c 2011 Society for Industrial and Applied Mathematics BOOTSTRAP AMG

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1 SIAM J. SCI. COMPUT. Vo. 0, No. 0, pp c 2011 Society for Industria and Appied Mathematics BOOTSTRAP AMG A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Abstract. We deveop an agebraic mutigrid (AMG) setup scheme based on the bootstrap framework for mutiscae scientific computation. Our approach uses a weighted east squares definition of interpoation, based on a set of test vectors that are computed by a bootstrap setup cyce and then improved by a mutigrid eigensover and a oca residua-based adaptive reaxation process. To emphasize the robustness, efficiency, and fexibiity of the individua components of the proposed approach, we incude extensive numerica resuts of the method appied to scaar eiptic partia differentia equations discretized on structured meshes. As a first test probem, we consider the Lapace equation discretized on a uniform quadriatera mesh, a probem for which mutigrid is we understood. Then, we consider various more chaenging variabe coefficient systems coming from covariant finite-difference approximations of the two-dimensiona gauge Lapacian system, a commony used mode probem in AMG agorithm deveopment for inear systems arising in attice fied theory computations. Key words. bootstrap agebraic mutigrid, east squares interpoation, mutigrid eigensover, adaptive reaxation AMS subject cassification. 65F10, 65N55, 65F30 DOI / Introduction. We deveop bootstrap agebraic mutigrid (BAMG) techniques for soving systems (1.1) Au = f, where A C n n is assumed to be Hermitian and positive definite. AMG approaches for soving (1.1) typicay invove a stationary inear iterative method (smoother), appied to the fine-grid system, and a coarse-grid correction. The corresponding twogrid method gives rise to the error propagation operator (1.2) E TG = (I MA)(I P(P H AP) 1 P H A)(I MA), where P : C nc C n with n c < n is the interpoation operator, M is the approximate inverse of A that defines the smoother, and for any matrix B C n m, B H denotes the conjugate transpose. A mutigrid agorithm is then obtained by recursivey soving the coarse-grid probem, invoving A c = P H AP, using the two-grid method. Submitted to the journa s Methods and Agorithms for Scientific Computing section March 16, 2009; accepted for pubication (in revised form) December 13, 2010; pubished eectronicay DATE. Department of Mathematics, University of Caifornia Los Angees, Los Angees, CA (abrandt@math.uca.edu). Department of Mathematics, Pennsyvania State University, University Park, PA (brannick@psu.edu). This author s work was supported by the Nationa Science Foundation under grants OCI and DMS Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wupperta, D Wupperta, Germany (kkah@math.uni-wupperta.de). This author s work was supported by the Deutsche Forschungsgemeinschaft through the Coaborative Research Centre SFB-TR 55 Hadron physics from Lattice QCD. Department of Mathematica Sciences, Ba State University, Muncie, IN (iivshits@bsu. edu). This author s work was supported by DOE subcontract B and NSF DMS

2 2 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS The efficiency of such an approach depends on the proper interpay between the smoother and the coarse-grid correction. Typicay, the AMG smoothing operator, M, is fixed and the coarse-grid correction is formed to compensate for its deficiencies. The primary task in defining a mutigrid method is then the seection of the sequence of interpoation operators. A genera two-grid process for constructing P is described by the foowing generic agorithm: 1. Given the set of n variabes on the fine grid, choose a set of n c coarse variabes such that n c < n. 2. Choose a sparsity pattern for interpoation, P C n nc. 3. Define the weights of the interpoation operator, i.e., the entries of P. Cassica AMG as originay introduced in [4, 5] and its widey used impementation by Ruge and Stüben [20] can be seen as important miestones in the deveopment of such agebraic setup agorithms. The Ruge Stüben agorithm exhibits optima efficiency for many chaenging probems, often substantiay outperforming traditiona iterative methods. However, the agorithm and, hence, its effectiveness depend on the foowing probem-specific properties: The notion of strength of connection used in coarsening variabes and forming interpoation can be defined from the entries of the system matrix. The eigenvectors with sma in absoute vaue eigenvaues (owest eigenvectors) are ocay smooth in directions of such strong connections. These owest eigenvector(s) of the system matrix provide a sufficienty accurate oca representation of the other ow eigenvectors not effectivey treated by the MG reaxation scheme. For probems where a or some of these properties are vioated, efficiency of Ruge Stüben AMG deteriorates. This oss in efficiency is often due to the ow accuracy of interpoation for vectors yieding sma normaized residuas [1], i.e., vectors x such that Ax 0. For simpe pointwise reaxation schemes, such as the Gauss Seide iteration we use in our tests, these components coincide with the error not effectivey reduced by the reaxation scheme and, hence, are often referred to as the agebraicay smooth components of the error. The bootstrap framework for mutiscae scientific computation [3] provides genera techniques for anaysis, deveopment, and practica appication of robust mutieve methods. In the context of MG agorithms, the bootstrap process 1 aows the user to input any features of the probem at hand (e.g., nested grids and kerne components) and then use this knowedge to iterativey improve itsef unti optima performance is achieved. The genera setup agorithm used in this scheme incorporates severa practica toos and measures derived from the evoving MG sover, incuding the foowing: compatibe reaxation, used in identifying suitabe smoothing schemes and coarse variabe sets [2, 8, 12, 14]; oca weighted east squares approximations of a set of test vectors, used in a greedy agorithm to determine an agebraic distance-based measure of strength of connection [3] and to define AMG interpoation [14, 17, 19]; the bootstrap cycing scheme, used to compute sufficienty accurate sets of test vectors [14, 15]; 1 Generay, the term bootstrap refers to the idea that a process coud better evove by improving the process used for its improvement (thus obtaining a compounding effect over time.) As a computing term, bootstrapping (from an od expression pu onesef up by one s bootstraps ) has been used since at east 1958 to refer to a technique by which a simpe computer program activates a more compex system of programs that utimatey ead to a sef-sustaining process that proceeds without externa hep from manuay entered instructions.

