Theoretically supported scalable FETI for numerical solution of variational inequalities

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1 Theoretically supported scalable FETI for numerical solution of variational inequalities Zdeněk Dostál and David Horák Abstract The FETI method with a natural coarse grid is combined with recently proposed optimal algorithms for the solution of bound and/or equality constrained quadratic programming problems in order to develop a scalable solver for elliptic boundary variational inequalities such as those describing equilibrium of a system of bodies in mutual contact. A discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints. The latter is then modified by means of orthogonal projectors to the natural coarse grid introduced by Farhat, Mandel and Roux. Finally the classical results on linear scalability for linear problems are extended to boundary variational inequalities. The results are validated by numerical experiments. The experiments also confirm that the algorithm enjoys the same parallel scalability as its linear counterpart. Keywords: Domain decomposition, variational inequality, scalability, parallel algorithms, FETI Acknowledgment: This research is supported by grant No. 0/04/45 of the Grant Agency of the Czech Republic and by projects of the Ministry of Education No. ET and ME64. Introduction The FETI (Finite Element Tearing and Interconnecting) domain decomposition method was originally proposed by Farhat and Roux [26] for parallel solving of the linear problems described by elliptic partial differential equations. Its key ingredient FEI VŠB-Technical University of Ostrava, CZ Ostrava, Czech Republic

2 is decomposition of the spatial domain into non-overlapping subdomains that are glued by Lagrange multipliers, so that, after eliminating the primal variables, the original problem is reduced to a small, relatively well conditioned, typically equality constrained quadratic programming problem that is solved iteratively. The time that is necessary for both the elimination and iterations can be reduced nearly proportionally to the number of the processors, so that the algorithm enjoys parallel scalability. Observing that the equality constraints may be used to define so called natural coarse grid, Farhat, Mandel and Roux [25] modified the basic FETI algorithm so that they were able to adapt the results by Bramble, Pasciak and Schatz [5] to prove its numerical scalability, i.e. asymptotically linear complexity. The comprehensive review of the mathematical results related to the FETI methods may be found in the monograph by Tosseli and Widlund [38]. If the FETI procedure is applied to an elliptic variational inequality, the resulting quadratic programming problem has not only the equality constraints, but also the non-negativity constraints. Even though the latter is a considerable complication as compared with linear problems, it seems that the FETI procedure should be even more powerful for the solution of variational inequalities than for the linear problems. The reason is that FETI not only reduces the original problem to a smaller and better conditioned one, but it also replaces for free all the inequalities by the bound constraints. Promising experimental results by Dureisseix and Farhat [22] supported this claim and even indicated numerical scalability of their method. Similar results were achieved also for the FETI DP (Dual Primal) method introduced by Farhat et al. [24]. The FETI DP method is very similar to the original FETI, the only difference is that it enforces the continuity of displacements at corners on primal level. A new Lagrange multipliers algorithm, FETI C, based on FETI DP and on active set strategies with additional planning steps and preconditioning, was introduced by Farhat et al. [, 22]. Its scalability was demonstrated experimentally. Another approach yielding an experimental evidence of scalability was proposed by Dostál, Friedlander, Gomes, Santos and Horák [2, 3, 4]. The algorithm combined FETI with a special variant of the augmented Lagrangian method [0]. Scalability was later proved for an algorithm that enforced the equality constraints by the optimal dual penalty [5, 6] and solved the resulting bound constrained problem by recent in a sense optimal algorithms [7, 2]. Using the same algorithms, Dostál, Horák and Stefanica then proved numerical scalability for a FETI DP algorithm applied to the coercive problems discretized by means of either nodal [8] or mortar [9] Lagrange multipliers. Most recently, the scalability results were proved also for semicoercive problems [20]. The rate of convergence was given in terms of the effective condition number of the dual Schur complement of the stiffness matrix which was proved to be bounded by CH 2 /h 2, where C is a constant independent of the discretization and decomposition parameters h and H, respectively. The estimates did not assume any preconditioning. Indeed, numerical experiments by the present authors, V. Vondrák and M. Lesoinne indicated that the performance of our FETI DP based algorithms may be considerably improved by preconditioning. 2

