### Table 8 Number of V-cycles using linear and energy-minimizing interpolations for the Helmholtz problem.

1998

"... In PAGE 16: ... For this problem, we obtained H i from solving the local PDEs (7), not from the minimization problem (12), since constant functions are not in the kernel of Ah. The convergence results of the multigrid methods using linear and energy-minimizing interpolations are shown in Table8 . The in the rst column indicates that standard multigrid takes more than 100 V-cycles to convergence.... ..."

Cited by 28

### Table 8 Number of V-cycles using linear and energy-minimizing interpolations for the Helmholtz problem.

1998

"... In PAGE 17: ... kernel of Ah. The convergence results of the multigrid methods using linear and energy-minimizing interpolations are shown in Table8 . The in the rst column indicates that standard multigrid takes more than 100 V-cycles to convergence.... ..."

Cited by 28

### Table 6 Number of V-cycles using bilinear and energy-minimizing interpolations for the oscillatory coe - cient problem. = 0:1; 0:01. More than 100 V-cycles required for convergence.

1998

Cited by 28

### Table 4 Number of V-cycles using bilinear and energy-minimizing interpolations for the discontinuous coe - cient problem. The jump a+ = 10; 102; 103; 104. More than 100 V-cycles required for convergence.

1998

"... In PAGE 15: ... We x a? = 1 and vary a+ from 10 to 104. The convergence results are given in Table4 . Same notations are used as in Example 1.... ..."

Cited by 28

### Table 6 Number of V-cycles using bilinear and energy-minimizing interpolations for the oscillatory coe cient problem. = 0:1; 0:01. More than 100 V-cycles required for convergence.

1998

Cited by 28

### Table 4 Number of V-cycles using bilinear and energy-minimizing interpolations for the discontinuous co- e cient problem. The jump a+ = 10; 102; 103; 104. More than 100 V-cycles required for convergence. Example 4: We solve another PDE to demonstrate the robustness of the energy-minimizing multigrid 15

1998

"... In PAGE 15: ... We x a? = 1 and vary a+ from 10 to 104. The convergence results are given in Table4 . Same notations are used as in Example 1.... ..."

Cited by 28

### Table 1: Notations used in the paper. process they used a training data that was dependently obtained by other assignment methods. The performance of their algorithm is totally depend on the used training data. Also in [12] only the co-channel constraint was considered. In this paper, a modi ed discrete Hop eld neural network algorithm for channel assignment problem is proposed in order to improve the convergence rate and to reduce the number of iterations. Our experimental results are also compared with that of Funabiki and Takefuji [11]. In this paper, the channel assignment problem is formulated as an energy minimization problem such that the energy is at its minimum when all the constraints are satis ed and the number of assigned frequencies are the same as the required channel numbers (RCNs) in each cell. Three conditions are considered in this paper as in references [3] and [11]: 1. co-site constraint (CSC) : any pair of frequencies assigned to a cell should have a minimal distance between frequencies; 2. co-channel constraint (CCC) : for a certain pair of radio cells, the same frequency cannot be used simultaneously;

1994

Cited by 7

### Table 2 Number of V-cycles using bilinear and energy-minimizing interpolations when a(x) = 1 + x exp(y). Example 3: We compare the multigrid method using bilinear interpolation with that using energy- minimizing interpolation by solving the following discontinuous coe cient problem [1, modi ed Exam- ple I]:

1998

"... In PAGE 14: ... Example 2: In this example, we verify numerically that the convergence rate does not depend on the number of levels. Here we consider the following PDE with a smooth coe cient: ?r (1 + x exp(y))ru = 1: Table2 shows the number of multigrid iterations to convergence. We denote the multigrid method with bilinear interpolation by MGBL and our energy-minimizing multigrid method by MGE( ), where speci es the stopping criterion for the conjugate gradient (CG) method applied to the Lagrange multiplier equation (16).... ..."

Cited by 28

### Table 1: Dependency of the minimal surface area and the cmc surface period on the number of iterations for a xed discretization of the fundamental patch with 291 trian- gles. The minimization algorithm converges rapidly during the rst iterations; when the surface is close to its minimum the vertices try to move tangential to further decrease the energy and this motion is very slow. For the period problem and the nal surface these last minimization steps seem to have qualitatively no in uence, the intermediate time at which zero period occurs is very stable w.r.t. increasing number of iterations.

1997

Cited by 19

### Table II confirms the increased efficiency of CAM with constraints. Finally, as an illustrative example, we present the performance of CAM and CAMC on a real-world global optimization problem, taken from the field of computational chemistry. The problem consists in finding a stable conformation of a small protein (met-enkaephalin), which minimizes its potential energy as a function of dihedral angles [15]. This is a benchmark problem for testing optimization algorithms in chemistry. Earlier we reported our numerical experiments with CAM on this challenging problem [14]. As the global variables we took 10 backbone dihedral angles, with all otheranglestreated aslocal variables.The objectivefunction has in order of 1011 local minimizers. The values of the potential energy were computed using ECEPP/3 software package. The detailed configuration and system setup are described in Ref. [14].

2003