AN OVERVIEW OF MULTILEVEL METHODS WITH AGGRESSIVE COARSENING AND MASSIVE POLYNOMIAL SMOOTHING

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1 Eectronic Transactions on Numerica Anaysis Voume 44, pp , 2015 Copyright c 2015, ISSN ETNA AN OVERVIEW OF MULTILEVEL METHODS WITH AGGRESSIVE COARSENING AND MASSIVE POLYNOMIAL SMOOTHING JAN BROUSEK, PAVLA FRANKOVÁ, MILAN HANUŠ, HANA KOPINCOVÁ, ROMAN KUŽEL, RADEK TEZAUR, PETR VANĚK, AND ZBYNĚK VASTL Abstract We review our two-eve and mutieve methods with aggressive coarsening and poynomia smoothing These methods can be seen as a ess expensive and more fexibe in the mutieve case) aternative to domain decomposition methods The poynomia smoothers empoyed by the reviewed methods consist of a sequence of Richardson iterations and can be performed using up to n processors, where n is the size of the considered matrix, thereby aowing for a higher eve of paraeism than domain decomposition methods Key words mutigrid, aggressive coarsening, optima convergence resut AMS subject cassifications 65F10,65M55 1 Introduction In genera, domain decomposition methods see, eg, [1, 2, 3, 4, 10, 12, 16] and for a comprehensive ist of references see the monograph [18]) can be, in a mutigrid terminoogy [5, 13, 26]), viewed as two-eve mutigrid methods with a sma coarse space and a massive smoother that uses oca subdomain sovers The size of the subdomains and the size of the coarse space are cosey and inversey) reated, which has important impications for the parae scaabiity of such methods From both a theoretica and practica viewpoint, domain decomposition methods are sought whose convergence rate is independent or neary independent of the mesh size and the subdomain size Some of the most successfu types of domain decomposition methods substructuring methods such as BDDC [17], FETI [12], and FETI-DP [11] cass methods and overapping Schwarz methods satisfy this property There are, however, severa disadvantages common to these domain decomposition agorithms when it comes to parae scaabiity, and conficting objectives are encountered The oca subdomain sovers are, especiay for 3D probems, reativey expensive and scae unfavoraby with respect to the subdomain size Since ony a sma number of processors can be used efficienty by each subdomain sover, massive paraeism may aso be imited Both of these issues ead to a preference for sma subdomains However, using sma enough subdomains gives rise to an undesiraby arge coarse-space probem, which destroys parae scaabiity The goa of this paper is to review our two-eve and mutieve methods with aggressive coarsening ie, eading to sma coarse probems) and massive smoothing that do not have the above disadvantages Instead of smoothers that are based on oca subdomain sovers, we use specia poynomia smoothers that are based on the properties of Chebyshev poynomias and are a sequence of Richardson iterations In our most genera resut, assuming that the mesh size on the evecan be characterized byh and empoying a carefuy designed poynomia smoother as a mutigrid reaxation process, we prove for a genera mutigrid V-cyce) a convergence resut independent of the ratioh +1 /h, provided that the degree of our smoother is greater than or equa to Ch +1 /h A Richardson iteration can be performed in parae using n processors, where n is the size of the considered matrix Thus, our methods are asymptoticay much ess expensive than the domain decomposition methods that use direct subdomain sovers and aow for a finer-grained paraeism Received Juy 19, 2014 Accepted May 19, 2015 Pubished onine August 21, 2015 Recommended by M Donatei Department of Mathematics, University of West Bohemia, Univerzitní 22, Pisen, Czech Repubik {brousek, frankova, mhanus, kopincov, rkuze, ptrvnk, zvast}@kmazcucz) Department of Aeronautics and Astronautics, Standford University, Stanford, CA 94305, US rtezaur@stanfordedu) 401

2 402 J BROUSEK, et a It is generay beieved that when assuming no reguarity, it is not usefu to perform more thano1) mutigrid smoothing steps One of the key objectives of this paper is to demonstrate that this is not the case Our earier methods are based on smoothed aggregation and show a significant acceeration compared to the use of O1) smoothing steps if the number of mutigrid smoothing steps is approximatey equa to the number of proongator smoothing steps Finay, our most genera resut is mutieve and fuy independent of the smoothing of the proongator This review paper covers a reativey arge time span of research and is organized as foows Four methods frameworks) are described in four key sections Two-eve methods Sections 3 and 4) are based on the smoothed-aggregation concept It is shown that it makes sense to perform a number of mutigrid smoothing iterations that is approximatey equa to the number of proongator smoothing steps The method reviewed in Section 3 shows a significant acceeration if mutipe proongator smoothing and mutigrid smoothing steps are performed, but it does not achieve a fuy optima convergence, that is, independent of both the fine resoution characterized by the mesh size h and the coarse-space resoution characterized by the subdomain sizeh for the cost ofoh/h) mutigrid smoothing steps On the other hand, the two-eve method of Section 4 is fuy optima Section 5 reviews a twoeve method that is fuy optima and aso aows for a mutieve extension A convergence rate independent of the resoutions of the first two eves is estabished That is, aggressive coarsening between eve 1 and eve 2 can be performed and is fuy compensated by the smoothing) The method presented here is in spirit cose to smoothed aggregation The most genera framework is described and anayzed in Section 6 A genera abstract) mutigrid method with our poynomia smoother is considered We stress that the anayzed method does not have to be based on smoothed aggregation As stated above, for the cost of a number of mutigrid smoothing iterations that is greater than or equa to ch +1 /h, a convergence rate independent of the resoutions h on a eves is proved The convergence estimates are confirmed numericay in Section 7 The resuts are summarized in Section 8 2 Two-eve variationa mutigrid The soution of a system of inear agebraic equations 21) Ax = f, where A is a symmetric positive definite n n matrix that arises from a finite eement discretization of an eiptic boundary vaue probem, is considered To define a variationa twoeve method, two agorithmic ingredients are required: a inear proongator, P : R m R n, m < n, and a smoothing procedure In this paper, poynomia smoothers that can be expressed as a sequence of Richardson iterations 22) x I ωa)x+ωf are considered Note that a particuar case of interest is that of a sma coarse space, that is, m n Letν 1 andν 2 be integers A variationa two-eve method proceeds as foows: ALGORITHM 1 1 Fori = 1,,ν 1 dox I α i A)x+α i f 2 Set d = Ax f 3 Restrictd 2 = P T d 4 Sove the coarse probema 2 v = d 2, wherea 2 = P T AP, 5 Correctx x Pv, 6 Fori = 1,,ν 2 dox I β i A)x+β i f