3 It should be noted that the effort to develop scalable solvers for variational inequalities was not restricted to FETI. For example, using ideas related to Mandel [34], Kornhuber, Krause, Sander and Wohlmuth [30, 3, 40, 32, 33] gave an experimental evidence of numerical scalability of the algorithm based on monotone multigrid. Probably the first theoretical results concerning development of scalable algorithms were proved by Schöberl [36, 37]. In this paper, we use the FETI method with a natural coarse grid to develop a scalable algorithm for numerical solution of both coercive and semicoercive variational inequalities. The rate of convergence is again given in terms of the effective condition number of the dual Schur complement of the stiffness matrix, but this time it is bounded by CH/h. The paper is organized as follows. After describing a model problem, we briefly review the FETI methodology [2] that turns the variational inequality into the well conditioned quadratic programming problem with bound and equality constraints. Then we review our algorithms for solution of the resulting bound and equality constrained quadratic programming problem whose rate of convergence may be expressed in terms of bounds on the spectrum of the dual Schur complement matrix [2, 8, 9]. Finally we present the main results about optimality of our method and give results of numerical experiments with parallel implementation of the algorithm in PETSc [3]. 2 Model problem For the sake of simplicity, we shall reduce our analysis to a simple model problem, but our reasoning is valid also in more general cases, including contact problems of 2D and 3D elasticity, provided that the conditions exploited in the proof of the results by Farhat, Mandel and Roux [25] are satisfied. Let Ω = Ω Ω 2, Ω = (0, ) (0, ) and Ω 2 = (, 2) (0, ) denote open domains with boundaries Γ, Γ 2 and their parts Γ i u, Γ i f, Γi c formed by the sides of Ω i, i =, 2 as in Figure a or Figure b. Let H (Ω i ), i =, 2 denote the Sobolev space of the first order in the space L 2 (Ω i ) of the functions on Ω i whose squares are integrable in the sense of Lebesgue. Let V i = { } v i H (Ω i ) : v i = 0 on Γ i u denote the closed subspaces of H (Ω i ), i =, 2, and let V = V V 2 and K = { (v, v 2 ) V : v 2 v 0 on Γ c } denote the closed subspace and the closed convex subset of H = H (Ω ) H (Ω 2 ), respectively. The relations on the boundaries are in terms of traces. On H we shall 3

4 define a symmetric bilinear form and a linear form a(u, v) = 2 i= Ω i l(v) = ( u i x 2 i= v i x + ui y Ω i f i v i dω, ) v i dω y where f i L 2 (Ω i ), i =, 2 are the restrictions of for (x, y) (0, ) [0.75, ) f(x, y) = 0 for (x, y) (0, ) [0, 0.75) and (x, y) (, 2) [0.25, ) 3 for (x, y) (, 2) [0, 0.25) for coercive problem and 3 for (x, y) (0, ) [0.75, ) f(x, y) = 0 for (x, y) (0, ) [0, 0.75) and (x, y) (, 2) [0.25, ) for (x, y) (, 2) [0, 0.25) for semicoercive problem. Thus we can define a problem to find min q(u) = a(u, u) l(u) subject to u K. (2.) 2 f 3-3 f Ω 2 Ω Ω 2 Ω u f c f Γ Γ Γ Γ Γ u 2 Fig. a: Coercive model problem 2 u f c f Γ Γ Γ Γ Fig. b: Semicoercive model problem Figure : Model problems We shall consider two variants of the Dirichlet data. In the first case, both the membranes are fixed on the outer edges as in Figure a, so that Γ u = {(0, y) IR 2 : y [0, ]}, Γ 2 u = {(2, y) IR 2 : y [0, ]}. 4