3 BOOTSTRAP AMG 3 adaptive reaxation [2, 14] and amost zero modes [3], used to improve both the AMG setup cyce and the sover. Many of the bootstrap ideas themseves are appicabe to a wide range of mutiscae probems in computationa science. Our focus in this paper is on combining severa of these techniques to deveop a robust scheme for computing AMG interpoation. A preiminary form of the bootstrap process for defining AMG interpoation first appeared in [5], where the use of reaxation appied to the homogeneous system to produce a singe prototype to somehow define cassica AMG interpoation was discussed. More recenty, such bootstrap setup agorithms were deveoped for smoothed aggregation mutigrid (adaptive SA; see [9]), eement-free AMG (see [22, pp ]), and cassica AMG (adaptive AMG, see [7, 10]). The main new ingredient in these more recent BAMG approaches is the idea to appy the current AMG sover to the homogeneous system to both test its performance and improve the prototypes used in computing the sequence of AMG interpoation operators. The agorithm we consider here combines a BAMG cyce with a weighted east squares form of interpoation. An additiona feature unique to our approach is the use of an MG eigensover derived from the existing MG structure to enhance the prototypes needed in the definition of east squares interpoation. We mention that simiar MG techniques for using eigenvectors to define AMG interpoation were roughy outined in [18, 20]. In another reated work [11], an MG eigensover was deveoped to compute an initia SA hierarchy, after which it is abandoned and the usua adaptive SA setup process is invoked. An outine of the remainder of this paper is as foows. First, in section 2, we present the weighted east squares process for computing interpoation and the idea of adaptive reaxation. In addition, we derive sufficient conditions guaranteeing the uniqueness of the soution to the east squares probem and compute an expicit form of the minimizer. Our approach for computing the set of test vectors using bootstrap techniques is described in section 3. Then, in section 4, we present resuts of the method appied to the scaar Lapace equation discretized on a uniform mesh and severa chaenging variabe coefficient probems. We end with concuding remarks in section Least squares interpoation. The basic idea of the east squares (LS) interpoation approach is to approximate a set of test vectors, V = {v (1),..., v (k) } C n, minimizing the interpoation error for these vectors in an LS sense. In the context considered here, namey, appying the LS process to construct a cassica AMG form of interpoation, each row of P, denoted by p i, is defined as the minimizer of a oca LS functiona: For each i F find p i such that (2.1) L(p i ) = k κ=1 ω κ v (κ) {i} (p i ) j v (κ) {j} j C i 2 min, where C i C with C and F = Ω \ C denoting the coarse-grid and fine-grid variabes, respectivey. Here, the notation v Ω denotes the canonica restriction of the vector v to the set Ω Ω := {1,.., n}. In the definition of L in (2.1), for exampe, v {i} is simpy the ith entry of v. Simiary, we can define the canonica restriction of a matrix, V = ( v (1) v (k)), to a set Ω: V Ω = ( v (1) Ω ) v (k), Ω

4 4 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS where V Ω C Ω k. The weights ω κ > 0 can be chosen to refect the energy (e.g., in A-norm v A = Av, v ) of the test vectors. We give our specific choice in the numerica experiments section Uniqueness of the soution to the oca LS probem and an expicit form of its minimizer. To derive conditions on the uniqueness of the soution to minimization probem (2.1) and compute an expicit form of the minimizer, we consider a cassica inear agebra formuation of the LS probem. Let W = and V Ω = (v(1) Ω ω 1... k κ=1 ω k v (k) ). Then, Ω ω κ v (κ) C k k, V = ( v (1) v (k)) C n k, {i} (p i ) j v (κ) {j} j C i 2 = V {i} W 1 2 pi V Ci W , where the weights of the individua terms in the LS functiona are now represented by a scaing defined by the matrix W. An equivaent LS formuation of (2.1) is then (2.2) L(p i ) = V {i} W 1 2 pi V Ci W min. The minimizer of L in (2.2) and necessary conditions for its uniqueness are now easy to compute. The derivative of the LS functiona L with respect to (p i ) j is Setting L(p i ) = 0 yieds (p i ) j L(p i ) = 2 k κ=1 ω κ ( v (κ) {i} p iv (κ) C i ) ( v (κ) C i )j. p i V Ci WV H C i = V {i} WV H C i. Thus, if rank(v Ci W 1 2) = C i, then V Ci WVC H i is nonsinguar and the soution to the LS minimization probem is uniquey defined by p i = V {i} WV H C i ( VCi WV H C i ) 1. We note that if the restricted vectors V Ci form a basis for the oca inear space C ni, n i = C i, then the soutions to the oca LS minimization probems are unique. This in turn suggests setting a ower bound on the number of vectors, k, used in the LS fit: k max i F C i =: c. Further, as we show numericay in section 4, the accuracy of the LS interpoation operator and, hence, the performance of the resuting sover generay improve with increasing k, up to some vaue proportiona to c. Our numerica experience suggests that the number of test vectors need not be arger than 2c to obtain a sufficienty