3 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 403 Throughout Sections 3 and 4, the tentative proongator p is assumed to be constructed by a generaized unknowns aggregation method [22] This is stricty because of considerations of agorithmic compexity A simpe exampe of a non-generaized) aggregation method is presented beow in Exampe 21 For a standard finite eement discretization of a scaar eiptic probem, the generaized aggregation method coincides up to a scaing) with a standard unknowns aggregation method [22] The resuting proongatorpis an orthogona matrix [22] The fina proongator P is obtained by poynomia smoothing [19, 20, 22, 23, 24] 23) P = Sp, S = I ω 1 A) I ω ν A), where ν is a positive integer The coefficients α i,β i, and ω i are chosen carefuy and kept in a cose reationship The tentative proongator p is responsibe for the approximation properties of the coarse-space RangeP) The proongator smoother S enforces smoothness of the coarse-space functions EXAMPLE 21 Consider a one-dimensiona Lapace equation discretized on a uniform mesh that consists of n = mn nodes A simpe unknowns aggregation proongator can be constructed as foows Let the nodes be numbered in the usua way from eft to right The aggregates are formed as disjoint sets ofn consecutive nodes, ie, 24) {A i } m i=1 = { {1,2,,N},{N +1,N +2,,2N},, {m 1)N +1,,mN 1,mN} } The corresponding proongator is given by { 1 iff i A j, 25) p ij = 0 otherwise, that is, the j-th coumn is created by restricting a vector of ones onto the j-th aggregate with zeroes esewhere In matrix form, 1 1 A A 2 26) p = 1 A m 1 The action of the proongator corresponds to a disaggregation of the j-th R m -variabe into N R n -variabes forming the aggregate A j Thus, p can be thought of as a discrete piecewise constant interpoation The proongator becomes an orthogona matrix by the scaing p 1 N p

4 404 J BROUSEK, et a For scaar probems such as Exampe 21), the coumns of the proongator p have a disjoint nonzero structure This can aso be viewed as the discrete basis functions of the coarse-space Rangep) having disjoint supports For non-scaar eiptic probems, severa fine-eve vectors are restricted to each of the aggregates For exampe, for a discretization of the equations of inear easticity, six rigid-body modes are restricted to each of the aggregates giving rise to six coumns with the same nonzero structure Such a set of coumns is abeed a super-coumn and the corresponding set of coarse-eve degrees of freedom each associated with one coumn) a super-node The super-coumns have a disjoint nonzero structure corresponding to the disjoint nonzero structure of the aggregates Thus, in genera, it is assumed that the discrete coarse-space basis functions coumns of the proongator p) are non-overapping uness they beong to the same aggregate A key assumption to prove convergence of a two-eve method is that the proongator satisfies the weak approximation condition 27) e R n v R m : e pv 2 C A A) ) 2 H e 2 A h Here, h is a characteristic eement size of the fine-eve discretization assuming the quasiuniformity of the mesh), andh is a characteristic diameter of the aggregates understood as a set of finite eement noda points) For a scaar eiptic second-order probem, 27) was proved in [22] For the case of inear easticity in 3D, the reader is referred to [23] A simpe exampe is given beow EXAMPLE 22 To iustrate the verification of property 27), consider a one-dimensiona mode exampe: for a givenf L 2 0,1)), findu H 1 00,1)) such that u,v ) L20,1)) = f,v) L20,1)), v H 1 00,1)) The mode probem is discretized by P1-eements on a uniform grid of mesh size h = 1 mn+1 The discretization eads to a system 21) of n = mn inear agebraic equations Each equation corresponds to one unconstrained node in the interva 0,1) The tentative proongator p is based on an aggregation of groups of N neighboring nodes as depicted in Figure 21 and described in Exampe 21 This yieds 25) or equivaenty the matrix form 26) A 1 A m 1 2 N n Ω Ω 1 ff m FIG 21 Aggregates of N nodes Consider a finite eement interpoatorπ h : x = x 1,,x n ) T i x iϕ i The tentative proongator p appied to a vector v = v 1,,v m ) T returns the vaue v i in a the variabes of the aggregatea i Hence, for the corresponding finite eement function we have 28) Π h pv = v i on Ω i, where Ω i is the subinterva of0,1) corresponding to the aggregate A i ; see Figure 21

5 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 405 Let H denote the ength of the intervas Ω i The verification of 27) consists in a straightforward appication of the scaed Poincaré inequaity on the subintervasω i, 29) min q R u q L 2Ω i) CH u H1 Ω i), u H 1 Ω i ), and the equivaence of the discrete and continuousl 2 -norms for finite eement functions [9], 210) ch 1 Π h u 2 L 2Ω i) u 2 2Ω i) j A i u 2 j Ch 1 Π h u 2 L 2Ω i) From 28), 29), and 210), it foows that for every u R n, there exists a coarse-eve vectorv = v 1,,v m ) T such that u pv 2 = m u pv 2 m 2Ω i) Ch 1 Π h u v i 2 L 2Ω i) i=1 i=1 m = Ch 1 min Π hu q i 2 L q 2Ω i) CH2 i R h i=1 C H2 h Π hu 2 H 1 0,1)) m Π h u 2 H 1 Ω i) Therefore, since A) Ch 1 cf [9]) and Π h u H1 0,1)) = u A, it foows that there exists a mapping Q C : R n Rangep) such that i=1 H u Q C u C h A) u A, u R n This shows 27) The proof can be easiy extended to mutipe dimensions as we as to the case of unstructured domainsω i as ong as the domains are shape reguar the constant in the scaed Poincaré inequaity is uniformy bounded) This requirement can be aso weakened by encapsuating the domains Ω i into bas with bounded overap [22] The constant in the weak approximation condition 27) depends on the ratio H h As a resut, the convergence of the straightforward two-eve method depends on the same ratio More specificay, assuming 27), the variationa two-eve method with the proongatorpand a singe Jacobi post-smoothing step converges with the rate of convergence ) ) 2 h 211) Ee 2 A 1 C e 2 H A, where E is the error propagation operator of the method The objective of the methods reviewed in this paper is to eiminate this dependence of the convergence of a two-eve method on the ratio H h with a minima possibe cost Domain decomposition methods strive toward the same goa A typica domain decomposition method can be viewed as a two-eve mutigrid method with a sma coarse space whose oca resoution corresponds to the subdomain size and a bock-smoother that uses direct subdomain sovers The subdomain sovers are reativey expensive Here, methods with a much ower cost that aso open the room for a higher eve of fine-grain paraeism are described and anayzed A two-eve method is abeed optima if, for a second-order eiptic probem discretized on a mesh with the mesh size h, it yieds a sma, sparse coarse space characterized by the diameter H and an H/h-independent rate of convergence for the cost ofoh/h) eementary

