### Table 4: Number of preconditioned conjugate gradient iterations for the Poisson problem when the number of subdomains and the subdomain mesh size is varied.

"... In PAGE 10: ... In optimal situations, numerical scalability would mean that the convergence rate would not depend on the number of subdomains; this would lead to constant computing time when the overall size of the problem and the number of processors increase proportionally. Table4 is devoted to experiments on the Poisson problem, Table 5 to Problem 2, and Table 6 reports results on the heterogeneous and anisotropic problems. We can first observe that the problems with both heterogeneity and anisotropy are the most difficult to solve and that the Poisson problem is the easiest.... ..."

### Table 4: Number of preconditioned conjugate gradient iterations for the Poisson problem when the number of subdomains and the subdomain mesh size is varied.

"... In PAGE 10: ... In optimal situations, numerical scalability would mean that the convergence rate would not depend on the number of subdomains; this would lead to constant computing time when the overall size of the problem and the number of processors increase proportionally. Table4 is devoted to experiments on the Poisson problem, Table 5 to Problem 2, and Table 6 reports results on the heterogeneous and anisotropic problems. We can rst observe that the problems with both heterogeneity and anisotropy are the most di cult to solve and that the Poisson problem is the easiest.... ..."

### Table 4 Number of preconditioned conjugate gradient iterations for di erent choices of Bh p

1997

"... In PAGE 13: ... Lastly, to determine how the choice of Bh p a ects the preconditioner we x = 0:2 and vary the tolerance in the Poisson solver used to compute Bh. Corresponding results are summarized in Table4 . The data in this table suggests that preconditioner (42) is not very sensitive with respect to the quality of Bh p and that the overall performance of the conjugate gradient method depends more critically on .... In PAGE 16: ... Asymp- totic convergence rates for the L2-methods are estimated using approximate solutions computed on a pair of uniform grids with 17x17 and 33x33 grid lines, respectively. Corresponding results are summarized in Table4 . Like in Tables 1-2, bold face in this table is used to denote asymptotic rates for the error components which are included in (44) and (45).... In PAGE 16: ... Like in Tables 1-2, bold face in this table is used to denote asymptotic rates for the error components which are included in (44) and (45). The asymptotic rates of (19) in Table4 are in excellent agreement with (45) and are higher than the H1-norm rates of the negative norm method. There... In PAGE 17: ... Although the H1-norm rates for (14) are closer to the rates of the negative norm method, in the L2-norm the former converges twice as fast as (14). The data in Table4 leads to the unambiguous conclusion that for smooth solutions the augmented L2-method ranks rst, while the method (14) o ers the worst perfor- mance. The main cause for this dismal performance of (14) is in the lack of norm- equivalence of the underlying least-squares functional.... In PAGE 17: ... These results are consistent with numerical experiments performed using other non-norm equivalent functionals, where suboptimal convergence rates were also observed; see [6]. From the data in Table4 we can conclude that, although performance of the negative norm method is not dramatically inferior to that of (19), the less complicated and straightforward implementation of the L2-method makes it more convenient when the exact solution is smooth enough.... ..."

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### Table 5: Number of preconditioned conjugate gradient iterations for Problem 2 when the number of subdomains and the subdomain mesh size is varied.

"... In PAGE 10: ... In optimal situations, numerical scalability would mean that the convergence rate would not depend on the number of subdomains; this would lead to constant computing time when the overall size of the problem and the number of processors increase proportionally. Table 4 is devoted to experiments on the Poisson problem, Table5 to Problem 2, and Table 6 reports results on the heterogeneous and anisotropic problems. We can first observe that the problems with both heterogeneity and anisotropy are the most difficult to solve and that the Poisson problem is the easiest.... ..."

### Table 5: Number of preconditioned conjugate gradient iterations for Problem 2 when the number of subdomains and the subdomain mesh size is varied.

"... In PAGE 10: ... In optimal situations, numerical scalability would mean that the convergence rate would not depend on the number of subdomains; this would lead to constant computing time when the overall size of the problem and the number of processors increase proportionally. Table 4 is devoted to experiments on the Poisson problem, Table5 to Problem 2, and Table 6 reports results on the heterogeneous and anisotropic problems. We can rst observe that the problems with both heterogeneity and anisotropy are the most di cult to solve and that the Poisson problem is the easiest.... ..."

