### Table 1 Compare incomplete mesh offset methods

2007

"... In PAGE 7: ... The num- ber of faces of mesh is 4284 and the number of boundary edge is 1012. Table1 shows the time required for offsetting incomplete mesh, the number of faces of the CL surface, and the preci- sion. Between three incomplete mesh offset methods the method using virtual multiple normal vectors of vertices is fast, pre- cise, and required smallest memory.... ..."

### Table 6: Iteration count and CPU time (in seconds) for steady transonic ow at conver- gence, for various preconditioner/decomposition pairs, and employing Schwarz-based methods with explicit boundary conditions and incomplete factorizations on the ne- mesh case.

1996

"... In PAGE 20: ... First, the study of the choice of the overlap is performed. In Table 7 we represent the results corresponding to an overlap of two mesh size for the di erent Schwarz algorithms using the iterative subdomain solver (ILU/GMRES) that we com- pare to the results obtained using one mesh size overlap Table6 . The conclusions are similar to the coarse mesh case.... In PAGE 20: ... The replacement of the full subdomain solver by the preconditioned Krylov subdomain solver (ILU/GMRES) is then performed. Table6 illustrates the number of nonlinear iterations (time steps) and the CPU time at convergence (steady- state regime) for the di erent Schwarz-based methods and for various decompositions, employing this iterative subdomain solver. These calculations were performed with a CFL number equal to 4:5.... ..."

Cited by 7

### Table 2: Iteration count and CPU time (in seconds) for steady transonic ow at conver- gence, for various preconditioner/decomposition pairs, and employing Schwarz-based methods with explicit boundary conditions and incomplete factorizations and with an overlap of two mesh sizes on the coarse-mesh case.

1996

"... In PAGE 14: ....1.1 Study of the overlap To study the choice of the overlap for the Schwarz-based methods, we rst present in Table 1 the results for di erent Schwarz methods with an overlap of one mesh size. To see the e ect of the overlap on the Schwarz-based methods studied here, we present in Table2 the results corresponding to an overlap of two mesh sizes for di erent Schwarz- based methods using the iterative subdomain solvers (ILU/GMRES). We observe rst that, for a given subdomain number the number of nonlinear iterations (time steps) varies slightly as we change the subdomain decomposition and/or the Schwarz-based method.... ..."

Cited by 7

### Table 1: Iteration count and CPU time (in seconds) for steady transonic ow at conver- gence, for various preconditioner/decomposition pairs, and employing Schwarz-based methods with explicit boundary conditions and incomplete factorizations on the coarse- mesh case.

1996

"... In PAGE 14: ...1.1 Study of the overlap To study the choice of the overlap for the Schwarz-based methods, we rst present in Table1 the results for di erent Schwarz methods with an overlap of one mesh size. To see the e ect of the overlap on the Schwarz-based methods studied here, we present in Table 2 the results corresponding to an overlap of two mesh sizes for di erent Schwarz- based methods using the iterative subdomain solvers (ILU/GMRES).... In PAGE 14: ...1.3 corresponding to an overlap of one mesh size ( Table1 ). We observe that, when the subdomain number increases, the di erence between the CPU time cost of the Schwarz algorithms with two and one mesh size overlap increases.... In PAGE 17: ....1.3 The preconditioned Krylov subdomain solvers We shall study now, the Schwarz-based methods using the preconditioned Krylov meth- ods as subdomain solvers. In Table1 we present the number of nonlinear iterations (time steps) and the CPU time at convergence (steady state regime), for the di erent Schwarz-based methods and for various decompositions. These calculations were per- formed using a CFL number equal to 5.... In PAGE 17: ... Moreover, the above observations are valid for all of the various decompositions studied here. For a given subdomain solver, the number of nonlinear iterations (time steps) ( Table1 ), is also... In PAGE 18: ....1.4 The full nested dissection versus the preconditioned Krylov subdo- main solvers Next, we perform comparisons of the subdomain solvers studied above, and study the e ect of replacing the full nested dissection subdomain solver by the preconditioned Krylov subdomain solver. In Table 3, the results are obtained using the full nested dissection methods as subdomain solvers, while in Table1 , those results are obtained using the preconditioned Krylov methods (ILU/GMRES). We observe rst that, for a given subdomain solver the number of nonlinear iterations (time steps) (Table 3 and 1) is nearly the same for all of the di erent Schwarz-based methods and for the various decomposition types.... In PAGE 19: ....1.7 Study of the di erent decomposition strategies The above study shows that the use of the preconditioned Krylov methods as sub- domain solvers for the di erent Schwarz-based methods studied in this paper is more attractive than that of the full nested dissection methods. Therefore, we shall study the di erent decomposition strategies only, for the iterative solver ( Table1 ). We shall compare the three decomposition startegies for each class of Schwarz methods reported in this paper.... In PAGE 26: ...his in the conclusions of section 4.2.1. For the overlapped Schwarz-based methods the situation is quite di erent. For the additive method, the gain in terms of the CPU time realized using implicit boundary conditions is more than 25% ( Table1 and 8). This is even better for the multiplicative method where a gain of more than 32% is realized repectively for the decompositions 2 2, 4 4, and 8 8.... In PAGE 26: ....4.1 Performance of Schwarz-based methods combined with Newton-Krylov matrix-free methods We shall focus now, on the comparison of the performance of the di erent Schwarz- based methods combined with the Newton-Krylov matrix-free methodology. For the coarse mesh case ( Table1 0), we observe that the block Jacobi method outperforms the additive method only in the case of the rst two decompositions. For the third decom- position the latter prevails over the former.... ..."