6 406 J BROUSEK, et a TABLE 21 Fine-eve cost of an optima two-eve mutigrid method and a domain decomposition method based on direct banded and sparse) subdomain sovers The coarse-eve setup and sove are not incuded Method Space Subdomain Bandwidth Tota cost dim d probem size DD+banded 2 OH/h) 2 ) OH/h) OnH/h) 2 ) DD+banded 3 OH/h) 3 ) OH/h) 2 ) OnH/h) 4 ) DD+sparse 2 OH/h) 2 ) OnH/h)) DD+sparse 3 OH/h) 3 ) OnH/h) 3 ) Optima TMG any d OnH/h)) smoothing steps Generay, a Richardson iteration sweep given by 22) is considered as an eementary smoothing step One cannot possiby expect a better resut, ie, a coarse-space size independent convergence with fewer than OH/h) smoothing steps since that many steps are needed to estabish essentia communication within the discrete distanceoh/h), that is, the continuous distanceoh) An optima two-eve method is significanty ess expensive than a domain decomposition DD) method based on oca subdomain sovers A DD method needs to sove 1/H) d subdomain inear probems of size OH/h) d ) This is refected in a comparison of the compexity of the optima two-eve method shown in Tabe 21 Whether a direct banded or a direct sparse sover is used for the Choesky decomposition of the oca matrices in the DD method, the cost of the optima two-eve method is significanty ower in three dimensions Furthermore, an optima two-eve method is more amenabe to massive paraeism In this method, the smoothing using OH/h) Richardson sweeps 22), which constitutes a botteneck of the entire procedure, can be performed using up to n processors On the other hand, in a DD method, subdomain-eve- and therefore much coarser-grained paraeism is natura, where typicay onyom) processors can be utiized, wheremis the number of subdomains 3 Naive beginning In this section, a two-eve method of [14] is described and anayzed Even though it does not have a fuy optima convergence, it sti has a practica vaue For the cost ofoh/h) smoothing steps in the smoothed proongator andoh/h) standard mutigrid smoothing steps, the rate of convergence of the straightforward method 211) is improved as Ee 2 A 1 C h H ) e 2 A When the agorithm is used as a conjugate gradient method preconditioner, the convergence improves to ) h 31) Ee 2 A 1 C e 2 H A Thus, when the coarse-space characteristic diameter H is increased 10 times making the coarse probem 100 times smaer for a two-dimensiona probem and 1000 times smaer for a three-dimensiona probem) and the number of smoothing steps is increased 10 times, the expression H/h in the estimate 31) grows ony by a factor of 10 = 3162 On the theoretica side, the method justifies the use ofoh/h) mutigrid smoothing steps It was generay beieved that under the reguarity-free assumption 27), it is impossibe to improve the convergence by adding more pre- or post-smoothing steps One of the main objectives of this paper is to show that this is not the case The resuts in Sections 3 and 4

7 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 407 justify using as many pre- or post-smoothing steps as there are proongator smoothing steps in 23) In Section 6 it is proven that, for a genera mutieve method, it makes sense to use OH/h) smoothing steps even for a method that does not use proongator smoothing The two-eve method of [14] uses the components described in the previous section a tentative proongator p obtained by the generaized unknowns aggregation, a smoothed proongator P = Sp as in 23) with the foowing choice ω i = ω, i = 1,,ν, ω 0,2), A) and a post-smoother given by step 6 of Agorithm 1 with ν 2 = ν +1, β i = ω, i = 1,,ν +1 A) The number of steps is assumed to satisfy the condition ν C H h, and ν 1 H 2 h is a recommended vaue Here,H is a characteristic diameter of the aggregates We wi aso consider a symmetrized version of this method in which the pre-smoother step 1 of Agorithm 1) and the post-smoother coincide Under reasonabe assumptions on the aggregates, the proongator smoothing by S estabishes natura overaps of the coarse-space basis functions coumns ofp ) and eads to a sparse coarse-eve matrixp T AP [14]) This is iustrated in the foowing exampe EXAMPLE 31 Consider the discretization of the scaar eiptic probem given in Exampe 22, ie, the weak form of a one-dimensiona Poisson equation on0, 1) with homogeneous Dirichet boundary conditions and its discretization by P1-eements on a uniform mesh The resuting stiffness matrix A is tridiagona and its pattern foows, aside from the Dirichet boundary condition, the three-point scheme 1 h [ 1,2, 1] Let the aggregates ben consecutive nodes as given by 24) and iustrated in Figure 21, the tentative proongator be given by 25), the proongator smoother be S = I ω ) N/2 A) A, and the fina proongatorp = Sp The standard finite eement interpoatorπ h can be written as n Π h : x R n x i ϕ i, where {ϕ i } n i=1 is the standard P1- piecewise-inear) finite eement basis Then, interpoants of the coarse-space vectors given by the proongatorpare i=1 ϕ 2 i = Π h pe i, i = 1,,m, where e i is the i-th canonica basis vector of R m and m is the number of aggregates see the top of Figure 31) Simiary, continuous basis functions corresponding to the smoothed proongator are given by ϕ 2,S i = Π h Pe i = Π h I ω ) N/2 A) A pe i, i = 1,,m

8 408 J BROUSEK, et a The overap of the supports of the unsmoothed basis functions is ony minima as depicted in Figure 31 top) Each smoothing step, ie, the mutipication by I ω/ A)A, adds to the support one eement on each side SinceH = N 1)h, the recommended vaue impies ν 1 2H/h = N 1)/2 N/2 Figure 31 bottom) shows that for aggregates of N = 7 nodes, the recommended vaue is ν 3, and the smoothing steps then estabish a natura overap of the smoothed basis functions Finay, the coarse-eve matrixa 2 = P T AP satisfies {A 2 ) ij } m i,j=1 = {APe i,pe j )} m i,j=1 = { Π h Pe i, Π h Pe j ) L20,1)) { } m = ϕ 2,S i, ϕ 2,S j ) L20,1)) i,j=1 } m i,j=1 The entry A 2 ) ij can be nonzero ony if the supports of the basis functions ϕ 2,S i and ϕ 2,S j overap Thus, the coarse-space matrix is sparse This is true with the recommended choice ofν under reasonabe conditions on the aggregates in genera supp ϕ 2,S i FIG 31 Non-smoothed above) and smoothed beow) continuous basis functions The convergence proof beow takes advantage of the observation that the next smoothing iteration is never more efficient than the preceding one Consider a inear smoothing iteration method with the error propagation operator M that is A-symmetric and a nonzero error vector e such that Me 0 Comparing the contraction ratios in the A-norm of one iteration to the next, it hods that 32) q 1 Me A e A q 2 MMe) A Me A Indeed, thea-symmetry ofm and the Cauchy-Schwarz inequaity impy Me 2 A = AMe,Me = AMMe),e MMe) A e A Dividing the above estimate by Me A e A yieds 32) The observation can be appied recursivey to infer that the effect of ν + 1 smoothing iterations can be estimated using the effect of the east efficient ast smoothing iteration: assume for the time being thatm i e 0, for i = 0,, ν + 1 Thus, for the A-symmetric error propagation operator M, 32) gives that is, M i e A M i 1 e A Mν+1 e A M ν e A, i = 1,,ν, 33) M ν+1 e A e A = Mν+1 e A M ν e A M ν e A M ν 1 e A Me A e A M ν+1 ) ν+1 e A M ν e A If M ν e = 0, then M ν+1 e A / e A = 0, and the estimate above is not needed

