### TABLE 8 Errors for the unforced flow test problem for projection method III using the free boundary condition. Errors for PmIII with modified boundary value 96 96 128 128 192 192 rate

### Table 1. Comparison of training and prediction times on a high dimensional problem

2005

Cited by 2

### Table 1. Comparison of training and prediction times on a high dimensional problem

2005

Cited by 2

### Table 4: Position of the free boundary. nx - Number of grid lines D = 2 10?3;Q = 1 10?3

### Table 1: Comparison of typical execution time (in seconds) of the eigenfunction expansion (EE) method and interpolation (INT) method to determine the series coe cients for various values of N: The interpolation method will be applied to a free boundary problem arising in steady hillside seepage in the next section. An investigation into the accuracy of this method is also carried out, by comparing the results obtained with those of the eigenfunction expansion method.

"... In PAGE 10: ... The conditions (46) result in 2N linear equations for the 2N series coe cients ae and b e; and may be written as F ubae + F vbb e = 0e; F utae + F vtb e = het (47) where, for i; j = 0; 1; 2; : : : ; N ? 1 F ut ij = ut j(xi); F vt ij = vt j(xi) (48) F ub ij = ub j(xi); F vb ij = vb j(xi) (49) ht i = ht(xi): (50) These equations may be rearranged to give b e = ?(F vb)?1F ubae (51) and F ut ? F vt(F vb)?1F ub ae = het (52) which resemble Equations (35) and (38) for the eigenfunction expansion method, except for the appearance of the additional matrices F vt and F vb: As mentioned above, the distinct advantage of the interpolation approach, in terms of computational e ciency, is that no inner products need to be evaluated. To illustrate this advantage, Table1 compares typical execution times for the eigen- function expansion and interpolation methods for determining the series coe cients for various values of N: From this one can clearly see the signi cant potential in adopting the interpolation approach. The eigenfunction expansion method is con- siderably slower, with the majority of the execution time spent on evaluating the inner products in (41), (42) and (43).... ..."

### Table 3: Position of the free boundary. nx - Number of grid lines D = 4 10?3;Q = 0:1 10?3

### Table 1: Absolute error for s(t). Number of time-steps nt and grid lines nx: (A) nt=2000, nx=200; (C) nt=267, nx=200; (B) nt=4000, nx=400; (D) nt=288, nx=400 In the following examples we use physically relevant data which are taken from [39]. The coe cients and the dimensions of the calculated functions are listed in Table 2. Symbol Description Dimension Constants ci aqueous concentration of Mi

"... In PAGE 21: ... As our rst example we present an \academic quot; one-phase problem, where the source term and the time dependent boundary conditions are such, that the exact solution, u(t; x); and the free boundary, s(t); are given by s(t) = sin(t) and u(t; x) = cos(t) sin(x ? s(t)): This example was chosen to show the behavior of our method, if the free boundary is non monotonic and reaches the left boundary x = 0 again in a nite time. Table1 contains the absolute error of the free boundary function s(t) at several time levels for di erent step-sizes in space. Figure 7 shows a plot of the numerical solution.... ..."

### Table 19. Test error (in % ), high-dimensional data sets.

2006

"... In PAGE 95: ... Table19 Cont. NMC KNNC LDC QDC natural textures Original 54.... ..."

### Table I. Comparsion of Pressure Peak Position and Free Boundary Values

in High order discontinuous Galerkin method for elastohydrodynamic lubrication line contact problems

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### Table 3: Root mean square errors in the top and bottom boundary conditions, and the rms error in the (free boundary) stream function condition. The free boundary error is reduced during the iterative process, until it is ap- proximately the same magnitude as the top boundary error. The rms error quot; N in the stream function cost function (55) is given by quot; N = 1s Z s 0 [ [x; (x)] ? R(s ? x)]2 dx 1 2 ; (71) and for aquifers 1 and 2 are included in Table 3. For aquifer 1, roughly forty itera- tions were necessary to obtain su cient accuracy for the stream function condition, 14

"... In PAGE 14: ... As noted above, Laplace apos;s equation obeys a maximum principle, and so the maximum error in the series approximation can be determined by examining the boundary errors.The root mean square (rms) error quot;b N in the bottom boundary approximation (3) can be obtained directly from the residual errors along the bottom ow boundary (31): quot;b N = 1s Z s 0 Rb N(x; ae; b e) 2 dx 1 2 : (69) Similarly, the rms error quot;t N in the top boundary approximation (5) can be obtained from (32): quot;t N = 1s Z s 0 Rt N(x; ae; b e) 2 dx 1 2 : (70) Table3 lists the top and bottom boundary rms errors for both ow solutions obtained above by the interpolation method with N = 25:... ..."