Cited by 7

### Table 7: Iteration count and CPU time (in seconds) for steady transonic ow at conver- gence, for various preconditioner/decomposition pairs, and employing Schwarz-based methods with explicit boundary conditions and incomplete factorizations with an over- lap of two mesh sizes on the ne-mesh case.

1996

Cited by 7

### Table 1. Preconditioners and solvers comparison for a mesh of 2000 nodes on a PC equipped with an Intel Pentium IV/2 GHz processor. CG: Conjugate Gradient, GM- RES: Generalized Minimum RESidual, BICGSTAB: BIConjugate Gradient STABi- lized, IC: Incomplete Cholesky, ILUT: Incomplete Lower Upper with Threshold.

2003

Cited by 1

### Table 2: Timing data for three iterative solvers to reach 1e-3 relative residual. PCG denotes preconditioned conjugate gradient with incomplete Cholesky decomposition. PCG did not converge to this precision within a reasonable amount of time on the largest mesh. Our multigrid solver can reach such an intermediate level of precision almost one order of magnitude faster than Trilinos. It is also competitive with the back- substitution times of direct solvers on large meshes (Table 1). The ability to quickly generate good approximate solutions is especially important when interactivity is demanded.

2006

"... In PAGE 7: ... the same residual, the visual acceptability of the output of different solvers will vary. As shown in Table2 , our algorithm effectively distributes errors and rapidly produces approximate solutions ( BO 1e-3 relative residual) which may be sufficient for many applica- tions. In fact, all of the visual results reported in this paper and the accompanying video were generated at this level of numerical accuracy.... ..."

Cited by 16

### Table 1. Optimized single-detect tests for ISCAS85 circuits. Circ. Initial Lower Rand round Rec. round ILP name vect. bound Vect. CPU Vect. CPU Vect. CPU s s s

2007

"... In PAGE 3: ....1. Single-Detect Tests Our first results evaluate the relative merits of recursive rounding against randomized round- ing [4, 20] and ILP. Table1 gives optimized test set sizes for single-detection. The initial vectors in the second column were obtained from an ATPG pro- gram [15].... In PAGE 3: ... As stated before, the minimum value of the sum of variables provided by the LP is a lower bound (sometimes unattainable) on the size of the abso- lute minimum test set. This is given in the third column of Table1 . The next six columns give opti- mized test set sizes and CPU times (Sun Ultra-5) for randomized rounding, recursive rounding, and ILP,... ..."

Cited by 2

### Table 12: Event table for Incomplete state. Incomplete

1999

Cited by 2

### Table 2. Incomplete Decision System S

"... In PAGE 4: ... For a generality reason, we take an incomplete decision system. It is represented as Table2 . Our goal is to re-classify objects in S from (d; 2) to (d; 1).... ..."