On approximating the shape of one dimensional functions
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Consider an s-dimensional function being evaluated at n points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out (s-1) variables. Here, the emphasis is on accurately approximating the shape of such one-dimensional functions. Approximating this shape when the function is evaluated on a set of grid points instead is relatively straightforward. However, the number of grid points needed increases exponentially with s. LDS are known to be increasingly more efficient at integrating s-dimensional functions compared to grids, as s increases. Yet, a method to approximate the shape of a one-dimensional function when the function is evaluated using an s-dimensional LDS has not been proposed thus far. We propose an approximation method for this problem. This method is based on an s-dimensional integration rule together with fitting a polynomial smoothing function. We state and prove results showing conditions under which this polynomial smoothing function will converge to the true one-dimensional function. We also demonstrate the computational efficiency of the new approach compared to a grid based approach.
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