected by smoothing, can be arge if c 2 is cose to θ H, i.e., if th 0. At the same time, we can write (3.28) as ) g (11) = (1 τ t ) 2ν (1 c c2 s 2 which shows that g (11) approaches (1 τ t ) 2ν s 2 < 1/2 as th increases, i.e., smooth error components associated with away from c 2 are we damped. Thus, if used as a preconditioner, Agorithm 3.1 with B = p m (L c 2 I) = L c 2 I aims at custering the spectrum of the preconditioned matrix around 1 and 1, with a few possibe outiers that resut from the ampification of the smooth eigenmodes associated with cose to c 2. Unike in the case where B = L, the increase of ch does not additionay ampify the smooth error components distant from c 2 on the eft. Therefore, Agorithm 3.1 with B = p m (L c 2 I) = L c 2 I can be expected to provide a more accurate preconditioner for arger shifts. Athough our anaysis targets the idea but barey feasibe case where p m (L c 2 I) = L c 2 I, it motivates the use of poynomia approximations p m (L c 2 I) L c 2 I and provides a theoretica insight into the superior behavior of such an option for arger ch. In the rest of this subsection we describe a method for constructing such poynomia approximations. Our approach is based on the finding that the probem is easiy reduced to constructing poynomia fiters. We start by introducing the step function { 1, λ α, h α (λ) = 0, λ < α; where α is a rea number, and noting that sign(λ) = 2h 0 (λ) 1, so that (3.33) L c 2 I = ( 2h 0 (L c 2 I) I ) ( L c 2 I ). Here h 0 (L c 2 I) = V h 0 (Λ)V, where V is the matrix of eigenvectors of L c 2 I and h 0 (Λ) = diag{0,..., 0, 1,..., 1} is obtained by appying the step function h 0 (λ) to the diagona entries of the matrix Λ of the associated eigenvaues. Ceary the number of zeros on the diagona of h 0 (Λ) equas the number of negative eigenvaues of L c 2 I. Let q m 1 (λ) be a poynomia of degree at most (m 1), such that q m 1 (λ) approximates h 0 (λ) on the interva [a, b], where a and b are the ower and upper bounds on the spectrum of L c 2 I, respectivey. In order to construct an approximation p m (L c 2 I) of L c 2 I, we repace the step function h 0 (L c 2 I) in (3.33) by the poynomia q m 1 (L c 2 I). Thus, (3.34) L c 2 I p m (L c 2 I) = ( 2q m 1 (L c 2 I) I ) ( L c 2 I ). The matrix L c 2 I is readiy avaiabe on the fine grid. Therefore, we have reduced the probem of evauating the poynomia approximation p m of the absoute vaue operator to constructing a poynomia q m 1 that approximates the step t H,

16 14 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV function h 0. More specificay, since Agorithm 3.1 can be impemented without the expicit knowedge of the matrix B, i.e., B can be accessed ony through its action on a vector, we need to construct approximations of the form q m 1 (L c 2 I)v to h 0 (L c 2 I)v, where v is a given vector. The task of constructing q m 1 (L c 2 I)v h 0 (L c 2 I)v represents an instance of poynomia fitering, which is we known; see, e.g., [11, 32, 45]. In this context, due to the property of fitering out certain undesirabe eigencomponents, the step function h 0 is caed a fiter function. The approximating poynomia q m 1 is referred to as a poynomia fiter. State-of-the-art poynomia fitering techniques such as [32] woud first repace the discontinuous step function h 0 (λ) by a smooth approximation on [a, b] and then approximate the atter by a poynomia in the east-squares sense. In this paper, we foow a simper approach based on the direct approximation of h 0 (λ) using Chebyshev poynomias [30, 31]. The constructed poynomia q m 1 aows defining q m 1 (L c 2 I)v h 0 (L c 2 I)v and hence p m (L c 2 I)v L c 2 I v. Thus, the entire procedure provides means to repace a matrix-vector product with the unavaiabe L c 2 I by, essentiay, a few mutipications with L c 2 I. As we further show, the degree m of the