5 Since the Dirichlet conditions are prescribed on parts Γ i u, i =, 2 of the boundaries of the both membranes with positive measure, the quadratic form a is coercive which guarantees both existence and uniqueness of the solution [28, 27]. In the second case, only the left membrane is fixed on the outer edge and the right membrane has no prescribed displacement as in Figure b, so that Γ u = {(0, y) IR 2 : y [0, ]}, Γ 2 u =. Even though a is in this case only semidefinite, the form q is still coercive due to the choice of f so that it has again the unique solution [28, 27]. More details about this particular model problem may be found in [2]. The solution of the model problem may be interpreted as the displacement of two membranes under the traction f. The left edge of the right membrane is not allowed to penetrate below the right edge of the left membrane. 3 Domain decomposition and discretization In our definition of the problem, we have so far used only the natural decomposition of the spatial domain Ω into Ω and Ω 2. However, to enable efficient application of the domain decomposition methods, we can optionally decompose each Ω i into subdomains Ω i,..., Ω ip, p > as in Figure 2. The continuity in Ω and Ω 2 of the global solution assembled from the local solutions u ij will be enforced by the gluing conditions u ij (x) = u ik (x) that should be satisfied for any x on the interface Γ ij,ik of Ω ij and Ω ik. After modifying appropriately the definition of problem (2.), introducing regular grids in the subdomains Ω ij that match across the interfaces Γ ij,kl, indexing contiguously the nodes and entries of corresponding vectors in the subdomains, and using the Lagrangian finite element discretization, we get the discretized version of problem (2.) with auxiliary domain decomposition that reads min 2 u Ku f u s.t. B I u 0 and B E u = 0. (3.) In (3.), K denotes a block diagonal positive semidefinite stiffness matrix, the full rank matrices B I and B E describe the discretized non-penetration and gluing conditions, respectively, and f represents the discrete analog of the linear term l(u). The rows of B E and B I are filled with zeros except and - in positions that correspond to the nodes with the same coordinates on the artificial or contact boundaries, respectively. In particular, if b i denotes a row of B E or B I, then b i will not have more than four nonzero entries, and for any displacement vector u, b i u will denote the difference or jump between the displacements on each side of the boundary. Some more details may be found in [2]. Our next step is to simplify the problem, in particular to replace the general inequality constraints B I u 0 by the nonnegativity constraints using the duality 5

6 H h Ω,2 Ω, Ω,3 Ω,4 Ω 2,2 Ω 2,4 Ω 2, Ω 2,3 λ λ λ E I E λ E Figure 2: Domain decomposition and discretization theory. To this end, let us introduce the Lagrangian associated with problem (3.) by L(u, λ I, λ E ) = 2 u Ku f u + λ I B I u + λ E B E u, (3.2) where λ I and λ E are the Lagrange multipliers associated with inequalities and equalities, respectively. Introducing the notation [ ] [ ] λi BI λ = and B =, λ E we can observe that B is a full rank matrix and write the Lagrangian briefly as B E L(u, λ) = 2 u Ku f u + λ Bu. It is well known [4] that (3.) is equivalent to the saddle point problem Find (u, λ) so that L(u, λ) = sup inf L(u, λ). (3.3) λ I 0 u For fixed λ, the Lagrange function L(, λ) is convex in the first variable and the minimizer u of L(, λ) satisfies Equation (3.4) has a solution iff Ku f + B λ = 0. (3.4) f B λ ImK, (3.5) which can be expressed more conveniently by means of a matrix R whose columns span the null space of K as R (f B λ) = 0. (3.6) 6

7 The matrix R may be formed directly so that each floating subdomain is assigned to a column of R with ones in positions of the nodal variables that belong to the subdomain and zeros elsewhere. It may be checked that R B is a full rank matrix. The matrix R may also be extracted from K [23]. Now assume that λ satisfies (3.5) and denote by K any matrix that satisfies KK K = K. (3.7) Let us note that a generalized inverse K that satisfies (3.7) may be evaluated at the cost comparable with the Cholesky decomposition of regularized K [23]. It may be verified directly that if u solves (3.4), then there is a vector α such that u = K (f B λ) + Rα. (3.8) After substituting expression (3.8) into problem (3.3) and changing signs, we shall get the minimization problem to find min Θ(λ) s.t. λ I 0 and R (f B λ) = 0 (3.9) where Θ(λ) = 2 λ BK B λ λ BK f. (3.0) Once the solution λ of (3.9) is known, the vector u that solves (3.) can be evaluated by (3.8) and the formula [2] α = (R B BR) R B BK (f B λ), (3.) where B = [ B I, B E ], and the matrix B I is formed by the rows b i of B I that correspond to the positive components of the solution λ characterized by λ i > 0. 4 Natural coarse grid Even though the problem (3.9) is much more suitable for computations than (3.) and was used to efficient solving of the discretized variational inequalities [], further improvement may be achieved by adapting some simple observations and the results of Farhat, Mandel and Roux [25]. Let us denote F = BK B, d = BK f, G = R B, ẽ = R f and let T denote a regular matrix that defines orthonormalization of the rows of G so that the matrix G = T G 7