5 BOOTSTRAP AMG 5 accurate P. In fact, these sets of interpoation points can often be adequatey chosen by natura considerations. For exampe, they can be chosen as the sets of geometrica neighbors with i in their convex hu. If a chosen set is inadequate, the LS procedure wi show bad fitness (arge interpoation errors, i.e., vaues of the LS functiona) and the set must then be improved. The LS procedure can aso be used to detect variabes in the sets C i that can be discarded without significant accuracy oss. Thus, this approach aows creating interpoation with whatever needed accuracy, which is as sparse as possibe. Note that it is aso possibe to adjust the number of test vectors on the fy if rank deficiency of any of the operators V Ci is detected. Simiary, many rows of P wi typicay have fewer than c nonzero entries, that is, C i < c. In such cases, fewer prototypes may be used in the LS process. We do not pursue this idea in this paper because it is not expected to resut in a significant reduction in setup costs, and further, as our numerica resuts show, the quaity of the sover generay improves with an increased number of prototypes Equivaence of adaptive reaxation and a modified LS functiona. A main assumption of cassica AMG is that the reaxation scheme used in the MG agorithm efficienty reduces the residua when appied to the current approximation. This assumption is in fact centra to the definition of cassica AMG interpoation, which is derived by setting the oca (pointwise) residua to zero. In [2], the idea of appying additiona oca reaxations to the equations i F for which the corresponding vaue of the residua is arge was proposed as a possibe approach for improving the performance of cassica AMG for certain probems (e.g., probems with singuarities). Assuming a ii 0, a Jacobi version of this iteration reads as (2.3) v (κ) {i} = v(κ) {i} 1 r (κ) a {i}, ii with r (κ) = Av (κ). Simiar approaches can aso be empoyed for more genera reaxation schemes. In [14], it was observed that the impicit appication of the oca Jacobi reaxation in (2.3) to the prototypes used in the LS definition of interpoation is equivaent to an operator-induced form of LS interpoation constructed using an eement-free AMG type approach (see [22]). This equivaence of the LS approach with an additiona oca reaxation step was aso formuated and discussed in a sighty different scope in [17], where it was defined as a residua-based LS fit, (2.4) L(p i ) = k ω κ v (κ) {i} 1 r (κ) {i} a (p i ) j v (κ) {j} ii j C i κ=1 2 min, which in turn was shown to be consistent with a cassica AMG operator-induced form of LS interpoation. We mention that the work in [17] focuses on defining different versions of the LS interpoation operators using a set of reaxed test vectors; it does not address the question of how to use the bootstrap process to compute these vectors. Deveoping the bootstrap setup scheme for defining LS interpoation is the main focus and contribution of the work we present here. Another interesting observation regarding the above residua form of the LS functiona is that the eement-free AMG and cassica AMG forms of LS interpoation considered in [14, 17] differ from the origina definition ony when the test vectors with arge weights, ω k, are not sufficienty accurate; i.e., the residuas they produce

6 6 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS are not uniformy cose to zero. In fact, our idea of appying adaptive reaxation to the test vector with argest weight (2.4) and then ony to the equations with arge reative vaues of the residua was motivated by these observations. We incude resuts and additiona discussion of both approaches in the numerica experiments section. 3. The bootstrap agorithm. In its simpest form, the bootstrap process for computing the test vectors used in constructing the LS interpoation operators proceeds by appying reaxation to the homogeneous system, (3.1) A x = 0, on each grid, where = 0,..., L 1; assuming that a priori knowedge of the agebraicay smooth error is not avaiabe, these vectors are initiaized randomy on the finest grid, whereas on a coarser grids they are defined by restricting the test vectors computed on the previous finer grid. Given interpoation, the coarse-grid operators are computed using the variationa definition. Once an initia MG hierarchy has been computed, the current sets of prototypes are further enhanced on a grids using the existing mutigrid structure. Specificay, the given hierarchy is used to formuate a mutigrid eigensover which is then appied to an appropriatey chosen generaized eigenprobem to compute additiona test vectors. This overa process is then repeated with the current AMG method repacing reaxation as the sover for the homogeneous systems in (3.1). Severa subte detais must be addressed when formuating the mutigrid eigensover (MGE), namey, (1) the specific formuation for the MGE hierarchy as we as the sorting and fitering of the coarsest-grid eigenvectors within the cyce; (2) deciding on an efficient cycing strategy when integrating the MGE into the BAMG process; and (3) measuring and improving the accuracy of the test vectors computed by the MGE (eigenapproximations) as they are transferred to increasingy finer grids. We describe our formuation of the MGE next Defining the MGE hierarchy: The generaized eigenvaue probem. Given the computed BAMG hierarchy of operators A = A 0, A 1,..., A L and their corresponding interpoation operators P+1, = 0,..., L 1, define the composite interpoation operators (3.2) P = P 0 1 P 1, = 1,..., L and the sequence of subspaces of C n spanned by their coumns as For any vector x C n we then have C n range(p 1 ) range(p L ). x, x A = P x, P x A, where n is the probem size on grid. Defining T = P H P, it foows that (3.3) x, x A x, x T = P x, P x A P x, P x 2, impying that on any grid, given a vector w C n and λ R, such that (3.4) A w = λ T w,

7 BOOTSTRAP AMG 7 we have (3.5) RQ(P w ) = P w, P w A P w, P w 2 = λ. The resut foows from (3.3) appied to w, the eigenvector of the generaized eigenvaue probem (3.4), and the reations A = P H AP and T = P H P. Note that RQ( ) is simpy the Rayeigh quotient and, hence, (3.5) reates the eigenpairs on different grids and can be used to define an MGE. Our agorithm begins by computing the k e eigenvectors with the smaest eigenvaues of the coarsest-grid operator W L = {w (κ) L A Lw (κ) L = λ(κ) L T Lw (κ) L, λ(κ) L R, κ = 1,..., k e}. Since the size of A L is sma, these eigenpairs, ( w (κ) L, ) λ(κ) L, are computed directy. Then, on any given grid, the existing interpoation operator P 1 is used to transfer these vectors to the next finer grid. We note that if w is a soution to (3.4) on grid, then ( P 1 ) H A 1 P 1 w = λ ( P 1 ) H T 1 P 1 w, and so w 1 = P 1 w is an approximate soution to the generaized eigenvaue probem. Next, a smoothing iteration is appied to the homogeneous probem on grid 1: (3.6) (A 1 λ 1 T 1 )w 1 = 0, λ 1 = λ. Then, the approximation to λ 1 is recomputed: λ 1 = A 1w 1, w 1 2 T 1 w 1, w 1 2. We note that this procedure resembes an inverse Rayeigh quotient iteration found in eigenvaue computations (see [23]), with inversion repaced by severa reaxation steps. Agorithm 1 describes one variant of the MGE process for enriching the sets of test vectors Integrating the MGE and BAMG processes: Cycing strategies. Combining the bootstrap cyce with the MGE can be done using a variety of cycing strategies. The two types of cycing schemes we consider are outined in Figure 3.1. The top pot is a visuaization of two successive iterations of a V -cyce scheme, which we refer to as a doube V -cyce (or V 2 -cyce). In genera, a V µ -cyce denotes an agorithm that uses µ such V -cyces. We note that this scheme makes use of the MGE as defined by Agorithm 1, which does not update interpoation unti it reaches the finest grid. The pot in the bottom of the figure outines our second approach, in which the approximations produced by the MGE on grid are used to recompute the hierarchy at each of the coarser grids, +1, + 2,..., L, before advancing to the next finer grid 1. Note that it may not be necessary to recompute the hierarchy at each of the intermediate grids; in practice we recompute P and update the hierarchy ony when reaxation appied to the interpoated eigenvector approximations significanty reduces at east one of the Rayeigh quotients (3.5) associated with these approximations by some prescribed toerance.

