On the performance of the Algebraic Optimized Schwarz Methods with applications

On the performance of the Algebraic Optimized Schwarz Methods with applications Report 12-1-18 Department of Mathematics Temple University October 212 This report is available at http://www.math.temple.edu/~szyld

ON THE PERFORMANCE OF THE ALGEBRAIC OPTIMIZED SCHWARZ METHODS WITH APPLICATIONS LAHCEN LAAYOUNI AND DANIEL B. SZYLD Abstract. We investigate the performance of algebraic Optimized Schwarz methods used as preconditioners for solving discretized differential equations. The algebraic optimized Schwarz methods are based on the modification of the transmission blocks. The transmission blocks are replaced by new blocks to improve the convergence of the corresponding algorithms. In the optimal case, convergence in two iterations can be achieved. We are also interested in the asymptotic behavior of the algebraic Optimized Schwarz methods with respect to change in the problems parameters. In this paper, we focus on constructing preconditioners for different numerically challenging differential problems such as: Periodic and Torus problems; Meshfree problems; Three-dimensional problems. We present different numerical simulations corresponding to different type of problems in two- and three-dimensions. AMS subject classifications. 65F8, 65F1, 65N22, 65N55. Key words. Linear systems, banded matrices, block matrices, Schwarz methods, optimized Schwarz methods, iterative methods, preconditioners. 1. Introduction. We are interested in solving large scale linear systems of the form Au = f, (1.1) where A is a n n block banded matrix. The matrix of the linear system (1.1) can take the form of an augmented matrix, i.e. A+H, where A is block banded and H has entries outside of the diagonal blocks which are small in a sense that we will define later. Our approach is based on the recent study [13]. To present the general idea of the method, we consider the case where A is a block banded matrix of the following form A = A 11 A 12 A 21 A 22 A 23 A 32 A 33 A 34 A 43 A 44, (1.2) where A ij are block matrices of size n i n j, i,j = 1,...,4, and n = i n i. We suppose that n 1 n 2 and n 4 n 3. The structure of the matrix A in (1.2) can represent for example finite difference or finite element discretizations of partial differential problems. The Algebraic Optimized Schwarz method recently proposed consists of replacing portions of the transmission blocks A 22 and A 33 by modified blocks S 1 = A 33 +D 1 and S 2 = A 22 +D 2 [13]. The idea behind introducing the blocks D 1 and D 2 was inspired from the optimized Schwarz methods (OSM) [11], [14], [19]. Those blocks correspond to the overlap region between subdomains of the associated decomposition of the original domain. The strategy of finding the best possible blocks D 1 and D 2 is similar to finding the best parameters in optimized Schwarz methods. this version dated October 18, 212 School of Science and Engineering, Al Akhawayn University, Avenue Hassan II, 53 P.O. Box 163, Ifrane, Morocco (L.Laayouni@aui.ma). Department of Mathematics, Temple University (38-16), 185 N. Broad Street, Philadelphia, Pennsylvania 19122-694, USA (szyld@temple.edu). 1

Algebraic Optimized Schwarz methods were introduced to overcome some of the limitations of Optimized Schwarz methods. The analysis of the latter methods is based on Fourier transform and thus cannot be used on irregular and general shape domains. The algebraic version of OSM was proposed to construct a fully algebraic box solver for partial differential equations. There are many algebraic studies of the Classical Schwarz methods; see, e.g., [2], [1], [16], [26], [26]. The Algebraic OSM methods were inspired by these algebraic studies combined with Optimized Schwarz methods. In this paper, we focus on the computational performance of the algebraic optimized Schwarz methods when used as preconditioners for Krylov subspace methods. The present work represent a natural extension of the paper [13], since we consider different type of partial differential problems with more difficulties. We show how to construct preconditioners to solve linear systems steaming for the discretization of periodic problems, meshfree problems, and three-dimension differential equations. 2. Algebraic Block Preconditioners. We begin by reviewing the algebraic Optimized Schwarz methods from [13]. We consider an iterative method and the corresponding preconditioner for solving the linear system (1.1). For a given u, classical stationary iterative methods consist of computing u k+1 = Tu k +M 1 f, k =,1,..., (2.1) where T is the corresponding iteration operator and M is the associated preconditioner which can be used, e.g., to solve the preconditioned problem M 1 Au = M 1 f. We consider Restricted Schwarz Methods (with overlap). In this case of p = 2 blocks the iteration operators corresponding to the additive and multiplicative Schwarz iterations are given by where T RAS = I A 1 = 2 i=1 R T i A 1 i R i A and T RMS = A 11 A 12 A 21 A 22 A 23 A 32 A 33, A 2 = 1 (I R i T A 1 i R i A), (2.2) i=2 A 22 A 23 A 32 A 33 A 34 A 43 A 44 and the corresponding restriction and prolongation operators are defined by [ ] [ I R 1 = [I ], R1 = ; R 2 = [ I], R2 = I, (2.3) ], (2.4) here R 1 and R 1 are of size (n 1 +n 2 +n 3 ) n, and the identity in R 1 is of size n 1 +n 2. Similarly, R 2 and R 2 are of size (n 2 +n 3 +n 4 ) n, and the identity in R 2 is of size n 3 +n 4. The recently proposed method [13] is based on replacing the transmission blocks A 33 in A 1 and A 22 in A 2, by the following blocks S 1 = A 33 +D 1 and S 2 = A 22 +D 2. The modified blocks become à 1 = A 11 A 12 A 21 A 22 A 23 A 32 S 1, à 2 = S 2 A 23 A 32 A 33 A 34 A 43 A 44. (2.5) The corresponding additive and multiplicative iteration operators of the modified restricted Schwarz methods, for 2 blocks are given respectively by T MRAS = I 2 i=1 R T i à 1 i R i A, and T MRMS = 2 1 (I R i T à 1 i R i A). (2.6) i=2