approximating poynomia can be kept reasonaby ow. Moreover, in the MG framework discussed in the next subsection, the agorithm has to be invoked ony on sufficienty coarse grids The MG AV preconditioner. Now et us consider a hierarchy of s + 1 grids numbered by = s, s 1,..., 0 with the corresponding mesh sizes {h } in decreasing order (h s = h corresponds to the finest grid, and h 0 to the coarsest). For each eve we define the discretization L c 2 I of the differentia operator in (3.1), where L is the Lapacian on grid, and I is the identity of the same size. In order to extend the two-grid AV preconditioner given by Agorithm 3.1 to the mutigrid, instead of inverting the absoute vaue LH c 2 I H in (3.6), we recursivey appy the agorithm to the restricted vector R(r Bw pre ). This pattern is then foowed in the V-cyce fashion on a eves, with the inversion of the absoute vaue of the shifted Lapacian on the coarsest grid. The matrix B on eve is denoted by B. Each B is assumed to be SPD and is expected to approximate L c 2 I. In the previous subsections we have considered two choices of B for the two-grid preconditioner in Agorithm 3.1. In the MG framework, these choices give B = L and B = p m (L c 2 I ), where p m is a poynomia of degree at most m on eve. The advantage of the first option, B = L, is that it can be easiy constructed and the appication of B to a vector is inexpensive even if the size of the operator is very arge. According to our anaysis for the 1D mode probem in subsection 3.2, the approach is suitabe for ch sufficienty sma. Typicay this is a case for corresponding to finer grids. However, ch increases with every new eve. This may resut in the deterioration of accuracy of the overa MG preconditioning scheme, uness the size of the coarsest eve is kept sufficienty arge. The situation is different for the second option B = p m (L c 2 I ). In this case, appications of B may be expensive on finer grids because they require a sequence of matrix-vector mutipications with arge shifted Lapacian operators. However, on coarser eves, i.e., for arger ch, this is not restrictive because the invoved operators are significanty decreased in size compared to the finest eve. Additionay, as suggested by the anaysis in subsection 3.3, if p m (L c 2 I ) represent reasonabe approximations of L c 2 I on eves, one can expect a higher accuracy of the whoe preconditioning scheme compared to the choice B = L.

17 ABSOLUTE VALUE PRECONDITIONING 15 Our idea is to combine the two options. Let δ (0, 1) be a switching parameter, where for finer grids ch < δ. We choose { L, ch (3.35) B = < δ, p m (L c 2 I ), ch δ. The poynomias p m (L c 2 I ) are accessed through their action on a vector. Summarizing our discussion, if started from the finest grid = s, the foowing scheme gives the mutieve extension of the two-grid AV preconditioner defined by Agorithm 3.1. The subscript is introduced to match quantities to the corresponding grid. We assume that the parameters δ, m, ν, and the smoothers M are pre-specified. Agorithm 3.2 (AV-MG(r ): the MG AV preconditioner). Input r. Output w. 1. Set B by (3.35). 2. Presmoothing. Appy ν smoothing steps, ν 1: (3.36) w (i+1) = w (i) + M 1 (r B w (i) ), i = 0,..., ν 1, w (0) = 0, where M defines a smoother on eve. Set w pre = w (ν). 