### Table 1 The projected conjugate gradient algorithm Initialize

"... In PAGE 13: ... The FETI-DPI algorithm is a classical conjugate gradient applied to the prob- lem (11). The algorithm is then similar with the one in Table1 , but no more projection is needed. One has to notice that at each iteration, this algorithm requires the solution of only one global coarse problem consisting in nding the displacement of corner dof, during matrix-vector product F ?v; as previously, the choice of the preconditioner is discussed in the next sec- tion.... In PAGE 20: ... With the equations (14), the overall problem to be solved is: 2 6 6 6 6 6 4 F F P Q G QT P T F L 0 GT 0 0 3 7 7 7 7 7 5 2 6 6 6 6 6 4 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 d QT P T d e 3 7 7 7 7 7 5 with L = QT P T F P Q To build the corresponding algorithm, the following ways are equivalent: using the FETI2 framework [32] with the interpretation of an additional coarse space correction with the regular matrix L, or, as it is done herein, condensing the new unknowns on the other ones to get: 2 6 4 F ? G GT 0 3 7 5 2 6 4 3 7 5 = 2 6 4 d? e 3 7 5 with F ? = F F P QL 1QT P T F and d? = d F P QL 1QT P T d, and apply the previous FETI-I algorithm to this new problem. the overall algorithm is the same as in Table1 ; the di erence in the implementation is the need for the additional coarse problem (with matrix L) within the matrix-vector product F ?p. The preconditioning step allows to apply the eventually singular (because incompressible) Dirichlet preconditioner to the residual: if s is a subdomain with all of its boundary subjected to a prescribed displacement v(s) = B(s)T W P T P s B(s)u(s), the Dirichlet problem to be solved on this subdomain is: K(s) Hii 2 6 4v(s) i p(s) 3 7 5 = 2 6 4 f K(s) ib C(s) b T 3 7 5 v(s) b If K(s) Hii is singular, its kernel is exactly a uniform pressure 0 p(s) 1 T T .... ..."

### Table 3: The pre-conditioned conjugate algorithm as implemented in PSAS (Golub and van Loan, 1989)

"... In PAGE 6: ... This is mainly because of the ltering properties of the operator PfHT in (4) which attenuates the small scale details in the linear system variable y. 3 Overview of the Conjugate Gradient Algorithm This section describes the pre-conditioned conjugate gradient algorithm from a numerical point of view; the algorithm adopted is given in Table3 . The choice of pre-conditioner in PSAS is discussed in the next section, followed by a discussion of the current Fortran 90 implementation.... In PAGE 9: ... Usually the pre-conditioner is obtained by solving a simpli ed version of the problem. The pre- conditioned conjugate gradient algoritm implemented in PSAS is given in Table3 . The pre-conditioner amounts to solve an extra linear system A2zk = rk every iteration.... In PAGE 11: ...e., the step qk = Cpk in the algorithm shown in Table3 ; this complex aspect of the PSAS algorithm will be documented in a separate O ce Note. A block diagram of the getAIall( ) solve4x( ) getAinc( ) solve4x0( ) cg_main( ) cg_level2( ) cg_level1( ) Univariate Pre-conditioner Driver Pre-conditioner Regional Conjugate Gradient PSAS Fortran 90 Driver getAIall0( ) Figure 2: Block diagram of the higher level PSAS modules.... In PAGE 13: ... 5.3 The main conjugate gradient driver: cg main() This routine does a straightforward implementation of the pre-conditioned conjugate gradi- ent algorithm given in Golub and van Loan (1989) and reproduced in Table3 ; even variable names have been chosen to closely follow the book notation (with the exception perhaps, of the matrix name which we use C instead of A). The Basic Linear Algebra Subprograms (BLAS), which are often hand-coded in assembler and provided by several vendors, are used to perfom the basic linear algebra operations such as dot products, norms, vector additions, etc.... ..."

### Table 4: Average number of conjugate gradient iterations per fold in the CV procedure. CG: without preconditioning. PCG: using diagonal preconditioning. We use C = 64 and = 0:001.

2007

"... In PAGE 25: ... di+1 = ri+1 + idi. Note that in practical implementations we calculate ^ Hdi by a way similar to (7) P 1(P T di + C(XT (D(X(P T di))))): In Table4 , we present the average number of conjugate gradient iterations per fold in the CV procedure. The approach of using diagonal preconditioning reduces the number of iterations for only two problems.... ..."

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### Table 3: Timings statistics for different models using the non-linear Green strain measure. The multigrid solver is compared to a preconditioned conjugate gradient solver (CG)

"... In PAGE 6: ... The multigrid solver is compared to a preconditioned conjugate gradient solver (CG) Compared to the linear setting, in the non-linear strain setting real-time can only be achieved if the number of el- ements is significantly reduced. However, compared to the corotational setting the performance is only about a factor of 2 lower (see Table3 ). Explicit timings of the reassembling step in the non-linear setting are given in Table 4.... ..."