Drury, Stephen W.

Loisel, Sébastien

English

Domain decomposition methods are used to find the numerical solution of large boundary value problems in parallel. In optimized domain decomposition methods, one solves a Robin subproblem on each subdomain, where the Robin parameter a must be tuned (or optimized) for good performance.We show that the 2-Lagrange multiplier method can be analyzed using matrix analytical techniques and we produce sharp condition number estimates.</p

Heriot Watt Pure

Sharp condition number estimates forthe symmetric 2-Lagrange multipliermethodStephen W. Drury∗ and Se´bastien Loisel†Abstract Domain decomposition methods are used to find the numericalsolution of large boundary value problems in parallel. In optimized domaindecomposition methods, one solves a Robin subproblem on each subdomain,where the Robin parameter a must be tuned (or optimized) for good per-formance. We show that the 2-Lagrange multiplier method can be analyzedusing matrix analytical techniques and we produce sharp condition numberestimates.1 Introduction.Consider the model problem∆u = f in Ω and u = 0 on ∂Ω, (1)where Ω is the domain, f is a given forcing and u ∈ H10 (Ω) is the unknownsolution. In the present paper, we describe a symmetric 2-Lagrange multiplier(S2LM) domain decomposition method to solve elliptic problems such as (1).When we discretize (1) using e.g. piecewise linear finite elements, we obtaina linear system of the formAu = f , (2)where u ∈ Rn is the finite element coefficient vector of the approximation tothe solution u of (1).∗ Dept. of Mathematics, McGill University, 805 Sherbrooke Street West, Montreal, Que-bec, Canada, H3A 2K6, drury@math.mcgill.ca† Dept. of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom,S.Loisel@hw.ac.uk12 Stephen W. Drury and Se´bastien LoiselWe now consider the domain decomposition [Toselli and Widlund, 2005]Ω = Γ∪Ω1∪. . .∪Ωp, where Ω1, . . . , Ωp are the (open, disjoint) “subdomains”and Γ = Ω ∩⋃pk=1 ∂Ωk is the “artificial interface”. We introduce the “localproblems” ∆uk = f in Ωk, (PDE)uk = 0 on ∂Ωk ∩ ∂Ω, (natural b.c.)(a+Dν)uk = λk on ∂Ωk ∩ Γ, (artificial b.c.)(3)where a > 0 is the Robin tuning parameter and k = 1, . . . , p and Dν denotesthe directional derivative in the outwards pointing normal ν of ∂Ωk. The in-terface Γ is artificial in that it is not a natural part of the “physical problem”(1) but instead is introduced purely for the purpose of calculation.We again discretize the systems (3) using a finite element method. TheRobin b.c. in (3) gives rise to a mass matrix on the interface Γ ∩ ∂Ωk, whichis spectrally equivalent to aI. Hence, after a suitable “mild” change of basis,we obtain the discrete system[AIIk AIΓkAΓIk AΓΓk + aI] uk︷ ︸︸ ︷[uIkuΓk]=fk︷ ︸︸ ︷[fIkfΓk]+[0λk]. (4)The FETI-2LM algorithm was introduced in [Farhat et al., 2000] for caseswithout cross-points, while the general case including cross points was intro-duced and analyzed in [Loisel, 2011a]. The method consists of finding thevalue of λ = [λT1 , . . . ,λTp ]T which yields solutions u1, . . . ,up to (4) in sucha way that u1, . . . ,up meet continuously across Γ and glue together into theunique solution u of (2).The main result of the present paper is a new estimate the condition num-ber of FETI-2LM algorithms using matrix analytical techniques. This newidea produces sharp condition number estimates with much more straightfor-ward proof techniques than the techniques used in [Loisel, 2011a] (where theestimates are not sharp). As a result, the present paper is a logical follow-upto [Loisel, 2011a].The present paper focuses on 1-level algorithms which are known not toscale. Scalable algorithms are considered in [Loisel, 2011b] and [Drury andLoisel, 2011].Our paper is organized as follows. In Section 2, we give the symmetric 2-Lagrange multiplier method for general domains with cross points. In Section3, we give spectral estimates including our main result on the condition num-ber of the symmetric 2-Lagrange multiplier system. in Section 4, we verifythis Theorem with some numerical experiments.Sharp condition estimates of S2LM 32 The symmetric 2-Lagrange multiplier method.We now describe the 2-Lagrange multiplier method that we analyze in thepresent paper. Consider the local problems (4) and eliminate the interiordegrees of freedom to obtain the relationauG︷ ︸︸ ︷u1...up =Q︷ ︸︸ ︷a(S1 + aI)−1. . .a(S1 + aI)−1g︷ ︸︸ ︷g1...gp+λ︷ ︸︸ ︷λ1...