The optimal choice for D 1 and D 2 for the fastest asymptotic convergence of (2.1) corresponds to minimizing the spectral radii ρ(t MRAS ) and ρ(t MRMS ). We need some notation. We introduce the matrices E 1 and E 3 of size (n 2 + n 3 + n 4 ) n 2 and (n 1 +n 2 +n 3 ) n 3, respectively, defined by E1 T = [ I O O ] and E3 T = [ O O I ]. We define the following block matrices A 1 1 E 3 =: B (1) 3 = B 31 B 32 B 33, A 1 2 E 1 =: B (2) 1 = B 11 B 12 B 13, (2.7) which correspond to the last block column of A 1 1 and the first block column of A 1 2, respectively. We introduce the matrices [ ] [ ] B (1) B31 B12 3 =, B B(2) 1 =, (2.8) 32 B 13 where B (1) 3 is of size (n 1 +n 2 ) n 3, and B (2) 1 is of size (n 3 +n 4 ) n 2. We define the column matrices Ē T 1 = [ I O ] and ĒT 2 = [ O I ], (2.9) which are of size (n 1 +n 2 ) n and (n 3 +n 4 ) n, respectively. The core result is based on the following lemma from [13]. where Lemma 2.1. The iteration matrix T MRAS has the following form T = T MRAS = I 2 i=1 [ R i T Ã 1 O K i R i A = L O ], (2.1) K = (1) B 3 (D 1 1 +B 33 ) 1[ Ē1 T D 1 1 A ] 34ĒT 2, L = B(2) 1 (D 1 2 +B 11 ) 1[ D2 1 A ] 21ĒT 1 +ĒT 2. (2.11) A similar result holds for the restricted multiplicative Schwarz case. The structure of the corresponding iteration operator T MRMS has the following form T MRMS = 1 (I R i T Ã 1 i R i A) = i=2 [ K LK ]. (2.12) Based on the forms (2.1) and (2.12) of the iteration matrices, one may think that the optimal choice for D 1 and D 2 would correspond to the minimization of the norms K and L. But, as it was shown in [13], even for very simple cases, that minimal values of K and L may give T MRAS > 1. Thus, an alternative is to look at higher powers of the iteration operator. For example, in the additive case, if we consider T 2 MRAS = [ KL LK then one can show the following asymptotic behavior result of the modified restricted additive Schwarz method. 3 ],