8 8 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Agorithm 1. for = L,..., 1 do if the current grid is the coarsest grid L then Take W L = {w (κ) L A Lw (κ) L = λ(κ) L T Lw (κ) L, κ = 1,..., k e} ese Given W +1 and P+1 from the initia setup w (κ) = P +1 w(κ) +1, λ(κ) for κ = 1,..., k e do Reax on end for end if end for Cacuate λ (κ) ( A λ (κ) T ) w (κ) = 0 = λ (κ) +1, κ = 1,..., k e = A w (κ), w (κ) 2 T w (κ), w (κ) 2 {Mutigrid eigensover} Reax on Av = 0,v V Reax on (A λt)w = 0,w W Compute W, s.t., Aw = λtw,w W Reax on Av = 0,v V and (A λt)w = 0,w W Fig Bootstrap AMG V 2 -cyce and W-cyce setup schemes Updating the MG eigendecomposition. Consider the generaized eigenvaue probem A j w j = λ j T j w j, on two subsequent grids, j =, 1, with λ denoting the Rayeigh quotient of a given approximation of an eigenvector on the coarser grid and λ 1 denoting the Rayeigh quotient of the vector obtained by appying reaxation to this vector interpoated to

9 BOOTSTRAP AMG 9 the next finer one. Define the eigenvaue approximation measure τ (, 1) λ (3.7) τ (, 1) λ = λ λ 1. λ 1 At any stage of the MGE iteration, a arge vaue of τ (, 1) λ indicates that the hierarchy shoud be recomputed to incorporate this reaxed eigenvector approximation. Otherwise, reaxation has not significanty changed this vector, and it is accuratey represented by the existing interpoation operator P 1. In this way, the MGE serves as a technique for efficienty identifying components that must be interpoated accuratey (e.g., the ow modes of A) and for determining if the current P approximates them sufficienty we. 4. Numerica resuts. In this section, we present numerica tests of our BAMG- MGE setup agorithm appied to a series of test probems. We consider isotropic systems defined on two-dimensiona (2D) equidistant quadriatera meshes. Fixing the coarse grids and the sparsity pattern of interpoation, we study the performance of the LS and MGE techniques for computing the entries in P. We begin with tests for the Lapace operator discretized using finite eements (FE Lapace) and finite differences (FD Lapace). We then transition to a sighty more difficut test probem of symmetric diagona scaings of the Lapace operator. Then, we proceed to tests of our method appied to the gauge Lapacian system (see Appendix A). In a tests, the sets of coarse variabes are defined by fu coarsening; that is, the coarse grids are obtained by doubing the mesh spacing in each spatia dimension of the reated finer mesh; the coarsening is continued unti the probem size on the coarsest grid is 7 7 or 8 8, depending on the size of the probem on the finest mesh. We imit interpoation to the nearest neighbors (in terms of the graph of the matrix), and the maxima number of interpoatory points for each i F is thus bounded by four for the probems we consider (see Figure 4.1). We use the weighted LS approach to define the entries of the interpoation operators, with the weights defined by (4.1) ω κ = T v (κ), v (κ) A v (κ), v (κ), and test the origina LS definition of interpoation (LS interpoation) as we as the approach that appies additiona adaptive oca reaxations to the test vectors prior to computing the interpoation operator (LSR interpoation). For the resuts reported in Tabe 4.10, we appy the oca reaxation scheme (2.3) to every test vector; for a other tests we appy oca reaxation ony to the singe test vector with the argest weight, ω max, and then ony to 20% of the F-points for which the associated vaue of the residua is argest in absoute vaue. Let k r = V denote the number of test vectors computed using reaxation in the bootstrap cyce, et k e = W denote the number of additiona eigenvector approximations computed by the MGE, and et η be the number of Gauss Seide (GS) iterations used in the BAMG cyce and MGE to compute them. The LS and LSR forms of interpoation are then computed on each eve using the combined sets of k = k r + k e test vectors. On the finest grid, the vectors, v (1),..., v (kr), used to initiaize the bootstrap process are generated randomy with a norma distribution with expectation zero and variance one, N(0, 1); on a other grids they are defined by restricting the reaxed test vectors from the reated finer grid. by

10 10 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS i C i F Fig Interpoation reations for our choice of fu-coarsening for i F \ C. We use a V (2, 2) MG sover with GS reaxation for a tests. The reported estimates of the asymptotic convergence rates are computed as foows: ρ = eν A e ν 1 A, where e ν is the error after ν MG iterations. The sover terminates if the method reduces the initia error by a factor 10 8 or if ν = 100 iterations are appied and the method fais to converge to this toerance Lapace s equation. We start the numerica experiments by appying the BAMG cyce and MGE to the 2D Lapace equation u xx u yy = 0, (x, y) (0, 1) 2, u(x, y) = 0, (x, y) Γ, discretized using biinear finite eements (see [21]) on an (N 1) (N 1) equidistant quadriatera mesh. The corresponding discretization yieds the symmetric and positive definite operator A given by the stenci A = For this probem our choices of fu coarsening and sparsity pattern of interpoation yied variationa coarse-grid operators with at most nine nonzero entries per row, impying that the sparsity structure of the finest-grid operator A is preserved on a grids. The resuts reported in Tabe 4.1 are for the LS- and LSR-based two-grid methods appied to the FE Lapace system on a grid. Overa, we observe that increasing the number of test vectors, k r, used in defining P and increasing the number of smoothing iterations, η, used to compute them both improve the performance (reduce the convergence rates) of the resuting two-grid sovers. Further, the resuts for the LSR scheme are consistenty better than the resuts for the LS approach. In Tabe 4.2, we report resuts obtained by appying a two-grid method to the FE Lapace system for different probem sizes. The sover is constructed using the LS and LSR formuations for defining P with a fixed numbers of test vectors k r = 8 and smoothing iterations η = 4 independent of the choice of probem size. Here, we see