λp , (5)whereSk = AΓΓk − AΓIkA−1IIkAIΓk and gk = fΓk −AΓIkA−1IIkfIkare the “Dirichlet-to-Neumann maps” and “accumulated right-hand-sides”.The matrices Sk are symmetric and semidefinite. Since Q = a(S + aI)−1,we find that the spectrum σ(Q) is contained in the set [ǫ, 1 − ǫ] ∪ {1} forsome ǫ > 0. The eigenvalue 1 of Q comes from the kernel of S and hencethe kernel of Q− I is spanned by the indicating functions of the subdomainsthat “float”. We define E to be the orthogonal projection onto the kernel ofQ− I.2.1 Relations between (4) and (2) and continuity.We define the boolean restriction matrix Rk by selecting rows of the n × nidentity matrix corresponding to those vertices of Ω that are in Ω¯k ∩Ω. Asa result, from a finite element coefficient vector v corresponding to a finiteelement function v ∈ H10 (Ω), we can define a finite element coefficient vectorvk = Rkv, which corresponds to a finite element function v ∈ H1(Ωk) ∩H10 (Ω), which is obtained by restricting v to Ωk.The identity∫Ω=∑pk=1∫Ωkinduces the following relations between (4)and (2):A =p∑k=1RTk[AIIk AIΓkAΓIk AΓΓk]Rk and f =p∑k=1fk. (6)Each interface vertex xi ∈ Γ is adjacent to mi ≥ 2 subdomains. As aresult, the “many-sided trace” uG defined by (5) contains mi entries corre-sponding to xi, one per subdomain adjacent to xi. We define the orthogonalprojection matrix K which averages function values for each interface vertexxi. A many-sided trace uG corresponds to local functions u1, . . . ,up that4 Stephen W. Drury and Se´bastien Loiselmeet continuously across Γ if and only ifKuG = uG. (7)2.2 A problem in λ.The symmetric 2-Lagrange multiplier (S2LM) system is given by(Q−K)λ = −Qg. (8)We further let E be the orthogonal projection onto the kernel of Q− I.Lemma 1. Assume that ‖EK‖ < 1. The problem (2) is equivalent to (8).Proof. In order to solve (2) using local problems (4), one should find Robinboundary values λ1, . . . ,λp which result in local solutions u1, . . . ,up thatmeet continuously across Γ . As a result, we impose the condition (7), whichwe multiply by a > 0 and convert to an expression in λ using (5) to obtainKa(S + aI)−1(λ + g) = a(S + aI)−1(λ + g) or(I −K)Qλ = (K − I)Qg (9)With this continuity condition, there is clearly a unique u which restricts tothe uj :uj = Rju, j = 1, . . . , p. (10)Imposing continuity is not sufficient, we must also ensure that the “fluxes”match. Indeed, if we impose on the solution u of (10) that the equation (2)should hold, one obtainsf = Au(6)=p∑j=1RTΓjANjRΓju(10)=p∑j=1RTΓjANjuj (11)(4),(6)= f −p∑j=1RTj(0λj − auΓj)(12)Canceling the f terms on each side and multiplying by K, we obtain Kλ −KauG = 0. Using (5), we obtainK(Q− I)λ = −KQg. (13)We add (9) and (13) to obtain (8).To see that the solution of (8) is unique, observe that the ranges of Eand K intersect trivially by the hypothesis that ‖EK‖ < 1. As a result, theSharp condition estimates of S2LM 5eigenspace of Q of eigenvalue 1 intersects trivially with the range of K andQ−K is nonsingular. ⊓⊔We will further discuss the choice of the parameter a in Section 3.1.3 Spectral estimates.If we use GMRES or MINRES on the symmetric indefinite system (8), theresidual norm can be estimated as a function of the condition number ofQ −K, cf. [Driscoll et al., 1998]. In order to estimate the condition numberof Q −K, we begin by giving a canonical form for the pair of projections Eand K.Lemma 2. Let E and K be orthogonal projections. There is a choice of or-thonormal basis that block diagonalizes E and K simultaneously and suchthat the blocks Ek and Kk of E and K satisfyEk ∈{0, 1,[1 00 0]}and Kk ∈{0, 1,[c2k ckskcksk s2k]}, (14)where ck = cos θk > 0, sk = sin θk > 0 and θk ∈ (0, π/2) is a “principalangle” relating E and K.The canonical form (14) can be obtained from the CS decomposition [Davisand Kahan, 1969] by starting from E = diag(I, 0) and picking orthonormalbases for the range and kernel of K. Due to space constraints, we omit thisargument.We also give a technical lemma which describes the spectrum of a sum ofcertain symmetric matrices.Lemma 3. Let X, Y be symmetric matrices of dimensions m×m. Let 0 <ymin < ymax and assume that |σ(Y )| ⊂ [ymin, ymax]. Denote by ρ(X) thespectral radius of X and assume that ρ(X) < ymin. Then,|σ(X + Y )| ⊂ [ymin − ρ(X), ymax + ρ(X)]. (15)Proof. This follows from a Theorem of Weyl [Horn and Johnson, 1990, The-orem 4.3.1, pp 181–182]. ⊓⊔3.1 Condition number of Q−K.We now come to our main result.6 Stephen W. Drury and Se´bastien LoiselTheorem 1. Let ǫ > 0. Assume that σ(Q) ⊂ [ǫ, 1 − ǫ] ∪ {1}. Let E,K beorthogonal projections and assume that ‖EK‖ < 1. Then we have the sharpestimates|σ(Q −K)| ⊂[ǫ+√(1 + ǫ)2 − 4‖EK‖2ǫ− 12, 1], and (16)κ(Q−K) ≤ 2ǫ+√(1 + ǫ)2 − 4‖EK‖2ǫ− 1 = O((1 − ‖EK‖)−1ǫ−1).