Theorem 2.2. [13] The asymptotic convergence factor of the modified RAS method given by (2.2) is bounded by the product of the following two norms (I +D 1 B 33 ) 1 [D 1 B 12 A 34 B 13 ], (I +D 2 B 11 ) 1 [D 2 B 32 A 21 B 31 ]. (2.13) A similar result holds also in the multiplicative case, that is, the asymptotic convergence factor of the modified restricted multiplicative Schwarz method is bounded by the product of the same norms as in (2.13). The goal is to find D 1 and D 2 which minimize the convergence factors of the proposed methods. Finding those minima may be too expensive in general. Instead we look for minima in a smaller sets of matrices (with fewer parameters), i.e., we solve the following minimization problems min D 1 Q 1 (I +D 1 B 33 ) 1 [D 1 B 12 A 34 B 13 ], min D 2 Q 2 (I +D 2 B 11 ) 1 [D 2 B 32 A 21 B 31 ], (2.14) where Q 1, and Q 2 are some suitable subspaces of matrices. Suppose that the optimal is attained in (2.14) and the norms can actually be zero. In this case, we can rewrite the matrix of the first minimization problem as follows (I +D 1 B 33 ) 1 [D 1 B 12 A 34 B 13 ] = B33 1 (B 1 33 +D 1) 1 [D 1 B 12 A 34 B 13 ] = B33 1 (B 1 33 +D 1) 1[ (B33 1 +D 1)B 12 B33 1 B ] 12 A 34 B 13 = [ B33 1 B 12 B33 1 (B 1 33 +D 1) 1 (B33 1 B 12 +A 34 B 13 ) ] Therefore, if the norm in (2.14) is zero, we have that = [ B 1 33 B 12 (I +D 1 B 33 ) 1 (B 1 33 B 12 +A 34 B 13 ) ]. B 1 33 B 12 = (I +D 1 B 33 ) 1 (B 1 33 B 12 +A 34 B 13 ), (I +D 1 B 33 )B 1 33 B 12 = B 1 33 B 12 +A 34 B 13, D 1 B 12 A 34 B 13 =. (2.15) In a similar way, we can show that if the optimal of minimizing the second equation in (2.13) is attained and the norm is zero, then D 2 B 32 A 21 B 31 =. Hence, if the optimal is attained and the norms are equal to zero, then solving the nonlinear problems produce the same result as solving the following linear problems min D 1 B 12 A 34 B 13, min D 2 B 32 A 21 B 31. (2.16) Therefore, when the optimal in (2.16) is reached and both norms vanish, and under the assumptions that n 2 = n 3 and B 12 and B 32 are nonsingular matrices, then we can solve for D 1 and D 2, i.e., then the modified blocks in (2.5) become D 1 = A 34 B 13 B 1 12, and D 2 = A 21 B 31 B 1 32, (2.17) S 1 = A 33 +A 34 B 13 B 1 12, and S 2 = A 22 +A 21 B 31 B 1 32. (2.18) We substitute by D 1 and D 2 in (2.13), we obtain TMRAS 2 =, so the modified restricted additive Schwarz method converges in no more than two iterations. Analogously, for the modified multiplicative Schwarzmethod(MRMS),using(2.18),itcanbeshownthatitalsoconvergesinatmosttwoiterations. 4

For the solution of the linear system (1.1) using a preconditioned minimal residual method such as GMRES or MINRES; see, e.g., [28], [29], we use the Modified Restricted Schwarz preconditioners defined by M 1 MRAS = 2 i=1 R T i à 1 i R i, M 1 MRMS = [I ] 1 (I R i T à 1 i R i A) A 1. (2.19) Using the matrices from (2.17), or equivalently (2.18), it is shown in [13] that these preconditioned problems converge in at most two iterations as well. Thus we call these choices of modification matrices (2.18) optimal. In general, instead of (2.16) a more computationally tractable case is to consider min D 1 Q 1 D 1 B 12 A 34 B 13, i=2 min D 2 Q 2 D 2 B 32 A 21 B 31, (2.2) for suitable classes of matrices Q 1 and Q 2, e.g., scalar, diagonal, or tridiagonal matrices. Those three cases require in general the solution of linear least squares (LLS) problems. Inspired by the optimized Schwarz methods (e.g., [11], [12]) those approaches are called the,, and methods, respectively. The optimal transmission matrices in (2.18) involve blocks of the inverseof A 1 and A 2. This could be computationally expensive especially in the case of multiple diagonal overlapping blocks [13]. An alternative is to approximate the blocks using some approximation methods, e.g., the incomplete LU factorizations (ILU) or the use of sparse approximate inverse factorizations [2]. Now, let us consider the case when the matrix of the linear system (1.1) can take the form K = A + H where A is the block banded matrix and H has entries outside of the diagonal blocks which are small. In particular, we consider matrices of the following structure K = A+H := A 11 A 12 A 21 A 22 A 23 A 32 A 33 A 34 A 43 A 44 + H 14 H 41. (2.21) Such matrices arise, e.g., from the discretization of two- or three-dimensional periodic or torus problems using finite elements or finite difference methods. The matrix H has small blocks H 14 and H 41 in the sense that the size of those blocks are small or equal to the size of the blocks corresponding of the overlap region, i.e., A 22 and A 33. In Figure 2.1 we illustrate the form of the matrix K = A+H corresponding to the case of a two-dimensional torus problem defined on unit square [,1] 2. Other examples of structures of the block H are shown in Figure 2.2, where the size of the offdiagonal block of H can be large to some extent. At the end of the paper we will analyze the impact of the size of the off-diagonal block on the convergence of the Algebraic Optimized Schwarz Methods and the classical Schwarz methods. In solving the preconditioned problem M 1 Ku = M 1 f where K has the structure (2.21) we construct first the modified restricted Schwarz preconditioners associated to the block banded matrix A as we did in (2.19), where, as before, A i and Ãi, i = 1,2 are defined in (2.3) and (2.4). Then, we solve the following preconditioned problems M 1 MRAS (A+H)u = M 1 MRAS f, M 1 MRMS (A+H)u = M 1 MRMSf. (2.22) In the next section we will explore how wide can the applicability of the preconditioners (2.19) be. In particular how do they perform when the parameters of the problem change. 5