11 BOOTSTRAP AMG 11 Tabe 4.1 Asymptotic convergence rate estimates, ρ, of the LS- (LSR)-based two-grid methods appied to the FE Lapace probem. The sover is constructed using various choices of η and k r. k r η (.962).973 (.958).971 (.955).967 (.949).963 (.944) (.866).916 (.827).904 (.806).878 (.786).861 (.759) (.679).803 (.637).775 (.606).734 (.566).709 (.535) (.524).681 (.453).648 (.403).608 (.269).575 (.316) (.374).553 (.326).518 (.287).484 (.165).457 (.138) (.294).451 (.242).425 (.205).381 (.154).305 (.136) (.227).364 (.186).346 (.155).268 (.089).257 (.098) (.169).308 (.140).287 (.123).238 (.054).216 (.061) Tabe 4.2 Asymptotic convergence rate estimates, ρ, of the LS- (LSR)-based two-grid methods appied to the FE Lapace probem. The sover is constructed using the LS (LSR) schemes in a V -cyce setup with η = 4 and k r = 8. N ρ.267 (.075).648 (.403).886 (.758).966 (.917).990 (.977) that the convergence rates of both the LS and LSR based two-grid methods increase as the size of the probem is increased, suggesting that appying a constant number of reaxation steps to compute a fixed number of test vectors does not produce a sufficienty accurate oca representation of the agebraicay smooth error for defining LS interpoation. In Tabe 4.3, we present resuts for the two-grid method obtained by adding the constant vector, 1, to V, again for a fixed number of test vectors k r = 7+1 and η = 4 reaxation steps used to compute them. Here, we see that incuding the constant vector in V significanty improves the LS- and LSR-based two-grid sovers. In fact, the method obtained using the LSR approach performs optimay for the probem sizes considered. This demonstrates the fexibiity of the LS formuation and its abiity to incorporate a priori knowedge of the ow modes of the probem at hand. It aso motivates the use of the MGE in the BAMG process for computing test vectors, as a way to improve their approximation of ow eigenmodes. Tabe 4.3 Asymptotic convergence rate estimates, ρ, of the LS- (LSR)-based two-grid methods appied to the FE Lapace probem. The sover is constructed using the LS (LSR) schemes using a V -cyce setup with η = 4 GS iterations to compute the seven initiay random test vectors; the additiona test vector is the constant vector chosen a priori. N ρ.089 (.039).121 (.040).130 (.042).148 (.042).150 (.043) Next, we consider using the V 2 -cyce setup (see Figure 3.1). We appy η = 4 iterations of the smoother to k r = 8 initiay random test vectors to generate the MG hierarchy and η = 4 reaxation steps to the k e = 8 additiona test vectors generated by the MGE. As the resuts reported in Tabe 4.4 show, using the MGE to enhance the test vectors consistenty improves the performance of the resuting sovers when compared to the resuts reported in Tabe 4.2, especiay for the LSR scheme. Here, the LSR approach yieds an optima method, whereas for the LS scheme, as the probem size goes from N = 255 to N = 511, the convergence rate of the sover

12 12 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Tabe 4.4 Asymptotic convergence rate estimates, ρ, of the LS- (LSR)-based two-grid methods appied to the FE Lapace probem. The sover is constructed using a V 2 -cyce setup with η = 4 and k r = k e = 8. N ρ.041 (.038).062 (.041).075 (.043).125 (.043).971 (.043) increases substantiay from ρ.125 to ρ.971. These resuts in turn suggest that the goba weights, ω κ, used in computing interpoation, bias the LS fit to some inadequate oca representations of the agebraicay smooth error, utimatey eading to a poor choice of interpoation for the associated fine-grid variabes. The LSR scheme is, however, abe to compensate for this in a very efficient manner. An aternative approach woud be to define the weights used in the LS fit ocay (i.e., for each i F) we mention that athough we have studied this idea extensivey, we have not yet derived an effective strategy for defining oca weights. To further iustrate the efficacy of the MGE when combined with the LSR form of P for this probem, we report the eigenvaue approximation measures τ (L,1) λ defined in (3.7), computed using the eight smaest eigenvaues of the coarsest-grid operator and the associated Rayeigh quotients of the finest-grid test vectors generated using the V 2 -cyce setup agorithm. Additionay, we report the two-norms of the differences between the eight eigenvectors with the smaest eigenvaues of the coarsest-grid system interpoated to the finest grid using the composite interpoation operators defined in (3.2) and the eigenvectors of the finest-grid system computed directy. The resuts are reported in Tabe 4.5. Here, we see that the eigenvaue approximation measures are uniformy sma for the k e = 8 computed eigenvector approximations and further that the eigenvector approximations interpoated form the coarsest grid to the finest one approximate we the targeted eigenvectors of the finest grid system, i.e., the eigenvectors associated with the smaest eigenvaues of A. We mention further that the eigenvector approximation measures consistenty detect the accuracy of the eigenvector approximations computed using the MGE. Tabe 4.5 Reative eigenvaue approximation measures τ (L,1) λ and eigenvector approximation estimates of i the eight smaest eigenvaues for the FE Lapace probem, computed within a V 2 -cyce setup with η = 4 and k r = k e = 8. i τ (L,1) λ i P L v L i v i Next, we provide pots of the eigenvector approximations computed using a V - cyce setup agorithm with η = 4 and k r = k e = 8 and of the associated eigenvectors of the finest grid operator computed directy. Figure 4.2(a) contains pots of the eigenvector with the smaest eigenvaue on each eve of the MGE computed using a singe V -cyce setup, and Figure 4.2(b) contains a pot of the eigenvector with the smaest eigenvaue of the system matrix on the finest grid computed directy. The pots demonstrate the abiity of the MGE (buit using smoothed random test vectors) to recover the smaest eigenvector of the finest-grid FE Lapace operator. In addition, we provide pots comparing the eigenvector approximations on the finest grid computed using the same V 2 -cyce setup and the eigenvectors of the finest-