8 has orthonormal rows. After denoting problem (3.9) reads e = T ẽ, min 2 λ F λ λ d s.t λi 0 and Gλ = e. (4.) Next we shall transform the problem of minimization on the subset of the affine space to that on the subset of the vector space by looking for the solution of (4.) in the form λ = µ + λ, where G λ = e. The following Lemma shows that we can even find λ such that λ I = 0. Lemma 4.. Let B be such that the negative entries of B I are in the columns that correspond to the nodes in the floating subdomain Ω 2. Then there is λ I 0 such that G λ = ẽ. Proof: See [5] (coercive problem) or [6] (semicoercive problem). To carry out the transformation, denote λ = µ + λ, so that 2 λ F λ λ d = 2 µ F µ µ ( d F λ) + 2 λ F λ λ d and the problem (4.) is, after returning to the old notation, equivalent to min 2 λ F λ λ d s.t Gλ = 0 and λ I λ I (4.2) with d = d F λ and λ I 0. Our final step is based on observation that the problem (4.2) is equivalent to min 2 λ (P F P + ρq)λ λ P d s.t Gλ = 0 and λ I λ I, (4.3) where ρ is arbitrary positive constant and Q = G G and P = I Q denote the orthogonal projectors on the image space of G and on the kernel of G, respectively. The regularization term is introduced in order to simplify the reference to the results of quadratic programming that assume regularity of the Hessian matrix of the quadratic form. The problem (4.3) turns out to be a suitable starting point for development of an efficient algorithm for variational inequalities due to the classical estimates of the extreme eigenvalues. To formulate them, we shall denote by α min (A) 8

9 and α max (A) the smallest and the largest eigenvalue of a given symmetric matrix A, respectively. Theorem 4.2. There are constants C > 0 and C 2 > 0 independent of the discretization parameter h and the decomposition parameter H such that α min (P F P ImP ) C and α max (P F P ImP ) P F P C 2 H h. Proof: See Theorem 3.2 of Farhat, Mandel and Roux [25]. Note: The statement of Theorem 3.2 of Farhat, Mandel and Roux [25] gives only an upper bound on the spectral condition number κ(p F P ImP ). However, the reasoning that precedes and substantiates their estimate proves both bounds of (4). 5 Optimal solvers to bound and equality constrained problems We shall now briefly review our in a sense optimal algorithms for the solution of the bound and equality constrained problem (4.3). They combine our semimonotonic augmented Lagrangian method [8] which generates approximations for Lagrange multipliers for the equality constraints in the outer loop with the working set algorithm for bound constrained auxiliary problems in the inner loop [2]. If a new Lagrange multiplier vector µ is used for the equality constraints, the augmented Lagrangian for problem (4.3) can be written as L(λ, µ, ρ) = 2 λ (P F P + ρq)λ λ P d + µ Gλ + ρλ Qλ. The gradient of L(λ, µ, ρ) is given by g(λ, µ, ρ) = (P F P + ρq)λ P d + G T (µ + ρgλ). Let I denote the set of the indices of the bound constrained entries of λ λ. The projected gradient g P = g P (λ, µ, ρ) of L at λ is given componentwise by { gi P g = i for λ i > λ i or i / I g i for λ i = λ i and i I where g i = min{g i, 0}. Our algorithm is a variant of that proposed by Conn, Gould and Toint [6] for identifying stationary points of more general problems. Its modification by Dostál, Friedlander and Santos [0] was used by Dostál and Horák to 9

