Assuming nominal values of a0 = 1, a1 = 4, a2 = ??’5.5, (x1, y1) = (30, 30),
(x2, y2) = (35, 65), we first divide the Gaussian center??™s search space into an
N ?— N grid. We constrain our nonlinear optimization routine to searching
inside all grid pairs where the centers are located (in order to avoid local
minima results). The optimal location of the grid center moves according to
the graphs shown in Figure 5. The figure also shows the convergence of the
three linear field coe?±cients to their actual values. For the simulation results
in Figure 5, N = 2, and the search space was a0, a1, a2 = [??’10 10], (x1, y1),
(x2, y2) in random subsquare pairs.
Because we cannot use the LSE estimate to find all the unknown parameters,
the sampling algorithm used is di?®erent than (AS-1) and summarized
below:
Numerical Gradient Search Adaptive Sampling Algorithm (AS-2)
Assumptions: The field model is nonlinear but parametric, sampling locations
xj belong to a discrete sampling space ??– and do not have uncertainty.
The field depends linearly on parameter set A, and nonlinearly on parameter
set B (in the previous example these are the centers and variance of the
Gaussian functions). The sample parameter set B is part of a discrete coarse
grid in its own q-dimensional space. The sample measurements have constant
uncertainty ??2
z = var(zi). The sampling space ??– can be a grid. Constants
" > 0, p ?‰? 1 are given.
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