13 BOOTSTRAP AMG 13 (a) Mutigrid representation of the coarsest-grid eigenvector (b) Finest grid Fig Comparison of the eigenvector corresponding to the smaest eigenvaue in (a) computed in a V -cyce setup with η = 4 and k r = k e = 8 and the associated finest grid eigenvector in (b) corresponding to the smaest eigenvaue of the FE Lapace probem computed directy. (a) Finest grid representation of the coarsest-grid eigenvectors (b) Finest grid Fig Visuaization of the finest-grid eigenvector approximations of the four smaest eigenvaues computed using a V 2 -cyce setup with η = 4 and k r = 8 in (a) and the associated exact finest-grid eigenvectors corresponding to the four smaest eigenvaues in (b). grid operator computed directy. Figure 4.3(a) contains pots of the finest-grid eigenvector approximations for the four smaest eigenvaues computed using a V 2 -cyce setup, and Figure 4.3(b) contains the pots of the eigenvectors of the finest-grid operator with the four smaest eigenvaues computed directy. Again, the pots iustrate that the MGE is abe to recover the ow eigenvectors of the fine-grid system. We now consider tests of the W-cyce setup, iustrated at the bottom of Figure 3.1. In Tabe 4.6, we present resuts for this scheme again with k r = k e = 8. We notice a marked improvement in the performance of the LS-based sover constructed using a W-cyce setup over that of the sover constructed using a V -cyce setup and note that the LSR scheme again scaes optimay with probem size.

14 t 14 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Tabe 4.6 Asymptotic convergence rate estimates, ρ, of the LS- (LSR)-based MG sovers appied to the FE Lapace probem. The sover is constructed using a W-cyce setup (see Figure 3.1) with η = 4 and k r = k e = 8. N ρ.042 (.038).062 (.041).073 (.043).130 (.043).161 (.044) The ast resut we present for the FE Lapace system is reated to the computationa compexity of the BAMG-MGE setup. In Figure 4.4, we show the time, t, spent in the setup phase as a function of the number of the finest-grid variabes, N 2. The numerica tests were performed using an 2.53GHz Inte Core 2 Duo processor with 4GB 1066MHz DDR3 SDRAM. The resuts suggest that the computationa compexity for both setup cycing strategies is neary optima (grows ineary with the probem size) when appied to this probem W cyce doube V cyce N 2 x 10 5 Fig Tota wa cock times for the V 2 - and W-cyce setup agorithms with η = 4 and k r = k e = 8 appied to the FE Lapace probem versus the probem size N The gauge Lapacian system. In this section, we consider soving the 2D gauge Lapacian system [6, 13, 16] with periodic boundary conditions, a probem that is typicay used for testing potentia AMG agorithms for soving the Dirac equation in more genera attice gauge theories, e.g., quantum chromodynamics (QCD). A detaied discussion of the GL system and its spectra properties is provided in Appendix A. The 2D GL can be viewed as a stochastic variant of the standard Lapace operator discretized using centra finite differences. Specificay, using stenci notation, the GL operator is given by (4.2) A(U) = where U z ex U z y x 4 + m Ux z, U z ey y U = {U z µ U(1), µ = x, y, z Ω} and m R. Here, z denotes a grid point on the computationa domain Ω. The off-diagona entries of the system matrix are often referred to as gauge configurations which vary

15 BOOTSTRAP AMG 15 according to a specific probabiity distribution. In our tests, we consider coections of constant gauge variabes, namey, U constant, as we as stochastic distributions. Our abeing of the unknowns and gauge variabes is iustrated in Figure A.1, where e µ denotes the unit vector in the µ-direction; i.e., it describes a shift on the attice by one attice site in the µ-direction. In a tests, we use the same coarsening and sparsity structure of P as for the FE Lapace system (see Figure 4.1) and coarsen the equations unti the dimension of the coarsest system is 8 8. We note that the GL operator is Hermitian, and we define m such that λ min = N 2, resuting in positive definite yet i-conditioned system matrices Numerica resuts for the GL with constant U. In section 4.1, we demonstrated the effectiveness of the BAMG-MGE approach for the biinear FE discretization of Lapace s equation. Here, we test the method for the GL system with various choices of U constant. In Tabe 4.7, we report resuts for a V 3 -cyce setup agorithm appied to the GL system with U 1. Taking m = 0, this yieds the FD Lapace operator discretized using centra finite differences, up to a scaing by h 2. As the resuts reported in Tabe 4.7 show, we obtain a very efficient mutigrid sover for this probem with the LSR approach it reduces the error by an order of magnitude at each iteration, whereas the performance of the LS scheme deteriorates as the probem size is increased. Tabe 4.7 Asymptotic convergence rate estimates, ρ, of the MG method obtained by using the LS (LSR) schemes in a V 3 -cyce setup with η = 4 and k r = k e = 8. N ρ.080 (.055).090 (.056).094 (.055).089 (.053).727 (.052) As a next test, we appy this same approach for U 1. This operator resuts from appying a symmetric diagona scaing to the FD Lapace probem, such that the kerne of the scaed operator aternates between 1 and 1. The resuts of our experiments are reported in Tabe 4.8. Again the resuts suggest that the LSR scheme with a V 3 -cyce setup is abe to sove this probem efficienty. Tabe 4.8 Asymptotic convergence rate estimates, ρ, of the MG method obtained by using the LS (LSR) schemes in a V 3 -cyce setup with η = 4 and k r = k e = 8. N ρ.074 (.059).080 (.057).095 (.055).126 (.053).998 (.052) Next, in Tabe 4.9, we report resuts for the GL operator with U e i π 7, giving the stenci e i π 7 A(U) = e 7 (4 + m) e i π 7. e i π 7 Again, we use a V 3 -cyce setup. We note that the resuting system is not equivaent to a diagona scaing of the FD Lapace operator (see Appendix A for detais). Nonetheess, the LSR approach with a V 3 -cyce setup yieds an efficient sover for this system as we.