(17)Proof. Let X = Q− 12I−ǫE and Y = 12I+ǫE−K. Then, Q−K = X+Y andwe are in a position to use Lemma 3. We now estimate the spectral propertiesof X and Y .Spectral properties of X: Recall that E projects onto the eigenspaceof Q with eigenvalue 1. As a result, after some orthonormal change of basis,we find that Q = diag(Q0, I) and E = diag(0, I) and henceρ(X) ≤ 12− ǫ. (18)Spectral properties of Y : Lemma 2 shows that E and K block diago-nalize simultaneously and Y is also block diagonal in the same basis. Using(14), we find that the kth block Yk of Y is given byYk =12 if Ek = Kk = 0,− 12 if Ek = 0, Kk = 1,12 + ǫ if Ek = 1, Kk = 0,[12 + ǫ− c2k −cksk−cksk 12 − s2k];(19)where the case Ek = Kk = 1 is excluded by the hypothesis that ‖EK‖ < 1.As a result, the eigenvalues of Yk are in the set {± 12 , 12 + ǫ, λ±(c2k)}, whereλ±(c2k) =ǫ±√(1 + ǫ)2 − 4c2kǫ2. (20)Note that ‖EK‖ =√ρ(EKE) = ck and that the functions λ±(c2k) aremonotonic in c2k. Hence, we find the following bounds for the modulus of aneigenvalue of Y :|σ(Y )| ⊂[ ymin︷ ︸︸ ︷√(1 + ǫ)2 − 4‖EK‖2ǫ− ǫ2,ymax︷ ︸︸ ︷12+ ǫ]. (21)Sharp condition estimates of S2LM 710−5 10−4 10−3 10−2 10−1 1001001011021031041051−||EK||κ10−5 10−4 10−3 10−2 10−1 100100101102103104105106107εκFig. 1 Comparing random Q−K (points) versus the estimate (17) (solid). Top: ǫ = 0.1,varying ‖EK‖, 3000 repetitions. Bottom: ‖EK‖ = 0.99, varying ǫ, 3000 repetitions.Combining (15), (18) and (21) gives (16).The sharpness of the estimate is shown by considering the example Q =diag(1, 1− ǫ) and K =[c2 c√1− c2c√1− c2 1− c2]for c = 0 and c = ‖EK‖. ⊓⊔8 Stephen W. Drury and Se´bastien LoiselIn view of Theorem 1, the Robin parameter a should be chosen so as tomake ǫ as large as possible. This occurs precisely when a is the geometricmean of the extremal positive eigenvalues of S. More details can be found in[Loisel, 2011a].4 Numerical verification.We verify numerically the validity of Theorem 1 by generating random 5× 5matrices Q and E as follows. We set Q = diag(ǫ, q, 1 − ǫ, 1, 1) where q ischosen randomly between ǫ and 1− ǫ. We generate randomly a 2-dimensionalspace and set K to be the orthogonal projection onto that space. We comparethe resulting condition number κ = κ(Q−K) against (17), cf. Fig. 1.ReferencesChandler Davis and W. M. Kahan. Some new bounds on perturbation ofsubspaces. Bulletin of the American Mathematical Society, pages 863–868,1969.Tobin A. Driscoll, Kim-Chuan Toh, and Lloyd N. Trefethen. From potentialtheory to matrix iterations in six steps. SIAM Review, pages 547–578,1998.Stephen W. Drury and Se´bastien Loisel. The performance of opti-mized Schwarz and 2-Lagrange multiplier preconditioners for GMRES.Manuscript, 2011.Charbel Farhat, Antonini Macedo, Michel Lesoinne, Francois-Xavier Roux,Fre´de´ric Magoule`s, and Armel de La Bourdonnaie. Two-level domain de-composition methods with lagrange multipliers for the fast iterative solu-tion of acoustic scattering problems. Computer Methods in Applied Me-chanics and Engineering, 184:213–239, 2000.Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge Uni-versity Press, 1990.Se´bastien Loisel. Condition number estimates for the nonoverlapping opti-mized Schwarz method and the 2-Lagrange multiplier method for generaldomains and cross points. Submitted to SIAM Journal on Numerical Anal-ysis, 2011a.Se´bastien Loisel. Condition number estimates and weak scaling for 2-level2-Lagrange multiplier methods for general domains and cross points. Sub-mitted, 2011b.Andrea Toselli and Olof B. Widlund. Domain Decomposition Methods –Algorithms and Theory, volume 34 of Springer Series in ComputationalMathematics. Springer Berlin Heidelberg, 2005.

Sharp condition number estimates for the symmetric 2-lagrange multiplier method

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Sharp condition number estimates for the symmetric 2-lagrange multiplier method

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Heriot Watt Pure

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