10 develop a scalable FETI based algorithm, as shown experimentally in [4]. The key to proving optimality results is to combine the adaptive precision control of auxiliary problems in Step with the new update rule for the penalty parameter ρ in Step 4. All the necessary parameters are listed in Step 0, and typical values of these parameters for our model problem are given in brackets. Algorithm 5.. Semi-monotonic augmented Lagrangian method for bound and equality constrained problems (SMALBE). Step 0. {Initialization of parameters} Given η > 0 [η = P d ], β > [β = 0], M > 0 [M = ], ρ 0 > 0 [ρ 0 = 00], and µ 0 [µ 0 = 0], set k = 0. Step. {Inner iteration with adaptive precision control.} Find λ k such that λ k I λ I g P (λ k, µ k, ρ k ) min{m Gλ k, η}. Step 2. {Stopping criterion.} If g P (λ k, µ k, ρ k ) and Gλ k are sufficiently small, then λ k is the solution. end if. Step 3. {Update of the Lagrange multipliers.} µ k+ = µ k + ρ k Gλ k Step 4. {Update the penalty parameter.} If k > 0 and L(λ k, µ k, ρ k ) < L(λ k, µ k, ρ k ) + ρ k Gλ k 2 /2 then ρ k+ = βρ k else ρ k+ = ρ k end if. Step 5. Increase k and return to Step. Step may be implemented by any algorithm for minimization of the augmented Lagrangian L with respect to λ subject to λ I λ I which guarantees convergence of the projected gradient to zero. More about the properties and implementation of SMALBE algorithm may be found in [8]. The unique feature of the SMALBE algorithm is its capability to find an approximate solution of problem (4.3) in a number of steps which is uniformly bounded in terms of the bounds on the spectrum of P F P + ρq [8]. To get bound on the number of matrix multiplication, it is necessary to have algorithm which can solve the problem minimize L(λ, µ, ρ) subject to λ I λ I (5.) with the rate of convergence in terms of the bounds on the spectrum of the Hessian matrix of L. To describe such algorithm, let us recall that the unique solution λ = λ(µ, ρ) of 0

11 (5.) satisfies the Karush-Kuhn-Tucker conditions g P (λ, µ, ρ) = 0. (5.2) Let A(λ) and F(λ) denote the active set and free set of indices of λ, respectively, i.e., A(λ) = {i I : λ i = λ i } and F(λ) = {i : λ i > λ i or i / I}. To enable an alternative reference to the KKT conditions [4], let us define the free gradient ϕ(λ) and the chopped gradient β(λ) by ϕ i (λ) = { gi (λ) for i F(λ) 0 for i A(λ) and β i (λ) = { 0 for i F(λ) (λ) for i A(λ) g i so that the KKT conditions are satisfied if and only if the projected gradient g P (λ) = ϕ(λ) + β(λ) is equal to zero. We call λ feasible if λ i λ i for i I. The projector P to the set of feasible vectors is defined for any λ by by P (λ) i = max{λ i, λ i } for i I, P (λ) i = λ i for i / I. Let A denote the Hessian of L with respect to λ. The expansion step is defined λ k+ = P ( λ k αϕ(λ k ) ) (5.3) with the steplength α (0, A ]. This step may expand the current active set. To describe it without P, let ϕ(λ) be the reduced free gradient for any feasible λ, with entries ϕ i = ϕ i (λ) = min{λ i /α, ϕ i } for i I, ϕ i = ϕ i for i E such that If the inequality P (λ αϕ(λ)) = λ α ϕ(λ). (5.4) β(λ k ) 2 Γ 2 ϕ(λ k ) ϕ(λ k ) (5.5) holds, then we call the iterate λ k strictly proportional. The test (5.5) is used to decide which component of the projected gradient g P (λ k ) will be reduced in the next step. The proportioning step is defined by λ k+ = λ k α cg β(λ k ). The steplength α cg is chosen to minimize L(λ k αβ(λ k ), µ k, ρ k ) with respect to α, i.e., α cg = β(λk ) g(λ k ) β(λ k ) Aβ(λ k ).

