Spline Methods Using Low Discrepancy Designs
نویسندگان
چکیده
Splines are minimum-norm approximations of functions that interpolate the given data, (xl, f(xl)), l = 0, . . . , N − 1. In this paper, we consider the problem of approximating functions in certain reproducing kernel Hilbert spaces, with special choices of the points xl, l = 0, . . . , N−1. The designs considered here are node sets of integration lattices and digital nets for functions f defined over the unit cube. The worst case errors (in the L∞ norm) of the splines are shown to depend on the smoothness of the functions being approximated.
منابع مشابه
Centered L2-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs
In this paper properties and construction of designs under a centered version of the L2-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is als...
متن کاملUniform Designs Limit Aliasing
When fitting a linear regression model to data, the effects not included in the model can confound those included in the model, resulting in incorrect estimates of the regression coefficients and incorrect inferences as to whether a term is significant. This paper shows how uniform designs can reduce this aliasing. The discrepancy is a quantitative measure of how uniformly design points are pla...
متن کاملLower bounds and stochastic optimization algorithms for uniform designs with three or four levels
New lower bounds for threeand four-level designs under the centered L2-discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L2-discrepancy. Besides the threshold accep...
متن کاملA New Approach to Construction of Nearly Uniform Designs
The uniform design is one of space filling designs and has been widely used in computer and industrial experiments. Many methods for construction of uniform designs or nearly uniform designs, such as the glp method, optimization method etc. have been proposed. A nearly uniform design is a design with low-discrepancy, where the discrepancy is a measure of uniformity. Various discrepancies have b...
متن کاملOptimized U-type designs on flexible regions
The concept of a flexible region describes an infinite variety of symmetrical shapes to enclose a particular region of interestwithin a space. In experimental design, the properties of a function on the region of interest are analyzed based on a set of design points. The choice of design points can be made based on some discrepancy criterion. The generation of design points on a flexible region...
متن کامل