Design Problems for Optimal Surface Interpolation.
WISCONSIN UNIV-MADISON DEPT OF STATISTICS
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We consider the problem of interpolating a surface given its values at a finite number of points. We place a special emphasis on the question of choosing the location of the points where the function will be sampled. Using minimal norm interpolation in reproducing kernel Hilbert spaces, equivalently Bayesian interpolation, and N-widths, we provide lower bounds for interpolation error relative to certain error criteria. These lower bounds can be used when evaluating an existing design, or when attempting to obtain a good design by iterative procedures to decide whether further minimization is worthwhile. The bounds are given in terms of the eigenvalues of a relevant reproducing kernel and the asymptotic behavior of these eigenvalues for certain tensor product spaces in the unit d-dimensional cube is obtained.
- Statistics and Probability