9 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 409 It is we-known [6] that the error propagation operator of the variationa two-eve method with a proongator P and ν +1 post-smoothing iterations with the error propagation operator M is given by E = M ν+1[ I PP T AP) 1 P T A ] The foowing abstract emma provides an estimate for E A LEMMA 32 Assume there is a constant C > 0 such that the tentative proongator p satisfies 34) for a fine u R n, there is a coarse v R m : u pv C A) u A Then the variationa two-eve method with the smoothed proongator P = M ν p and ν +1 post-smoothing steps with the error propagation operatorm = I ω/ A)A satisfies 35) M ν+1[ I PP T AP) 1 P T A ] [ 2 A 1 ω2 ω) ] ν+1 C 2 Moreover, for the symmetrized version, the foowing estimate hods 36) M ν+1[ I PP T AP) 1 P T A ] M ν+1 [ 2 A 1 ω2 ω) ] 2ν+1) C 2 Proof The matrix PP T AP) 1 P T A is an A-orthogona projection onto RangeP) The compementary projection Q = I PP T AP) 1 P T A is an A-orthogona projection onto RangeP) A, where the symbo RangeP) A denotes an A-orthogona compement of RangeP) It hods that 37) RangeQ) = RangeP) A = KerP T A) The operator norm s submutipicativity, 37), and the fact that Q A = 1, since Q is an A-orthogona projection, give 38) M ν+1 Q A M ν+1 x A M ν+1 x A sup Q A sup x KerP T A)\{0} x A x KerP T A)\{0} x A Note that the above estimate originates in [6] The key idea of the proof foows Since P = M ν p, wherem is a poynomia ina,m andacommute, hencekerp T A) can be aso characterized as KerP T A) = KerM ν p) T A) = Kerp T M ν A) = Kerp T AM ν ) Thus, M ν x Kerp T A) for x KerP T A) This fact together with the estimates 33) and 38) yied 39) M ν+1 Q A M ν+1 x A M ν+1 x A sup sup x KerP T A)\{0} x A x KerP T A)\{0} M ν x A ) ν+1 Mx A = = sup x Kerp T A)\{0} x A sup x Kerp T A)\{0} Mx A x A ) ν+1 ) ν+1

10 410 J BROUSEK, et a The rest of the proof is again standard [6] The term Mx A sup, x Kerp T A)\{0} x A M = I ω A) A, is estimated using an orthogonaity trick from the proof of Céa s emma First, for anyx R n, it hods that Mx 2 A = A I ω ) A) A x, I ω ) A) A x 310) = x 2 A 2 ω A) Ax,Ax + ω A) ) 2 A 2 x,ax x 2 A 2 ω ω2 Ax,Ax + A) A) Ax,Ax [ = 1 ω2 ω) ) ] 2 Ax x 2 A) x A A Thus, the smoother is forω 1) efficient provided that Ax 2 x 2 A A) Next, et x Rangep) A = Kerp T A) The assumption 34) guarantees the existence of v such that x 2 C A = Ax,x = Ax,x pv Ax x pv Ax x A, A) which foows aong the ines of the proof of Céa s emma and using the Cauchy-Schwarz C inequaity Dividing the above inequaity by x A) 2 A, squaring the resut, and substituting into 310) yieds Mx 2 A [ 1 ω2 ω) ] C 2 x 2 A, x Rangep) A This and 39) give the resut 35) SinceQis ana-orthogona projection, it hod thatq 2 = Q, and therefore it foows that M ν+1 QM ν+1 = M ν+1 Q 2 M ν+1 and M ν QM ν e 2 A e 2 A = Mν Q 2 M ν e 2 A e 2 A M ν e 2 QM ν e 2 A A e 2 A = M ν Q 2 AQM ν e,qm ν e A e 2 = M ν Q 2 AQM ν e,m ν e A A e 2 A = M ν Q 2 AM ν QM ν e,e A e 2 M ν Q 2 M ν QM ν e A e A A A e 2 A = M ν Q 2 M ν QM ν e A A e A

11 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 411 Dividing the above inequaity by M ν QM ν e A / e A yieds 311) M ν QM ν A M ν Q 2 A The symmetrized resut 36) foows from this estimate and 35), thus competing the proof The estimate for the symmetrized method, ie, that with the error propagation operator E sym = M ν+1[ I PP T AP) 1 P T A ] M ν+1, is up to a constant the same as that of the nonsymmetric one with the error propagation operatore = M ν+1[ I PP T AP) 1 P T A ] However, the symmetrized method can be used as a conjugate gradient method preconditioner For ν = OH/h), the estimate 36) and the assumption 27) give M ν+1[ I PP T AP) 1 P T A ] M ν+1 2 A [ ] 2ν+1) [ 1 ω2 ω) C34) 2 1 ω2 ω) C 1 ) ] H 2 C2 h h H 1 ω2 ω) C 3 If the symmetrized method with the error propagation operator E sym is used as a conjugate gradient method preconditioner, the resuting condition number of the preconditioned system is ) 1 H cond = O, 1 E sym A h and the convergence of the resuting method is guided by the factor cond 1 cond+1 = 1 1 C 4 h H h H 4 A fuy optima two-eve method In this section, an optima two-eve method of [23] is described and anayzed In this method, the fina proongator is given byp = Sp, where S is a proongator smoother and p is a tentative proongator given by the generaized unknowns aggregation The proongator smoother is a poynomia in A that is an error propagation operator of an A-non-divergent smoother Denote A S = S 2 A Two reated agorithms are considered, namey a nonsymmetric and an A S -symmetric variant The nonsymmetric agorithm proceeds as foows: ALGORITHM 2 nonsymmetric) Repeat 1 Pre-smoothx Sx,f), wheres,) is a singe step of a inear iteration with the error propagation operators 2 Sove P T APv = P T Ax f) 3 Update x x Pv 4 Post-smooth 41) x I ω ) A S ) A S x+ ω A S ) S2 f unti convergence in thea S -norm Post-process x Sx,f)

12 412 J BROUSEK, et a By reordering the terms, the step 41) can be performed more efficienty as x x ω A S ) S2 Ax f) The symmetric variant modifies the first step of Agorithm 2 to make ita S -symmetric: ALGORITHM 3 A S -symmetric) Repeat 1 Pre-smooth a) x Sx,f), b) x I ω A S ) A S c) x Sx,f) 2 Sove P T APv = P T Ax f) 3 Updatex x Pv 4 Post-smooth x ) x+ ω A S ) S2 f, I ω ) A S ) S2 A x+ ω A S ) S2 f unti convergence in thea S -norm Post-processx Sx,f) In order to justify the use of the above agorithms, the we-posedness of their coarse-eve probems needs to be inspected In summary, the coarse-eve probems of Agorithms 2 and 3 have a soution for a reevant right-hand sides, and the proongated and smoothed coarse-eve probem soution does not depend on the choice of the pseudo-inverse since Kerp T SASp) = KerSp) In more detai, it is cear from Agorithms 2 and 3 that a admissibe right-hand sides have the form Sp) T ASe = p T SASe, e R n Such right-hand sides are orthogona to the possibe kerne of the coarse-eve matrix since for u Kerp T SASp) = KerSp), it hods that p T SASe,u = ASe,Spu = 0 Thus, the coarse-eve probem has a soution This soution is unique up to a component that beongs tokerp T S 2 Ap) = KerSp) Thus, the proongated and smoothed soution that is, Pv = Spv) is unique and independent of the choice of the pseudoinverse Denote S = I ω A S ) A S By a direct computation, the error propagation operator of Agorithm 2 can be written as 42) E = S I PP T AP) + P T A)S = S I SpSp) T ASp) + Sp) T A)S = S SI psp) T ASp) + Sp) T AS) = S SI pp T A S p) + p T A S ) since A commutes withs Simiary, the error propagation operator of Agorithm 3 is 43) E sym = ESS = S SI pp T A S p) + p T A S )SS Since S and S are symmetric and commute with A S, making them aso A S -symmetric, and I pp T A S p) + p T A S is ana S -symmetric projection, the error propagation operatore sym is asoa S -symmetric