5 5 1 1 15 15 2 2 25 25 3 3 35 35 4 1 2 3 4 nz = 192 4 1 2 3 4 nz = 4 Fig. 2.1. On the left the structure of a block banded matrix A. On the right the matrix H has entries outside the diagonal blocks which are small. 5 5 1 1 15 15 2 2 25 25 3 3 35 35 4 1 2 3 4 nz = 16 4 1 2 3 4 nz = 4 Fig. 2.2. Other examples of the off-diagonal block H. 3. Performance and Applications. We consider a variety of applications from two- and threedimensional problems. 3.1. Two-dimensional case. We begin with an example from [13]. We consider the advection diffusion problem ηu (a u)+b u = f, (3.1) where a = a(x), b = [ b1 (x) b 2 (x) ], η = η(x), (3.2) and b 1 = y 1/2, b 2 = (x 1/2), η = x 2 cos(x + y) 2, a = (x + y) 2 e x y. The domain is either a square or an L-shaped region. We use finite differences to discretize the differential equation (3.1). 6

For the square region we use h = 1/32 in each direction. We preprocess the resulting matrix using the reverse Cuthill-McKee algorithm; see, e.g., [15]. This results in a matrix of size 124 124 of the same structure as the first matrix in Figure 2.1. In the same figure, we show also the partition of the block banded matrix as in (1.2), with n 1 = n 4 = 48 and n 2 = n 3 = 32. 1 2 Nonoverlapping 1 (luinc) (luinc) 1 2 (luinc) (luinc) 1 4 1 Nonoverlapping (luinc) 1 2 (luinc) (luinc) 1 4 (luinc) 1 6 1 6 1 8 1 2 3 4 5 1 8 5 1 15 2 25 3 Fig. 3.1. The convergence history of the additive version of the modified block methods used as an iterative method (left) or as a preconditioner (right) in solving an advection-convection equation on the square and with a two-block decomposition. In Figure 3.1, we present the convergence history of the additive version of modified block methods used as an iterative method or as preconditioner to find the solution of the advection-convection problem (3.1) with f = and using a two-block decomposition. As an initial vector u for all the numerical experiments presented in this paper we use a vector with all entries equal to one. We consider the optimal transmission matrices (2.18) and minima of (2.2) for suitable classes of matrices, i.e., scalar matrices D i = αi (), diagonal, and tridiagonal matrices. We approximate the blocks B ij involved in the optimal transmission matrices (2.18) by B (ILU) ij using incomplete LU factorization (ILU) of blocks of the block banded matrix A. Using those approximations, we computed (luinc), (luinc), (luinc) and (luinc) transmission conditions D 1 and D 2. For completeness, we also include the classical Schwarz method, both with non-overlapping and overlapping blocks. We can see that the Algebraic optimized Schwarz methods perform very well compared to the classical Schwarz methods either used as iterative method or as preconditioner. For instance when the methods are used as preconditioners, Figure 3.1 (right), while the optimal Schwarz method and its approximation (luinc) take only two iterations to reach the residual 1 8 the non-overlapping Schwarz method needs around 26 iterations. We present in Figure 3.2 the convergence history of the multiplicative version of the modified block methods compared also to the classical Schwarz method. Here again we obtain, as the theory expected, similar results as the additive version of the optimized methods. Using the multiplicative version of the optimal modified block method as preconditioner takes two iterations while the classical Schwarz method requires about 14 iterations. Similar results were obtained when the domain is an L-shaped region. Figures 3.1 and 3.2 present the performance of all the methods used as iterative and as preconditioners solvers. In this paper we concentrate on the performance of the preconditioning methods, thus the rest of the numerical simulations are with preconditioners only, but we note that in most cases we obtain similar results as in Figure 3.1 when the methods are used iteratively. 7