12 The purpose of the proportioning step is to remove indexes from the active set. The conjugate gradient step is defined by λ k+ = λ k α cg p k (5.6) where p k is the conjugate gradient direction [2] which is constructed recurrently. The recurrence starts (or restarts) with p s = ϕ(λ s ) whenever λ s is generated by the expansion step or the proportioning step. If p k is known, then p k+ is given by the formulae [2] p k+ = ϕ(λ k ) γp k, γ = ϕ(λk ) Ap k (p k ) Ap. (5.7) k The conjugate gradient steps are used to carry out the minimization in the face W J = {λ : λ i = 0 for i J } given by J = A(λ s ) efficiently. The algorithm that we use may now be described as follows. Algorithm 5.2. Modified proportioning with reduced gradient projections (MPRGP). Let λ 0 be an n-vector such that λ i λ i for i I, α (0, A ], and Γ > 0 be given. For k 0 and λ k known, choose λ k+ by the following rules: Step. If g P (λ k ) = 0, set λ k+ = λ k. Step 2. If λ k is strictly proportional and g P (λ k ) 0, try to generate λ k+ by the conjugate gradient step. If λ k+ i 0 for i I, then accept it, else generate λ k+ by the expansion step. Step 3. If λ k is not strictly proportional, define λ k+ by proportioning. The MPRGP algorithm has linear rate of convergence in terms of the spectral condition number of the Hessian A of L [2]. More about the properties and implementation of SMALBE algorithm may be found in [2] and [9]. 6 Optimality To show that Algorithm 5. with the inner loop implemented by Algorithm 5.2 is optimal for the solution of problem (or a class of problems) (4.3), we shall introduce new notation that complies with that used in [9]. We shall use T = {(H, h) IR 2 : H, 2h H and H/h IN} as the set of indices. Given a constant C 2, we shall define a subset T C of T by T C = {(H, h) IR 2 : H, 2h H, H/h IN and H/h C}. 2

13 For any t T, we shall define A t = P F P + ρq, b t = P d C t = G, l t,i = λ I and l t,e = by the vectors and matrices generated with the discretization and decomposition parameters H and h, respectively, so that the problem (4.3) is equivalent to the problem minimize Θ t (λ t ) s.t. C t λ t = 0 and λ t l t (6.) with Θ t (λ) = 2 λ A t λ b t λ. Using these definitions, Lemma 4. and GG = I, we obtain C t and l + t = 0, (6.2) where for any vector v with the entries v i, v + denotes the vector with the entries v + i = max{v i, 0}. Moreover, it follows by Theorem 4.2 that for any C 2 there are constants a C max > a C min > 0 such that a C min α min (A t ) α max (A t ) a C max (6.3) for any t T C. Moreover, there are positive constants C and C 2 such that a C min C and a C max C 2 C. In particular, it follows that the assumptions of Theorem 5 (i.e. the inequalities (6.2) and (6.3)) of [9] are satisfied for any set of indices T C, C 2 and we have the following result: Theorem 6. Let C 2 denote a given constant, let {λ k t }, {µ k t } and {ρ t,k } be generated by Algorithm 5. (SMALBE) for (6.) with b t η t > 0, β >, M > 0, ρ t,0 = ρ 0 > 0, and µ 0 t = 0. Let s 0 denote the smallest integer such that β s ρ 0 M 2 /a min and assume that Step of Algorithm 5. is implemented by means of Algorithm 5.2 (MPRGP) with parameters Γ > 0 and α (0, (a max + β s ρ 0 ) ], so that it generates the iterates λ k,0 t, λ k, t,..., λ k,l t = λ k t for the solution of (6.) starting from λ k,0 t = λ k t with λ t = 0, where l = l t,k is the first index satisfying g P (λ k,l t, µ k t, ρ t,k ) M C t λ k,l t (6.4) or g P (λ k,l t, µ k t, ρ t,k ) ɛ b t min{, M }. (6.5) Then for any t T C and problem (6.), Algorithm 5. generates an approximate solution λ kt t which satisfies M g P (λ k t t, µ k t t, ρ t,kt ) C t λ k t t ɛ b t (6.6) at O() matrix-vector multiplications by the Hessian of the augmented Lagrangian L t for (6.) and ρ t,k β s ρ 0. 3