13 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 413 The error e KerS) is eiminated by the first smoothing step of Agorithms 2 and 3 Therefore, it is sufficient to estabish convergence bounds for the errorse KerS) THEOREM 41 Letω [0,2], A S ) A S ) be an upper bound ands be a poynomia in A that satisfies S) 1 Assume there exists a constant C > 0 such that: for every fine vectoru R n, there is a coarse vectorv R m that satisfies 44) u pv 2 C A S ) u 2 A Then the error propagation operator E of one iteration of Agorithm 2 and the error propagation operator E sym of one iteration of Agorithm 3 satisfy 45) E 2 A S qc,ω), and 46) E sym AS qc,ω), respectivey, where Here, thea S -operator norm is defined as qc,ω) = 1 ω2 ω) C +ω2 ω) B AS = Bx AS max x R n KerS)) \{0} x AS Furthermore, for any positive integerν, it hods that 47) SE ν 2 A q ν C,ω) and SE ν sym A q ν C,ω) REMARK 42 The function q,ω) in Theorem 41 is minimized by ω = 1, which gives qc,1) = 1 1 C+1 REMARK 43 The estimates 47) justify the use of the post-processing step in Agorithms 2 and 3, respectivey Proof of Theorem 41 First consider the anaysis of Agorithm 2 It is easy to verify that Q = I pp T A S p) + p T A S is an A S -orthogona projection onto T Rangep)) A S Hence, Q AS = 1 Using 42) yieds E AS = S SQ AS max x T KerS)) \{0} S Sx AS = max x T KerS)) \{0} x AS S Sx AS Q AS x AS Since S,S, and A S are poynomias in A, they commute, and S and S are A S -symmetric Furthermore, since S) 1 and S ) 1, it hods that S Sx AS S x AS and S Sx AS Sx AS

14 414 J BROUSEK, et a Therefore, 48) E AS = S Sx AS max x T KerS)) \{0} x AS max min x T KerS)) \{0} { Sx AS x AS, S x AS x AS } It is shown beow that, for x T KerS)), when the smoother S fais to reduce the error in thea S -norm Sx AS / x AS 1), the performance of the smoothers is improved More specificay, for ax T KerS)),x 0, 49) S x 2 A S x 2 A S 1 ω2 ω) Sx 2 A S C x 2 A S First, the term S x 2 A S is estimated as foows 410) S x 2 A S = I ω A S ) A S)x 2 A S = x 2 ω A S 2 A S ) A Sx 2 + ) 2 ω A S x 2 A A S ) S x 2 ω A S 2 A S ) A Sx 2 + ω2 A S ) A Sx 2 = x 2 A S 1 ω2 ω) A S x 2 ) A S ) x 2 A S Reca thata S = AS 2 and A ands commute Hence S 2 x 2 A = Sx 2 A S, and 411) A S x 2 x 2 = A Sx 2 S 2 x 2 A A S S 2 x 2 A x 2 = AS2 x 2 Sx 2 A S A S S 2 x 2 A x 2 A S Next, a ower bound for the first fraction on the right-hand side of 411) is estabished To this end, an orthogonaity trick known from the proof of Céa s emma is used First, denote T = Rangep)) A S = Kerp) T A S = Kerp) T AS 2 Hence, x T : S 2 x Kerp) T A = Rangep)) A Thus, for anyx T, using assumption 44) withu = S 2 x and the Cauchy-Schwarz inequaity gives S 2 x 2 A = AS 2 x,s 2 x = AS 2 x,s 2 x pv AS 2 x S 2 x pv C1/2 A S ) 1/2 AS2 x S 2 x A Dividing both sides by S 2 x 2 A yieds the desired coercivity bound, 412) AS 2 x S 2 x A Combining 412), 411), and 410) proves 49) As ) C, x T KerS))

15 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 415 By substituting 49) into 48) and taking into account that 0 Sx 2 A S / x 2 A S 1, the fina estimate 45) is obtained as foows E 2 A S max x T KerS)) \{0} max t [0,1] min { Sx 2 min{ AS x 2,1 ω2 ω) A S C } t,1 ω2 ω) t C Sx 2 A S x 2 A S = 1 ω2 ω) C +ω2 ω) Equation 43) shows that the error propagation operator of Agorithm 3 is given by E sym = S SQSS Then, sinceqis an A S -orthogona projection, 311) yieds E sym AS E 2 A S, proving46) Finay, using S) 1 and 45), we have for anyx KerS) that } SE ν x A x A = Eν x AS x A = Eν x AS x AS x AS x A Eν x AS x AS q ν/2 C,ω), proving the first part of 47) The second part is obtained from 46) anaogousy In view of Theorem 41, a smoothing poynomia S is desired that is an error propagation operator of ana-non-divergent smoother and makes S 2 A) as sma as possibe A poynomia smoother S = poa) with such a minimizing property is described beow Specificay, the poynomia smoother satisfies 413) S 2 A) A) 1+2degS)) 2, S) 1 LEMMA 44 [7]) For any > 0 and integer d > 0, there is a unique poynomia p of degree d that satisfies the constraintp0) = 1 and minimizes the quantity This poynomia is given by 414) pλ) = 1 λr1 ) 0 λ p2 max λ)λ ) 1 λrd, r k = 1 cos 2 ) 2kπ, 2d+1 and satisfies 415) 416) max 0 λ p2 λ)λ = 2d+1) 2, max pλ) = 1 0 λ and Proof If p 2 λ)λ can be written as the ineary transformed Chebyshev poynomia of order 2n+1, 417) p 2 λ)λ = qλ) = c 2 1 T 2n+11 2λ/L)), c = max 0 λ L p2 λ)λ, then it foows from minimax properties of Chebyshev poynomias that p is the sought poynomia The zeros of qλ) are the points λ where T 2n+1 1 2λ/L) = 1, that is,