1 2 Nonoverlapping 1 (luinc) (luinc) 1 2 (luinc) (luinc) 1 4 1 Nonoverlapping (luinc) 1 2 (luinc) (luinc) 1 4 (luinc) 1 6 1 6 1 8 1 2 3 4 5 1 8 5 1 15 Fig. 3.2. The convergence history of the multiplicative version of the modified block methods used as an iterative method (left) or as a preconditioner (right) in solving an advection-convection equation on the square and with two-block decomposition. Table 3.1 Number of iterations for the additive and multiplicative versions of the block modified and the classical Schwarz precondtitioners for varying number of variables corresponding to the discretization. All methods are run with two-subdomains decomposition on the square. The size of the matrix A is N 2. Opt. Opt. (lui) (lui) (lui) (lui) Overl. Non-overl. N Additive version 8 2 3 6 6 7 7 8 8 8 13 16 2 3 7 7 8 8 1 1 1 19 32 2 3 7 7 8 8 12 12 12 26 64 2 3 8 8 1 1 18 18 16 33 128 2 3 8 8 11 11 24 24 2 44 256 2 3 9 9 12 12 33 33 26 57 Multiplicative version 8 2 2 4 4 4 4 4 4 4 7 16 2 2 4 4 4 4 6 6 5 1 32 2 2 4 4 4 4 7 7 7 14 64 2 2 4 4 5 6 1 1 9 17 128 2 2 4 4 5 7 13 13 12 22 256 2 2 4 4 4 9 17 17 16 29 3.2. Performance dependence on mesh size and Péclet number. Our next numerical experiments are devoted to the asymptotic behavior of the block modified methods and how they behave with respect to the number of variables of the discretization. We consider the advectiondiffusion problem on the L-shaped region and we obtain similar results to the case of a square region. In Figure 3.3 we show the number of iterations for convergence of the additive and multiplicative versions of the Optimized and classical Schwarz methods as we vary the number of variables corresponding to the discretization, the iteration stops when the residual norm is below 1 8. As one can see (see also Table 3.1) the algebraic Optimized Schwarz methods perform well with respect to the 8

1 2 (luinc) (luinc) (luinc) (luinc) 1 1 Non overlapping 1 2 1 1 (luinc) (luinc) (luinc) (luinc) Non overlapping 1 1 2 1 1 h 1 2 1 1 h Fig. 3.3. The asymptotic behavior of the convergence of the optimal methods, additive version (left) and multiplicative version (right), with respect to the discretization parameter h = 1/N. Table 3.2 Timing, in seconds, associated to all the methods used as additive preconditioners, where t blocks is the necessary time to compute the optimized blocks and t total is the total time including the time of solving. The size of the matrix A is N 2. Method N = 16 N = 32 N = 64 N = 128 t blocks t total t blocks t total t blocks t total t blocks t total 5.91 4 2.71 3 7.51 4 2.41 2 1.41 3 5.1 1 5.61 3 7.91 + (lui) 1.51 5 2.31 3 1.41 5 2.71 2 7.1 6 5.11 1 6.1 6 8.11 + 3.1 3 6.21 3 7.1 3 4.1 2 2.1 2 6.41 1 7.21 2 1.11 +1 (lui) 6.41 3 9.91 3 1.1 2 4.21 2 2.31 2 7.11 1 7.61 2 1.21 +1 1.81 3 5.1 3 4.1 3 3.71 2 1.21 2 6.51 1 4.41 2 1.11 +1 (lui) 3.11 3 7.1 3 5.41 3 4.21 2 1.31 2 7.21 1 4.61 2 1.21 +1 2.61 4 4.51 3 5.11 4 4.1 1.51 3 8.1 1 5.61 3 1.41 +1 (lui) 1.41 3 5.41 3 1.41 3 4.1 2 2.41 3 8.11 1 6.51 3 1.41 +1 Overlap. 4.81 3 3.91 2 8.1 1 1.41 +1 Non-overlap. 5.51 2 1.81 1 3.51 + 8.1 +1 change of the number of discretization variables. In particular the convergence of the optimal method and its (ILU) approximation are independent of the change of the meshsize. The other optimized Schwarz methods performed relatively well compared to the classical Schwarz methods. In tables 3.2 and 3.3 we present the timing corresponding to all the different methods associated to Table 3.1. The variable t blocks represents the time in seconds is the necessary time to compute the blocks D 1 and D 2 associated to each method. The variable t total is the total time for computing the blocks plus the time to run the corresponding GMRES algorithm. For all the numerical simulations as well as for the timing computations we use a Dell machine with Core i5 dual 2.4 GHz and 32 bit operating system. Observe that the set up time, i.e., the time to solve the minimization problems(2.2) are negligible or very small in relation to the overall computation. Note also that for the optimal case (even with ILU) the CPU time grows linearly with the number of variables (N 2 ). We consider now the asymptotic behavior of the convergence of the algebraic Optimized Schwarz 9