16 16 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Tabe 4.9 Asymptotic convergence rate estimates, ρ, of the MG method obtained using the LS (LSR) schemes in a V 3 -cyce setup with η = 4 and k r = k e = 8. N ρ.058 (.056).054 (.054).055 (.051).051 (.049).056 (.048) Numerica resuts for the GL with stochastic distributions of the gauge variabes. We concude our experiments with tests for the GL operators A(U) for various reaizations of the gauge variabes. In interesting cases, they are weaky correated among neighboring grid points. In genera, their distribution depends on a parameter β. The case β = yieds U z µ = 1 for µ = x, y and a z Ω. As we discuss in Appendix A, as the gauge variabes become ess correated, the support of the ow eigenvectors of A(U) in turn become increasingy oca (see Figure A.3). Further, the number of ocay supported ow modes of A(U) generay increases as the probem size is increased, making it difficut to define an effective MG interpoation operator for the system. We report asymptotic convergence rates of the stand-aone MG sover and the number of iterations it takes the associated MG preconditioned conjugate gradient method to reduce the initia residua by a factor of The resuts are contained in Tabe Tabe 4.10 Asymptotic convergence rate estimates, ρ, of the MG method obtained using the LS scheme (on the eft) and LSR scheme (on the right) from V 3 - and W-cyce setup agorithms with η = 4, k r = 8, and k e = 16. The provided integer vaues denote the number of MG preconditioned conjugate gradient iterations needed to reduce the reative residua by a factor of β \N W.242 W.416 W.286 W.648 W.478 W.658 V V W.284 W.264 W.225 W.576 W.389 W.672 V V W.120 W.225 W.223 W.586 W.349 W.433 V V An important observation here is that the improved sover performance observed for the LSR approach over the LS one for the GL system with stochastic distributions of the gauge variabes is far ess pronounced than it was for the FE Lapace system and the GL system with constant gauge variabes. Reca that for these tests we appy the adaptive reaxation scheme to a the test vectors before constructing the LSR form of P. We note that even when appying the adaptive reaxation technique to a test vectors, the convergence rates reported for the LSR-based sover are ony marginay better than those reported for the LS-based scheme. Overa, the ack of scaing of the stand-aone method when going to arger grid sizes can perhaps be expained by the fact that the coarsest grid is not fine enough to accuratey represent a of the ocay supported eigenvectors with sma eigenvaues. Due to the stochastic nature of the configurations used in defining the entries of the system matrix, it is not cear that a direct comparison of the method for different probem sizes is reevant. Perhaps a more meaningfu observation here is that for the AMG preconditioned conjugate gradient sover the number of iterations is roughy constant for a probem sizes and configurations. This in turn suggests that in the stand-aone mutigrid sover ony a few error components are not efficienty reduced.

17 BOOTSTRAP AMG 17 There are severa possibe ways to remedy this. First, as we did in the tests, we coud try to capture as many of the components in our BAMG-MGE setup as possibe and then treat the remaining components by recombining successive iterates, for exampe by using the mutigrid sover as a preconditioner for a Kryov subspace method (e.g., the conjugate gradient method). Aternativey, we coud treat these ocay supported vectors by a bock smoother, an idea coser to the mindset of geometric mutigrid. As a fina iustration, we demonstrate that even for these more chaenging tests the MGE is abe to efficienty compute eigenvector approximations of the eigenvectors of the finest-eve system. Figure 4.5 contains pots of the moduus of the eigenvector approximations with the smaest eigenvaue computed using a V 2 -cyce setup. Again, we observe that the BAMG-MGE cyce efficienty generates an accurate mutigrid representation of the eigenvector with the smaest eigenvaue. (a) Mutigrid representation of the coarsest-grid eigenvector (b) Finest grid Fig Comparison of the MGE representation of the eigenvector approximations with the smaest eigenvaue on each eve (a) with the associated finest-grid eigenvector with the smaest eigenvaue (b). 5. Concuding remarks. In this paper, we deveoped and tested a bootstrap approach for computing mutigrid interpoation operators. As in any efficient mutigrid sover, these operators have to be accurate for the owest eigenvectors of the probem s finest-grid operator. Here, this is achieved by defining interpoation to fit, in an LS sense, a set of test vectors that coectivey approximate the agebraicay smooth error. We have shown that using LS interpoation with a BAMG cyce, a MGE, and adaptive reaxation eads to an efficient AMG setup agorithm and sover for the scaar test probems considered. A numerica experiments presented in this paper were for scaar PDEs discretized on structured grids, using fu coarsening and interpoation with a fixed sparsity pattern. This aowed us to concentrate on deveoping and testing techniques for computing the interpoation and the impact of the BAMG setup, the MGE, and the adaptive reaxation on the accuracy of the resuting interpoation operators. Our future research wi focus on the use of compatibe reaxation as an efficient too for choosing coarse-grid variabes and the use of agebraic distance to define the sparsity structure of the corresponding interpoation operator.