16 416 J BROUSEK, et a 1 2λ/L = cos 2kπ 2n+1 The vauek = 0 gives the simpe rootλ = 0 ofq, whiek = 1,,n, yied doube roots ofq given by 414) This proves thatpis indeed the poynomia 414) To prove 415), we determine the quantitycin 417) from the condition thatp0) = 1 This condition impies that the inear term of p 2 λ)λ is one, hence from 417) it foows that 1 = p 2 λ)λ) 0) = c/l)t 2n+11) Therefore,c = L/T 2n+11) = L/2n+1) 2 It remains to prove 416) First, 416) is equivaent to p 2 λ) 1, for a λ, 0 λ 1 Using 417), this is equivaent to 418) L1 T 2n+1 1 2λ/L)) 2λT 2n+1 1) 1, for aλ, 0 λ L Using the substitution1 2λ/L = x, 418) becomes by a simpe manipuation T 2n+1 x) 1+T 2n+11)x 1), for ax, 1 x 1, which is the we-known fact that the graph of a Chebyshev poynomia ies above its tangent at x = 1 We chooses to be the poynomia 414) inawith = A) Using the spectra mapping theorem, 415), and 416), we get 413) THEOREM 45 Assume that the tentative proongator p satisfies 27) Let the fina proongator be constructed asp = Sp, where the proongator smoother iss = pa) andp ) is the poynomia 414) withρ = A) Assume the degree of S satisfies 419) degs) c H h, and ω 0,2) Then, Agorithms 2 and 3 converge with qω) = 1 1 Cω), wherecω) > 0 is a constant independent of both h andh cf Theorem 41) Proof The statement 413) and the assumption 419) give ) S 2 A) S 2 A) h A) 2 A) 1+2degS)) 2 C H Substituting this into 27) yieds e R n v R m : e pv 2 The resut now foows from Theorem 41 C S 2 A) e 2 A 5 An aternative with a mutieve extension The optima two-eve framework described in Section 4 requires the exact soution of the coarse-eve probem In this section, we describe a method that is a modification of the one given in [15] The framework we use here aows for empoying more than two eves with aggressive coarsening between the first two of them Let us consider first the two-eve case The content of this section is based on the foowing key observation: assume that for a particuar error e, the foowing weak approximation condition hods, 51) for the error e, there exists a coarse v such that e pv C ρa) e A

17 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 417 Then, assuming a reasonabe smoother, a singe iteration of a two-eve method with an error propagation operatore) satisfies Ee A 1 1 ) e A qc) Here qc) 1 is a monotonous function growing with growing C Thus we observe that the two-eve error estimate hods, so to speak, point by point If the weak approximation condition is satisfied for a particuar error e, then the error estimate gives the contraction number for this particuar error In what foows, we wi refer to this property of the estimate as a pointwise convergence property This is by no means easy to prove We wi provide such a proof, based on [5], in Lemma 54 The convergence proof of [5] satisfies, under some restrictions when certain arguments are avoided, the pointwise convergence property Our first goa is to guarantee a coarse-space size independent rate of convergence for a two-eve method with OH/h) smoothing iterations The convergence of a two-eve method is guided by condition 51) One obvious way how to make 51) weaker and thereby aowing for a sma coarse space) is to reduce ρa) This can be achieved by soving instead a transformed probem 52) S 2 Ax = S 2 f, where S = I α 1 A) I α d A) is a poynomia in A such that S) 1 and S 2 A) ρa) In other words, we use a poynomia S being an error propagation operator of a non-divergent poynomia iterative method such that S 2 A) ρa) In this case, however, the weak approximation condition 51) becomes, 53) e pv C ρs2 A) e S 2 A The norm on the right-hand side is generay smaer than the A-norm For a given continuous probem to be soved, such a condition is difficut to verify The goa of this section is to prove uniform convergence of a two-eve method assuming the weak approximation condition with a reduced spectra bound S 2 A) and the origina A-norm on the right-hand side, that is, 54) e pv In fact, assuming that C ρs2 A) e A ) h S 2 A) C 2 A), H which was proved in Section 4 assuming degs) ch/h), condition 54) foows from our genera assumption 27) To be abe to prove uniform convergence under assumption 54) that is, to prove 53) based on 54)), we have to compensate for the oss of the foowing coercivity condition: 55) e S2 A q e A, 0 < q < 1 Here, the numberq 0,1) is the threshod we choose Let the reader imagine, for exampe, q = 05 For a given error vector e R n, we consider two possibe cases: 1 the coercivity 55) hods, or 2 the coercivity 55) fais Given a threshodq 0,1), assume 55) hods

18 418 J BROUSEK, et a for a specific errore Then the weak approximation condition 53) foows from 54) with C 53) = C 54) /q In this case, the two-eve procedure for the transformed probem 52) is efficient Let us take a ook at the opposite case If e S 2 A q e A, q < 1, then equivaenty Se A q e A In other words the oss of coercivity 55) makes the iterative method with the error propagation operators efficient Summing up, if the coercivity 55) hods then 53) foows from 54) In case of oss of coercivity, the iterative method with the error propagation operator S is efficient So, under 54) either the two-eve method for soving the transformed probem 52) or the iterative method with an error propagation operators is efficient Thus, we wi base our method on two basic eements: 1 soving the transformed probem 52) by a two-eve method, or 2 empoying the outer iteration for soving the origina probem Ax = f using the iterative method with an error propagation operator S Reca that and S = I α 1 A) I α ν A), α 1,,α ν R n, 56) S) 1 We start with an investigation of the nonsymmetric agorithm The estimate for the symmetric version wi ater foow by a trivia argument In what foows, we wi prove a genera convergence bound for the foowing agorithm: ALGORITHM 4 1 Perform a singe iteration of the mutieve method for soving the probem 57) A S x = f S, A S = S 2 A, f S = S 2 f 2 Fori = 1,,d do x I α i A)x+α i f Note that in the context of a mutieve method, the matrix A S is never constructed and ony its action is evauated 51 Anaysis of the nonsymmetric agorithm The error propagation operator E of Agorithm 4 has the form 58) E = SE S, where E S is an error propagation operator corresponding to step 1 THEOREM 51 For every t [0,1], define a set 59) Vt) = {v R n : v AS t v A } Then 510) E A sup t [0,1] { t sup v Vt) E S v AS v AS REMARK 52 Note thats is a poynomia inasuch that S) 1, hence AS A, and Vt) is a set of vectorsv R n such that the norm equivaence t v A v S 2 A v A, hods Vt) {v R n : v AS t v A } = {v R n : t v A v AS v A }) Therefore, for vectorsu Vt), the desired weak approximation condition 53) foows from the weakened approximation condition 54) with a constantc 1 = C 2 /t }