Table 3.3 Timing, in seconds, associated to all the methods used as multiplicative preconditioners, where t blocks is the necessary time to compute the optimized blocks and t total is the total time including the time of solving. The size of the matrix A is N 2. Method N = 16 N = 32 N = 64 N = 128 t blocks t total t blocks t total t blocks t total t blocks t total 5.71 4 2.91 3 7.51 4 2.81 2 1.41 3 6.11 1 7.21 3 9.81 + (lui) 1.1 5 2.31 3 1.1 5 3.41 2 7.1 6 6.11 1 7.1 6 9.91 + 2.91 3 5.81 3 6.91 3 3.61 2 1.91 2 7.11 1 7.21 2 1.21 +1 (lui) 3.1 3 5.81 3 6.91 3 4.1 2 1.91 2 7.61 1 7.21 2 1.31 +1 1.71 3 4.71 3 4.1 3 4.41 2 1.21 2 7.31 1 4.41 2 1.21 +1 (lui) 1.71 3 4.61 3 4.1 3 3.71 2 1.11 2 7.51 1 4.41 2 1.31 +1 2.51 4 3.51 3 5.1 4 3.71 2 1.61 3 7.91 1 5.71 3 1.51 +1 (lui) 2.51 4 3.51 3 5.1 4 4.31 2 1.61 3 7.91 1 5.71 3 1.41 +1 Overlap. 3.51 3 3.81 2 8.21 1 1.51 +1 Non-overlap. 8.1 3 8.81 2 2.41 + 6.1 +1 methods with respect to the change of some parameters of the differential equation. An important quantity to study is the Péclet number, which is a dimensionless number that measures the rate of change of the advection in relation to the rate of change of diffusion of the equation. In our experiments we will multiply the function b(x,y) by a factor s in the range [1,1 4 ], thus changing the influence of the advection term. This interval will cover different type of differential equations from low Péclet numbers to high. The numerical experiments were performed on the unit squarebut similar results can be obtained in the caseof an L-shaped region. In Figure 3.4 we show the number of iterations needed to convergence of the block modified Schwarz methods and the classical Schwarz method with respect to the variation of the Péclet number. The number of iterations needed for the convergence of the optimal block Schwarz method is constant with respect to the variation of the Péclet number. The number of iterations needed for the convergence for all the other methods vary with respect to the change of the Péclet number. It is interesting to note that for moderate to high Péclet numbers the suboptimal optimized preconditioners offer advantages over standard Schwarz methods. Only when considering very high Péclet numbers, both types of method behave similarly. 1 1 (luinc) (luinc) (luinc) (luinc) Non overlapping 1 1 (luinc) (luinc) (luinc) (luinc) Non overlapping Factor s of Péclet number 1 1 1 1 2 1 3 1 4 Factor s of Péclet number 1 1 1 1 2 1 3 1 4 Fig. 3.4. The number of iterations needed to convergence for all the methods, additive version (left) and multiplicative version (right), vary by the Péclet number on the unit square with two-subdomains decomposition. 1