14 7 Numerical experiments In this section we report some results of numerical solution of the semicoercive model problem of Section 2 in order to illustrate the performance of the algorithm, in particular its numerical and parallel scalability. To this end, we have implemented Algorithm 5. with the solution of auxiliary bound constraints by Algorithm 5.2 in C exploiting PETSc [3] to solve problem (4.3) with varying decomposition and discretization parameters Fig. 3a: Coercive problem Fig. 3b: Semicoercive problem Figure 3: Solution of model problems The experiments were run on the Lomond 8-processor Sun HPC 6500 Ultra SPARC-II based SMP system with 400 MHz, 8 GB of shared memory, 90 GB disc space, nominal peak performance 4.4 GFlops, 6 kb level and 8 MB level 2 cache in EPCC Edinburgh, and on the Turing Cray T3E 200, 788 applications processors, each.2 GFlops with 256 MB, 209 GB memory, 28 command processors, 2TB disk space, high-speed network with low latency in the University of Manchester. All the computations were carried out with parameters M =, ρ 0 = 0, Γ =, λ 0 = max{ λ, 2 Bf}, ɛ = 0 4. The results of computations are summarized in Tables 3. Table illustrates numerical scalability of Algorithm 5.. In particular, for varying decompositions and discretization parameters, the upper row of each field of the table gives the corresponding primal dimension/ dual dimension/ times in seconds, while the number in the lower row gives a number of the conjugate gradient iterations that were necessary for the solution of the problem to the given precision. We can see that the number of the conjugate gradient iterations for a given ratio H/h (in rows) varies very moderately. 4

15 Table : Performance for varying decomposition and discretization H /2 /4 /8 H/h \ procs /29/ /287/ /6687/ /29823/ /65/ /647/ /3359/ /4975/ /33/ /327/ /695/ /755/ /7/ /67/ /863/ /3839/ /9/ /87/ /447/ /983/ /5/ /47/ /239/ /055/ Table 2 indicates that the algorithm presented enjoys high parallel scalability. The results for the largest problems are in Table 3. Table 2: Parallel scalability for 28 subdomains processors time[sec] Comments and conclusion We have presented the scalability results related to application of the augmented Lagrangians with the FETI based domain decomposition method using the natural coarse grid to the solution of variational inequalities by recently developed algorithms for the solution of special QP problems. In particular, we have shown that the solution of the discretized elliptic variational inequality to a prescribed precision may be found in a number of matrix vector multiplications bounded independently of the discretization parameter provided the ration of the decomposition and the discretization parameters is kept bounded. Numerical experiments with the model variational inequality are in agreement with the theory and indicate that the algorithm may 5

16 Table 3: Highlights h H prim. dual. num. of procs out. cg. time dim. dim. subdom. iter. iter. [sec] /024 / of Turing /2048 / of Lomond be efficient. The results remain valid also for the solution of frictionless 2D and 3D contact problems of elasticity and may be adapted to the solution of problems with Coulomb friction as indicated in [7]. The solution of auxiliary linear problems in the inner loop may be improved by standard preconditioners [29, 35, 38] and may be adapted to the mortar discretization [39]. We shall discuss these topics elsewhere. References [] P. Avery, G. Rebel, M. Lesoinne and C. Farhat, A umerically scalable dualprimal substructuring method for the solution of contact problems. I: the frictionless case. Comput. Methods Appl. Mech. Eng. 93, 23 26, (2004) [2] O. Axelsson, Iterative Solution Methods. Cambridge University Press, Cambridge, 994. [3] S. Balay, W. Gropp, L.C. McInnes and B. Smith, PETSc 2.0 Users Manual. Argonne National Laboratory, [4] D. P. Bertsekas, Nonlinear Optimization. Athena Scientific, Belmont 999. [5] J.H. Bramble, J. E. Pasciak, A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring I, Mathematics of Comput. 47, (986). [6] A. R. Conn, N. I. M. Gould and Ph. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM Journal on Numerical Analysis 28 (99) [7] Z. Dostál, A proportioning based algorithm for bound constrained quadratic programming with the rate of convergence. Numerical Algorithms 34, 2 4(2003) [8] Z. Dostál Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. SIAM J. Numerical Analysis 43, (2005)

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