3. Coarse grid correction. Restrict (R 1 ) r B w pre to the grid 1, recursivey appy AV-MG, and proongate (P ) back to the fine grid. This deivers the coarse grid correction added to w pre : { L0 (3.37) w 1 = c 2 I 0 1 R0 (r 1 B 1 w pre 1 ), = 1, AV-MG (R 1 (r B w pre )), > 1; (3.38) w cgc = w pre + P w Postsmoothing. Appy ν smoothing steps: (3.39) w (i+1) = w (i) + M (r B w (i) ), i = 0,..., ν 1, w (0) = w cgc, where M and ν are the same as in step 2. Return w = w post = w (ν ). The described MG AV preconditioner impicity constructs a mapping denoted by r w = T mg r, where the operator T = T mg has the foowing structure: (3.40) T mg = ( I M B ) ν P T (s 1) mg R ( I BM 1) ν + F, with F as in (3.10) and T (s 1) mg (3.41) T () T (0) defined according to the recursion mg = ( I M ) ν B P T mg ( 1) ( R 1 I B M 1 ) ν + F, mg = L0 c 2 I 0 1, = 1,..., s 1, where F = B 1 ( I M ) ν ( B B 1 I B M 1 ) ν. The structure of the mutieve preconditioner T = T mg in (3.40) is the same as that of the two-grid preconditioner T = T tg in (3.10), with LH c 2 I H 1 repaced by the recursivey defined operator T mg (m 1) in (3.41). Thus, the symmetry and positive definiteness of T = T mg foows from the same property of the two-grid operator through reations (3.41), provided that P = α R 1 and the spectra radii of I

18 16 EUGENE VECHARYNSKI AND ANDREW V. KNYAZEV M 1 B and I M B are ess than 1 throughout the coarser eves. We remark that preconditioner (3.40) (3.41) is non-variabe, i.e., it preserves the goba optimaity of PMINRES. The simpest possibe approach for computing w 0 in (3.37) is to expicity construct L 0 c 2 I 0 1 through the fu eigendecomposition of the coarse-eve Lapacian, and then appy it to R 0 (r 1 B 1 w pre 1 ). An aternative approach is to determine w 0 as a soution of the inear system (L 0 c 2 I 0 + 2V 0 Λ 0 V0 )w 0 = R 0 (r 1 B 1 w pre 1 ), where V 0 is the matrix of eigenvectors associated with the negative eigenvaues of L 0 c 2 I 0 contained in the corresponding diagona matrix Λ 0. In the atter case, the fu eigendecomposition of L 0 is repaced by the partia eigendecomposition targeting negative eigenpairs, foowed by a inear sove. Since we use Richardson s iteration with respect to p m (L c 2 I ) as a smoother on coarser grids, as motivated by the discussion in subsection 3.3, the guidance for the choice of the coarsest grid is given by condition (3.25). More specificay, in the context of the standard coarsening procedure (h 1 = 2h ), we seect hierarchies of grids satisfying ch < 1 for = s,..., 1, and ch 0 > 1. As shown in the next section, even for reasonaby arge c 2, the coarsest-eve probems are sma. The parameter δ in (3.35) shoud be chosen to ensure the baance between computationa costs and the quaity of the MG preconditioner. In particuar, if δ is reasonaby arge then the choices of B are dominated by the option B = L, which is inexpensive but may not be suitabe for arger shifts on coarser eves. On the other extreme, if δ is cose to zero then the common choice corresponds to B = p m (L c 2 I ), which provides a better preconditioning accuracy for arger shifts but may be too computationay intense on finer eves. In our numerica experiments, we keep δ [1/3, 3/4]. As we demonstrate in the next section, the degrees m of the occurring poynomias p m shoud not be arge, i.e., ony a few matrix-vector mutipications with L c 2 I are required to obtain satisfactory approximations of absoute vaue operators. For propery chosen δ, these additiona mutipications need to be performed on grids that are significanty coarser than the finest grid, i.e., the invoved matrices L c 2 I are orders of magnitude smaer than the origina fine grid operator. As confirmed by our numerica experiments, the overhead caused by the poynomia approximations appears to be margina and does not affect much the computationa cost of the overa preconditioning scheme. 