3.3. Periodic boundary conditions. Another interesting application is to find the solution of the differential problem (3.1) (3.2) on the square domain I = [ 1,1] [ 1,1] with periodic boundary conditions, i.e., we assume that the solution is the same on y = +1 and y = 1, and it is the same on the boundaries x = 1 and x = 1. The obtained boundary-value problem is known as the torus problem and the structure of the associated matrix has the form (2.21). Figure 3.5 summarizes the convergence history of GMRES preconditioned with the block algebraic Schwarz methods corresponding to the torus BVP on the unit square with two-subdomains. Here again the optimized methods perform well compared to the classical Schwarz methods in the additive and the multiplicative versions. In particular the methods preconditioned with optimal algebraic block converge in two iterations. Nonoverlapping 1 (luinc) 1 1 (luinc) (luinc) 1 2 (luinc) 1 3 1 4 1 5 Nonoverlapping 1 (luinc) 1 1 (luinc) (luinc) 1 2 (luinc) 1 3 1 4 1 5 1 6 1 6 1 7 1 7 1 8 2 4 6 8 1 12 14 16 18 1 8 1 2 3 4 5 6 7 8 9 1 Fig. 3.5. The convergence history of all the methods, additive version (left) and multiplicative version (right), corresponding to the torus problem on the unit square with two-subdomains decomposition. The structure of the matrix K = A+H, with H having additional entries, used in our numerical experiments is presented in Figure 3.6. We consider symmetric blocks H, where the entries are all equal to one. In Figure 3.7 we summarized the history of convergence of the algebraic OSM and the 1 2 3 4 5 6 7 8 9 1 2 4 6 8 1 nz = 6912 Fig. 3.6. The structure of the matrix K = A+H, where H has large size off-diagonal block. 11

classical Schwarz methods for the torus problem, where we assumed that the size of the off-diagonal block H has more than one diagonal (in the corner blocks) of larger and larger size. Based on the results shown in Figure 3.7 we can conclude that the size of the blocks H does not affect the good performance of the algebraic optimized Schwarz methods. The impact of the off-diagonal blocks on the convergence of the algebraic optimized Schwarz methods is indeed very small. 1 Nonoverlapping (luinc) 1 1 (luinc) (luinc) 1 2 (luinc) 1 3 1 4 1 Nonoverlapping (luinc) 1 1 (luinc) (luinc) 1 2 (luinc) 1 3 1 4 1 5 1 5 1 6 2 4 6 8 1 12 14 16 1 6 1 2 3 4 5 6 7 8 9 Fig. 3.7. The convergence history of all preconditioning methods, additive version (left) and multiplicative (right), for the torus problem on the square with two-subdomains decomposition, where the off-diagonal H is large. 3.4. Meshfree discretizations. Our next application corresponds to the solution of a linear system derived from a meshfree discretization of boundary value problem (BVP). The geometry of the corresponding problem is given in Figure 3.8, where the nodes are not connected as they would be in a finite element discretization. In general, the resulting matrix from the meshfree problem is an unstructured matrix, as shown in Figure 3.9 (left). In the right of Figure 3.9 is the structure of the meshfree matrix using the reverse Cuthill-McKee ordering. In Figure 3.1 we show the distribution 1.9.8.7.6.5.4.3.2.1.2.4.6.8 1 Fig. 3.8. The geometry of the meshfree problem in two-dimensional case. 12

values of the optimal blocks D1 and D2. As we can see the distribution is not uniform and the diagonal is not dominant. Therefore, our approach to compute the optimized blocks is slightly different for the meshfree problem. For instance, to calculate the blocks of the diagonal Optimized method, we keep, from D1 and D2, only important entries such that the size of the resulting blocks have the same number of nonzeros as a diagonal matrix. Similarly, we define Optimized blocks by keeping important entries of optimal blocks such that the number of nonzeros of the blocks is the same as tridiagonal matrices. The scalar optimized block (with scalar matrices) was computed in the same manner as before. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 1 2 4 6 nz = 1583 8 1 1 2 3 4 5 6 nz = 1583 7 8 9 1 Fig. 3.9. The structure of the matrix corresponding to the meshfree problem without ordering (left) and after ordering (right). Fig. 3.1. The distribution of values of the optimal blocks D1 (left) and D2 (right) of the meshfree problem. In Figure 3.11 we summarized the history of the convergence of the additive and the multiplicative versions of all the preconditioning methods corresponding to the meshfree problem. Once more, the optimal and the optimized Schwarz methods perform well. We note here that the convergence of scalar optimized method is similar to the convergence of the tridiagonal optimized Schwarz method. 13