19 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 419 Proof of Theorem 51 Ceary, sincee = SE S see 58)), it hods for everye R n that 511) Let us set Ee A = SE Se A Se A = E Se AS e A Se A e A e AS t = e A S e A e A S e A Then, triviay,e Vt) Hence by 511) and t [0,1] it foows that { Ee A e A S E Se AS E S v AS t sup max e A e A e AS v Vt) v AS t [0,1] t sup v Vt) E S v AS v AS proving 510) We further investigate the case when step 1 of Agorithm 4 is given by the foowing nonsymmetric two-eve procedure: ALGORITHM 5 Let A S ) be an upper bound of A S ) 1 Perform x x pp T A S p) + p T f S A S x), where x R n, p : R m R n is a fu-rank inear mapping, m < n, and f S = S 2 f Here, the symbo + denotes the pseudoinverse 2 Perform the foowing smoothing procedure: 512) x I ω ) A S ) A S x+ ω A S ) f S, ω 0,1) REMARK 53 Agorithm 5 is impemented as foows assuming the coarse-eve matrix is reguar): first, the auxiiary smoothed proongator p S = Sp = I α 1 A) I α ν A)p is evauated This is done by settingp 0 = p and evauating p i = I ω i A)p i 1, i = 1,,ν Then we set p S = p ν Further, we cacuate the coarse-eve matrix A S 2 = p S ) T Ap S Note thata S 2 = p T A S p = p T S 2 Ap We decompose the matrixa S 2 by a Choesky decomposition and, in each iteration, evauate the action of the inverse of A S 2 = p T A S p in the usua way by doube backward substitution Note that the appication that we have in mind dea withs = poa) having degree about 1 H 2 h and a proongator p given by generaized aggregation Here, h is the resoution of a finite eement mesh andh the characteristic diameter of the aggregates In Exampe 31 we have demonstrated on a mode exampe that such a construction eads to a sparse coarse-eve matrix p T S 2 Ap The error propagation operator of Agorithm 5 is E = I ω ) A S ) A S I pp T A S p) + p T A S ) The foowing emma is a two-eve variant of the mutieve estimate of [5] Its proof is basicay a routine simpification of the origina, where certain technicaities have been taken care of Unike the origina convergence theorem of [5], this emma gives an estimate satisfying the pointwise convergence property Further, our emma avoids the assumption of A - boundedness of the interpoation operatorq : u R n pv Rangep) in 54), that is, this },

20 420 J BROUSEK, et a emma gives the estimate based soey on the weak approximation condition and of course, the assumption on the smoother) Note that the simpest error estimate for a two-eve method, that is, the estimate by Achi Brandt in [6] based on the orthogonaity trick of Céa, does not satisfy the pointwise convergence property LEMMA 54 Let A be a symmetric positive semidefinite matrix, A) A), and p an n m fu-rank matrix with m < n Further, et R be a symmetric positive definite n n matrix such that 513) K I R 1 A is positive semidefinite in the A-inner product Note that from the symmetry of R, the A- symmetry of K foows) We assume there is a constant C R > 0 such that for a w R n it hods that 514) 1 A) w 2 C R R 1 w,w) LetV R n Assume further there is a constantc A = C A V) > 0 such that for everyu V, there exists a vectorv R m such that 515) u pv 2 C A A) u 2 A Then for every errore V and the error propagation operator of the nonsymmetric two-eve method, it hods that E = K[I pp T Ap) + p T A] = I R 1 A)[I pp T Ap) + p T A], 516) Ee 2 A C A C R ) e 2 A Proof We start with introducing some notations First, et P 2 be an A-orthogona projection ontorangep) Further, we set see 513) Note that T = I K = R 1 A, E 2 = I P 2 ; 517) E = I T)E 2 Let u V Then 515) hods foru Our goa is to prove that Eu 2 A 1 1 ) u 2 C A, wherec is a positive constant dependent onc A andc R Instead, we wi prove an equivaent estimate 518) u,u) A C[u,u) A Eu,Eu) A ] Our first step wi be to estabish 519) u,u) A Eu,Eu) A P 2 u,u)+te 2 u,e 2 u) A

21 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 421 Ceary, we have thati = E 2 +P 2 SinceP 2 is ana-orthogona projection, the decomposition w = w 1 +w 2, w 1 = E 2 w = I P 2 )w, w 2 = P 2 w is A-orthogona, that is, taking into account that P 2 u,p 2 u) A = P 2 u,u) A, 520) u,u) A = E 2 u,e 2 u) A +P 2 u,u) A Further, we notice that from 517) it foows that E 2 = TE 2 +E Using this identity, 517), and the fact that T = R 1 A is A-symmetric, we have E 2 u,e 2 u) A = TE 2 +E)u,TE 2 +E)u) A = Eu,Eu) A +2TE 2 u,eu) A +TE 2 u,te 2 u) A = Eu,Eu) A +2TE 2 u,i T)E 2 u) A +TE 2 u,te 2 u) A = Eu,Eu) A +2TI T)E 2 u,e 2 u) A +T 2 E 2 u,e 2 u) A = Eu,Eu) A +TI T)E 2 u,e 2 u) A +[TI T)+T 2 ]E 2 u,e 2 u) A = Eu,Eu) A +TI T)E 2 u,e 2 u) A +TE 2 u,e 2 u) A By assumption, K = I R 1 A = I T is positive semidefinite in the A-inner product Further, since R 1 is positive definite, R 1 A is positive semidefinite in the A-inner product Hence, it foows that the spectrum of T = R 1 A is contained in [0,1] and TI T) is A-positive semidefinite Therefore, TI T)E 2 u,e 2 u) A 0, and the previous identity becomes 521) E 2 u,e 2 u) A Eu,Eu) A +TE 2 u,e 2 u) A Now, 520) and 521) yied u,u) A Eu,Eu) A +TE 2 u,e 2 u) A +P 2 u,u) A, proving 519) Using 519), we wi now estabish the inequaity 522) u,u) A C[P 2 u,u) A +TE 2 u,e 2 u) A ], where C = CC A,C R ) > 0 Note that from 522), the desired resut 518) and in turn aso the fina statement 516) foow Indeed, sincep 2 = I E 2 ande 2 is ana-orthogona projection, we have P 2 u,u) A +TE 2 u,u) A = u,u) A E 2 u,u) A +TE 2 u,e 2 u) A = u,u) A E 2 u,e 2 u) A +TE 2 u,e 2 u) A = u,u) A I T)E 2 u,e 2 u) A = u,u) A Eu,E 2 u) A Further, sincei T = I R 1 A is ana-symmetric operator with the spectrum contained in the interva[0,1], we aso have Eu,Eu) A = I T)E 2 u,i T)E 2 u) A I T)E 2 u,e 2 u) A = Eu,E 2 u) A Thus, P 2 u,u) A +TE 2 u,u) A u,u) A Eu,Eu) A, and 518) foows from 522) Thus, it remains to estabish 522)

