We show some simple one-dimensional examples for illustrative purposes.

In the sequence of graphs that display our results, the objective *f* is
shown in blue, while the approximation *a* is shown in cyan.

The grid upon which we are operating is shown in ``hatch'' marks along
the *x*-axis

The points at which *f*(*x*) is known are denoted with red
circles.

When we have only two known points (which is what we use in the initial experimental design) we build a simple linear interpolant. We distinguish these two initial design sites with yellow stars.

When we have three known points (after the first optimization step), we build a quadratic interpolant.

For subsequent iterations we use a cubic spline interpolant [12], which for the one-dimensional examples we show requires a minimum of four data points.

The candidate *x _{t}* for which the approximation

**Warning:** The pictures that follow were generated using Matlab.
The sequence of graphs presented have been scaled by Matlab to best
portray the range of values specific to that graph; thus, different
scales for the *y*-axis are used for different graphs to
capture the changing features of the interpolant as additional points
are added. (The scale of the *x*-axis remains constant.)

** Next:** First Example
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