4. Numerica experiments. This section presents a numerica study of the MG preconditioner in Agorithm 3.2. Our goa here is twofod. On the one hand, the reported numerica experiments serve as a proof of concept of the AV preconditioning described in Section 2. In particuar, we show that the AV preconditioners can be constructed at essentiay the same cost as the standard preconditioning methods (MG in our case). On the other hand, we demonstrate that the MG AV preconditioner in Agorithm 3.2 combined with the optima PMINRES iteration, in fact, eads to an efficient and economica computationa scheme, further caed MINRES-AV-MG, which outperforms severa known competitive approaches for the mode probem. Let us briefy describe the aternative preconditioners used for our comparisons. Throughout, we use matab for our numerica exampes. The inverted Lapacian preconditioner. This strategy, introduced in [2], is a representative of an SPD preconditioning for mode probem (3.2), where the preconditioner is appied through soving systems Lw = r, i.e., T = L 1. As has been previousy discussed, for reativey sma shifts c 2, the Lapacian L constitutes

19 ABSOLUTE VALUE PRECONDITIONING 17 a good SPD approximation of L c 2 I. In this sense, the choice T = L 1 perfecty fits, as a specia case, into the genera concept of the AV preconditioning presented in Section 2. We refer to PMINRES with T = L 1 as MINRES-Lapace. Usuay, one wants to sove the system Lw = r ony approximatey, i.e., use T L 1. This can be efficienty done, e.g., by appying the V-cyce of a standard MG method [8, 39]. In our tests, however, we perform the exact soves using the matab s backsash, so that the reported resuts refect the best possibe convergence with the inverted Lapacian type preconditioning. The indefinite MG preconditioner. We consider a standard V-cyce for probem (3.2). Formay, it can be obtained from Agorithm 3.2 by setting B = L c 2 I on a eves and repacing the first equaity in (3.37) by the inear sove with L 0 c 2 I 0. The resuting MG scheme is used as a preconditioner for restarted GMRES and for Bi-CGSTAB. We refer to these methods as GMRES(k)-MG and Bi-CGSTAB-MG, respectivey; k denotes the restart parameter. A thorough discussion of the indefinite MG preconditioning for Hemhotz probems can be found, e.g., in [13]. Tabe 4.1 The argest probem sizes satisfying ch δ for different vaues of the shift c 2, switching parameters δ, and the standard coarsening scheme h 1 = 2h. The ast row (δ = 1) corresponds to the sizes of the coarsest probems for different c 2. c 2 = 300 c 2 = 400 c 2 = 1500 c 2 = 3000 c 2 = 4000 δ = 1/ δ = 1/ δ = 3/ δ = In our tests, we consider 2D mode probem (3.2) corresponding to (3.1) discretized on the grid of size h = 2 8 (the fine probem size n = 65025). The exact soution x and the initia guess x 0 are randomy chosen. The right-hand side b = (L c 2 I)x, which aows evauating the actua errors aong the steps of an iterative method. A the occurring MG preconditioners are buit upon the standard coarsening scheme (i.e., h 1 = 2h ), restriction is based on the fu weighting, and proongation on piecewise mutiinear interpoation [8, 39]. Let us reca that Agorithm 3.2 requires setting a parameter δ to switch between B = L and p m (L c 2 I) on different eves; see (3.35). Assuming standard coarsening, Tabe 4.1 presents the argest probem sizes corresponding to the condition ch δ for a few vaues of δ and c 2. In other words, given δ and c 2, each ce of Tabe 4.1 contains the argest probem size for which the poynomia approximation of L c 2 I is constructed. Uness otherwise expicity stated, we set δ = 1/3. Note that according to the discussion in subsection 3.4 (condition (3.25)), the row of Tabe 4.1 corresponding to δ = 1 deivers the sizes n 0 of the coarsest probems for different shift vaues. Tabe 4.1 shows that the coarsest probems remain reativey sma even for arge shifts. The poynomia approximations are constructed for coarser probems of significanty reduced dimensions, which in practica appications are negigiby sma compared to the origina probem size. As a smoother on a eves of Agorithm 3.2 we use Richardson s iteration, i.e., M 1 τ I. On the finer eves, where B = L, we choose τ = h 2 /5 and ν = 1. On the coarser eves, where B = p m (L c 2 I ), we set τ = h 2 /(5 c2 h 2 ) and ν = 5.