22 422 J BROUSEK, et a Let Q 2 : u V pv, v R m, be an operator fufiing 515), that is, Q 2 : u pv, u V, v R m, such that u pv 2 = I Q 2 )u 2 C A A) u 2 A SinceE 2 = I P 2, we havei = E 2 +P 2, and therefore u,u) A = P 2 u,u) A +E 2 u,u) A 523) = P 2 u,u) A +E 2 u,i Q 2 )u) A +E 2 u,q 2 u) A = P 2 u,u) A +E 2 u,i Q 2 )u) A, because it hods thate 2 u,q 2 u) A = 0, which is a consequence ofrangeq 2 ) = Rangep) andrangee 2 ) = RangeI P 2 ) = Rangep) A By the Cauchy-Schwarz inequaity and assumptions 515) and 514), we get E 2 u,i Q 2 )u) A AE 2 u I Q 2 )u C 1/2 A A) 1/2 u A AE 2 u C A C R ) 1/2 u A R 1 E 2 u,ae 2 u) 1/2 = C A C R ) 1/2 u A TE 2 u,ae 2 u) 1/2 By substituting the above estimate into 523), using the Cauchy-Schwarz inequaity a 1 b 1 +a 2 b 2 a 2 1 +a 2 2) 1/2 b 2 1 +b 2 2) 1/2, and the inequaity P 2 u,u) A u,u) A, we get u,u) A P 2 u,u) A +C A C R ) 1/2 u A TE 2 u,ae 2 u) u,u) 1/2 A P 2u,u) 1/2 A +C AC R ) 1/2 u A TE 2 u,ae 2 u) ) u A 1 P 2 u,u) 1/2 +C A C R ) 1/2 TE 2 u,u) A ) 1/2 u A 1+C A C R ) 1/2 [P 2 u,u) A +TE 2 u,e 2 u) A ] 1/2, proving 522) with C = 1+C A C R Using 519), the estimate 518) foows from 522), and in turn the statement 516) Next we anayze the nonsymmetric Agorithm 4 with step 1 given by Agorithm 5 Summing up, we anayze the foowing abstract method: ALGORITHM 6 Let A S ) A S ) be an avaiabe upper bound 1 Perform x x pp T A S p) + p T f S A S x), where x R n, p : R m R n is a fu-rank inear mapping, m < n, and f S = S 2 f Here the symbo + denotes the pseudoinverse 2 Perform the foowing smoothing procedure: x I ω ) A S ) A S x+ ω A S ) f S, ω 0,1) 3 Fori = 1,,d do x I α i A)x+α i f Here I α 1 A) I α d A) = S A

23 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 423 THEOREM 55 Let A S ) A S ) Assume that there is a constantc > 0 such that 524) u R n v R m : u pv 2 C A S ) u 2 A Then the error propagation operator E of the nonsymmetric Agorithm 4 with step 1 given by Agorithm 5 that is, Agorithm 6) satisfies E 2 A C < 1 ω Proof In view of Theorem 51, we need to estimate E A sup t [0,1] { t sup u Vt) wherevt) is given by 59) By 524) and 59), E S u AS u AS 525) u Vt) v R n : u pv 2 C/t2 A S ) u 2 A S Ceary, the smoother 512) can be written in the form x I R 1 A S )x+r 1 f S with the A-symmetric positive definite error propagation operator K = I R 1 A S = I ω A S ) A S, ω [0,1], }, and therefore, R 1 = ω A S ) I Thus, 514) is satisfied with a constant C R = 1/ω By this, a assumptions of Lemma 54 appied to Agorithm 5 are satisfied for a u Vt) with constants C A = C/t 2 see 525)) and C R = 1/ω Hence, by Lemma 54, choosingv = Vt), it foows that E S u 2 A sup S u Vt) u 2 A S C ωt 2, ω [0,1], and Theorem 51 gives E 2 A sup t [0,1] {t 2 1 )} 1 1+ C ωt 2 It remains to evauate the maximum of the function ) φt) = t C, t [0,1] ωt 2 By inspecting the first and second derivative, we find that the function φt), t R, does not attain its maximum in the interva 0,1) Thus, the function φt), t [0,1], attains its maximum for t = 1, proving our statement

24 424 J BROUSEK, et a 52 Anaysis of the symmetrization The error propagation operator of Agorithm 6 can be written as E = SE S = SS Q, S = I ω A S ) A S, Q = I pp T A S p) + p T A S The operatorqis ana S -orthogona projection ontorangep)) A S Our goa is to derive the operator E sym = E E, where E is thea-adjoint operator SinceS and S are poynomias ina, A S = S 2 A, and Q isa S -symmetric, we have for anyx,y R n, AE Ex,y = AEx,Ey = ASS Qx,SS Qy Thus, we concude that = A S S Qx,S Qy = A S QS ) 2 Qx,y = AS 2 QS ) 2 Qx,y E sym = S 2 QS ) 2 Q One can easiy see that the above error propagation operator corresponds to the foowing agorithm: ALGORITHM 7 Let A S ) A S ) be an avaiabe upper bound 1 Perform x x pp T A S p) + p T f S A S x), where x R n, p : R m R n is a fu-rank inear mapping, m < n, and f S = S 2 f Here the symbo + denotes the pseudoinverse 2 Perform twice the foowing smoothing procedure: x I ω ) A S ) A S x+ ω A S ) f S, ω 0,1) 3 Performx x pp T A S p) + p T f S A S x) 4 Perform twice: for i = 1,,d do x I α i A)x + α i f Here, I α 1 A) I α d A) = S The foowing convergence bound for E sym A is a trivia consequence of Theorem 55 THEOREM 56 Let A S ) A S ) Assume there is a constantc > 0 such that u R n v R m : u pv 2 C A S ) u 2 A Then the error propagation operatore sym of the symmetric Agorithm 7 satisfies E sym A = E 2 A C ω < 1 Proof By E sym = E E and Theorem 55 we have ) AE sym e,e = Ee 2 A C e 2 A ω Since the operatore sym isa-symmetric positive semidefinite, it immediatey foows that competing the proof E sym A = λ max E sym ) = E 2 A C, ω

25 MULTILEVEL METHODS WITH AGGRESSIVE COARSENING 425 THEOREM 57 Assume the proongatorpsatisfies the assumption 27) and the smoother S is given by S = pa), withp) defined by 414), where = A) We set A S ) = Assume that there is a constantc > 0 such that A) 1+2degS)) 2 degs) c H h Then the error propagation operator E of Agorithm 6 and the error propagation operator E sym of Agorithm 7 satisfy where C > 1 is independent of both h and H Proof By assumption 27), we have E sym A = E 2 A 1 1 C, 526) e R n v R m : e pv 2 C A A) Further, since we assume degs) c H h, the statement 413) gives S 2 A) S 2 A) Substituting the above estimate into 526) yieds ) 2 H e 2 h A ) A) h 2 A) 1+2degS)) 2 C H u R n v R m : u pv 2 C A S ) u 2 A The proof now foows from Theorem A mutieve extension In this section we investigate the case when step 1 of Agorithm 4 uses a mutigrid sover for soving the transformed probem 57) Our goa is to prove the resut independent of the first- and second-eve resoution Note that in this section we consider the nonsymmetric Agorithm 4 The extension to a symmetric agorithm is possibe by foowing the arguments of Section 52 We consider a standard variationa mutigrid with proongators fu-rank inear mappings) 527) P +1 : R n +1 R n, n 1 = n, n +1 < n, = 1,,L, that is, mutigrid with coarse-eve matrices A S = P 1 ) T A S P 1, P 1 = P 1 2 P 1 L, P1 1 = I, and restrictions given by the transpose of the proongators Here, L denotes the number of eves The particuar case of interest isn 2 n 1 For the sake of brevity we assume simpe Richardson pre- and post-smoothers 528) x I ω A S )AS ) x + ω A S )f, x,f R n,