PWP PWP

Minimization of Equation (6) with respect to P leads to

where WT is the transpose of matrix W . Solving the resulting system (of N equations with N unknowns) for P we obtain

An important advantage of this method is the sparsity of the matrix W . Since W has a high percentage of zero-valued elements, using a sparse matrix data only the nonzero elements and their indices are stored, reducing significantly the amount of memory for storage, and making the matrix inversion calculation process more efficient. Only a few seconds were necessary for the matrix inversion process, being W a 16075 by 396 matrix. Notice that d was not defined exactly as Euclidean distance, i.e. L2 norm, but a L245 norm. The explanation for this is illustrated in Fig. 8: plots (a) and (b) show the behaviour of function W for a 1D measurement domain Q = [0,4], and collocation points

{Xj} ={1,2,3} with Qj =[0,2], Q2 =[1,3] and Q3 =[2,4]; plots (c), (d), and (e) show the behaviour of function W for a 2D measurement domain Q = [0,3] x [0,3], and collocation points {X,} 3 4 = {(1,1), (2,1),(1,2),(2,2)} with Qx = [0,2] x [0,2],

Q2 = [1,3] x [0,2], Q3 = [0,2] x [1,3], and Q4 = [1,3] x [1,3] using L2 norm, and plots (f), (g), and (h) are the same as before except using L2.45 norm.

As can be seen from plot (b), in the 1D measurement domain case, the sum of magnitudes of the approximated measurements of each collocation point Xj with respect to a observation point X,, with d = - xj in the influence domain of point Xj is, as expected, a unit value, except in the boundaries of course. In the 2D measurement domain case, as can be seen from plot (d) and (e), the sum of magnitudes of the approximated measurements of each collocation point Xj with respect to a observation point Xt, with d = ^jx,. - Xj| + jy,. - yj| , the Euclidean distance, in the influence domain of point Xj , is not as should, a unit value, except of course in the boundaries.

Fig. 8. Behaviour of weight function W for 1D and 2D measurement domains, using norm

L2 and L245 in 2D case.

This problem is partially eliminated if a norm L245 is used, as show plots (g) and (h). This value was empirically adjusted so that the sum of magnitudes of the measurements of each collocation point be approximated a unit value. Cross sections for the 3D measurements domain case (not shown) were performed and the results are similar to the 2D case previously presented.

A "less visited collocation point" X. was defined as one whose sum of magnitudes (sum of elements of column vector j ) was less then the difference between the mean value of the all sums of magnitudes (SumMag) and three times the standard deviation of these sums lessvisitj = ^ Wj < (mean(SumMag) - 3 x std(SumMag)). (9)

To increase information on the desired variable in the vicinity of less visited collocation points Xj, a cell grid size update was performed

Finally, a finer meshgrid of the form [Ax Ay Az] = [2 2 0.2] was considered for the surface visualization generation. The desired variable on the M visualization points was calculated as follows

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