By E. Arge, M. Dæhlen, T. Lyche, K. Mørken (auth.), J. C. Mason, M. G. Cox (eds.)

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Extra resources for Algorithms for Approximation II: Based on the proceedings of the Second International Conference on Algorithms for Approximation, held at Royal Military College of Science, Shrivenham, July 1988

Example text

00295, respectively. In this example, the use of (1) gives a new set of knots which results in a very substantial improvement in the root mean square residual. Figure 3(c) shows the knots and corresponding spline fit after 4 iterations of adaptive knot placement. 73 X 10- 7 • Adaptive knot placement has reduced the root mean square residual by a factor of 10,000 without overfitting. We observe that much of this improvement results from the strategy repositioning the knots in such a way as to remove the severe oscillation that previously existed in the "shoulder" of the curve.

2 Convex approximation. I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I t I I I I I I I I I 100 80 60 40 20 100 Figure 3. The initial eonvex approximation to the turbine data. The dashed eurve shown in Figure 3 is a eubie spline approximation to a set of data points fori=1,2, ... ,50, 15 where Yi is measured electricity when 2i percent of maximum power is forced upon a particular water turbine. A large number of turbines have been tested and data measured, and all turbines have similar curves depending on the turbine type, the size of the turbine and various other factors.

As a result we hope to achieve a better, and certainly smoother, approximation. The strategy is adaptive since the approximation to j(n) is obtained from a spline fit to the data based on a current set of interior knots. Thus, one application of the criterion results in this current set being updated, and the new knot vector forms the current knots at the next application. In order to use (1) adaptively, we need (a) an initial interior knot vector A, (b) a means of approximating f(n) using the current set of knots, and ( c) a means of terminating the algorithm.

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