1 1 Nonoverlapping 1 1 Nonoverlapping This is due to the fact that the scalar diagonal approximation is accurate compare to the tridiagonal approximation of the no-dominant diagonal optimal blocks. 1 1 1 1 1 1 1 2 1 3 1 2 1 3 1 4 1 4 1 5 1 5 1 6 1 2 3 4 5 6 7 8 9 1 1 6 1 2 3 4 5 6 7 8 9 1 Fig. 3.11. The convergence history of all preconditioning methods, additive version (left) and multiplicative version (right), for the meshfree problem with two-subdomains decomposition. 3.5. Three-dimensional problems. The three-dimensional problem we consider here is (η )u = f, in Ω R 3, (3.3) where Ω is unit cube and we chose for our simulation η = 1. The structure of the associated matrix is shown in Figure 3.12, where the size of the matrix is 1 1. By linearity we take f to be zero function. We approximate the differential equation (3.3) using finite differences. The convergence history of optimal and optimized Schwarz preconditioners and the classical Schwarz methods are summarized in Figure 3.13. The optimal Schwarz preconditioner for the three-dimensional differential equation converges in two iterations as it was expected. The optimized Schwarz preconditioners for both additive and multiplicative versions have a good asymptotic convergence behavior, and these are comparable to all other applications covered earlier in this paper. Similar results wereobtained for a three-dimension problemwith a matrix ofsize 496and η = 1. The structure of the corresponding matrix is the same as that on Figure 3.12. The asymptotic behavior of the optimized Schwarz and the classical Schwarz preconditioners are summarized in Figures 3.14. These plots indicate that the optimized Schwarz preconditioners perform well independently of the size of the problem. Finally, we present here another three-dimensional example taken from [1]. It consists of computing the temperature distribution of a simplified piston see Figure 3.15. The model differential equation is given by u = f in Ω u = u D on Γ D (3.4) u n = g on Γ N We use a finite element discretization to compute the temperature distribution with 15111 tetrahedra elements illustrated in Figure 3.15. 14

1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 1 nz=64 Fig. 3.12. The structure of the matrix corresponding to three-dimensional problem on the cube. 1 1 Nonoverlapping 1 (luinc) (luinc) 1 1 (luinc) (luinc) 1 2 1 3 1 1 Nonoverlapping 1 (luinc) (luinc) 1 1 (luinc) (luinc) 1 2 1 3 1 4 1 4 1 5 1 5 1 6 1 2 3 4 5 6 7 8 1 6 1 2 3 4 5 6 7 8 Fig. 3.13. The convergence history of the preconditioning methods, additive version (left) and multiplicative version (right), for a three-dimensional problem with a matrix of size 1 1 and η = 1. The structure of the resulting matrix A, of size 3319 3319, without ordering and after RCM ordering is shown in Figure 3.16. For more details about the discretization method, the source functions f and g, and more, see [1]. The history of the convergence of additive and multiplicative preconditioning methods corresponding to the calculation of the temperature distribution of the piston is shown in Figure 3.17. Again the performance of the optimized Schwarz preconditioners outperform the performance of the classical preconditioners. 4. Conclusions. In this computational study, we extend the applicability of the Algebraic Optimized Schwarz methods to matrices which are band plus some small corner blocks. We analyze the performance of the methods and how this performance depends on various parameters of the problems studied. 15

1 2 Nonoverlapping 1 1 (luinc) (luinc) 1 (luinc) (luinc) 1 1 1 2 1 3 1 1 Nonoverlapping 1 (luinc) (luinc) 1 1 (luinc) (luinc) 1 2 1 3 1 4 1 4 1 5 1 5 1 6 1 2 3 4 5 6 7 8 9 1 1 6 1 2 3 4 5 6 7 8 Fig. 3.14. The convergence history of the preconditioning methods, additive version (left) and multiplicative version (right), for a three-dimensional problem with a matrix of size 496 496 and η = 1. Fig. 3.15. The temperature distribution of a piston. Acknowledgements. Part of this research was performed during a visit of the first author to Temple University which was supported by a Fulbright fellowship. The second author was supported in part by the U.S. Department of Energy under grant DE-FG2-5ER25672 and NSF. We would like to thank Martin Gander for fruitful discussions concerning the paper. We also thank Benjamin Seibold for providing us with the data of the meshfree examples. 16

5 5 1 1 15 15 2 2 25 25 3 3 35 5 1 15 2 nz=45884 25 3 35 5 1 15 2 nz=45884 25 3 Fig. 3.16. Left: Structure of the matrix A without ordering. Right: A after RCM ordering. Nonoverlapping (luinc) (luinc) (luinc) (luinc) 2 1 1 1 1 2 2 1 1 4 4 1 1 6 6 1 1 8 1 Nonoverlapping (luinc) (luinc) (luinc) (luinc) 2 8 2 4 6 8 1 12 14 16 18 2 1 2 4 6 8 1 12 14 16 18 2 Fig. 3.17. The convergence history of the preconditioning methods, additive version (left) and multiplicative version (right), for the second three-dimensional problem with a matrix of size 3319 3319. 17

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