18 18 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS Appendix A. The GL system. The GL operator is a commony used test probem in AMG agorithm deveopment for attice formuations of the Dirac equation arising in attice gauge theories. The attice Dirac operator describes a discretized system of couped PDEs. For the sake of definiteness, Wison s origina discretization gives the nearest-neighbor couping of the unknowns which, for a 2D space and a given U(1) background gauge fied, can be written in spin-permuted ordering as ( ) A(U) B(U) D = B(U) H, A(U) where U denotes a discrete reaization of the so-caed gauge fied. The gauge configuration U can be understood as a coection of ink variabes of U(1), i.e., compex numbers with moduus one: U = {U z µ U(1), µ = x, y, z Ω}. As stated earier, our abeing of the unknowns and gauge inks is iustrated in Figure A.1. Herein, e µ is the unit vector in the µ-direction; i.e., it describes a shift on the attice by one attice site in the µ-direction. U z+ey y U z+ex+ey y U z ex+ey x z + e y U z+ey x z + e x + e y U z+ex+ey x U z y U z+ex y U z ex x z U z x z + e x U z+ex x U z ey y U z+ex ey y Fig. A.1. Naming convention on the attice. The diagona bocks, A(U), are referred to as GLs. The action of A(U) on a vector ψ C N2 at a attice site z Ω reads as (A.1) (A(U)ψ) z = (4 + m)ψ z Ux z ex ψ z ex Uy z ey ψ z ey Ux zψ z+e x Uy zψ z+e y, and the action of B(U) on ψ at site z Ω is given by (B(U)ψ) z = Uxψ z z+ex Ux z ex ψ z ex + i ( ) Uy z ψ z+ey Uy z ey ψ z ey. In our numerica experiments, we consider soving the GL system A(U)ψ = ϕ

19 BOOTSTRAP AMG 19 Fig. A.2. Moduus, rea, and imaginary parts of the eigenvector to the smaest eigenvaue for β = 5 on a grid. Fig. A.3. Moduus, rea, and imaginary parts of agebraicay smooth error after 50 GS iterations appied to a random initia guess for β = 5 on a grid, which for our choice of simpe pointwise smoother is a inear combination of the ow eigenvectors of the fine-grid system. for different choices of the shift m and configurations of U. In interesting cases, the off-diagona entries of A(U) are weaky correated among neighboring grid points. In genera, their distribution depends on a parameter β. The case β = yieds U z µ = 1 for µ = x, y and a z Ω. As β 0, the gauge variabes θ z µ in Uz µ := eiθz µ become ess correated 2 and the support of the ow eigenvectors becomes increasingy oca. A.1. Spectra properties of the GL. From (A.1), it foows that the GL matrix is Hermitian. We define the constant, m, so that the resuting matrix A has as its smaest eigenvaue λ min = N 2, with N denoting the number of grid points in a given direction of the 2D grid. This choice in turn yieds positive definite yet i-conditioned system GL systems. An important issue to consider when deveoping sovers for the GL is the oca character of its agebraicay smooth error. Figure A.2 contains pots of the moduus, rea, and imaginary parts of the eigenvector with the smaest eigenvaue of the system matrix in (A.1). In Figure A.3, we provide a pot of the error for β = 5 on a grid computed using 50 GS reaxations. Here, we see two main reasons why standard AMG approaches break down when appied to this probem the agebraicay smooth error is ocay supported, and it is not smooth among neighboring grid points in regions where it is nonzero. To further anayze the properties of the GL operator, we consider the notion of a gauge transformation. A gauge transformation g : Ω U(1), z g z 2 In the reported resuts, we use gauge variabes generated using a code suppied to us by R. Brower from Boston University. The gauge data is avaiabe in ASCII format onine from

20 20 A. BRANDT, J. BRANNICK, K. KAHL, AND I. LIVSHITS of a gauge configuration U = {U z µ} U(1) is defined by U z µ ḡ z U z µg z+eµ. That is, a gauge transformation can be represented as a diagona matrix: G = diag(g z ), z Ω. Thus, the action of the gauge transformation on a gauge covariant operator Z(U) is given by (A.2) Z(U) G H Z(U)G. In the case of U U(1), we obtain the gauge transformation of U under the gauge transformation g : z e iψz by (A.3) U z µ = e iθz µ g e iψz e iθz µ e iψ z+eµ. Note that the U(1) gauge transformation G in (A.2) fufis G H G = I and G.,i 2 = 1, i = 1,..., m; that is, the gauge transformation is a unitary simiarity transformation of the matrix Z(U). Hence, if x 1,..., x m are the eigenvectors of Z(U) corresponding to the eigenvaues λ 1,..., λ m, then G H x 1,..., G H x m are the eigenvectors of G H Z(U)G corresponding to the same eigenvaues. Accordingy, we define an equivaence reation between two operators Z(U 1 ), Z(U 2 ) with gauge configurations U i, i = 1, 2, as Z(U 1 ) Z(U 2 ) there exists G such that Z(U 1 ) = G H Z(U 2 )G. Appying this definition, we have that for a given constant configuration (U z µ e iθ ) on an N N equidistant attice, if θ fufis (A.4) θ = 2πk N for some k {0,..., N 1}, then A(U) A(U 0 ), where U 0 denotes the case when U 1. Thus, for certain distributions of U, A(U) is simpy a diagonay scaed Lapacian, whereas, in others, the system is not equivaent. We note that an effective AMG agorithm shoud sove the Lapace-equivaent systems with a performance simiar to its performance for the standard Lapacian, whereas the performance in the nonequivaent case is not predictabe. A more detaied study of this system can be found in [14]. REFERENCES [1] A. Brandt, Agebraic mutigrid theory: The symmetric case, App. Math. Comput., 19 (1986), pp [2] A. Brandt, Genera highy accurate agebraic coarsening, Eectron. Trans. Numer. Ana., 10 (2000), pp [3] A. Brandt, Mutiscae scientific computation: Review 2001, in Mutiscae and Mutiresoution Methods: Theory and Appications, T. J. Barth, T. F. Chan, and R. Haimes, eds., Springer, Heideberg, 2001, pp