关于一维函数形状的逼近
On approximating the shape of one dimensional functions
摘要：
考虑在低差异序列（LDS）的$n$点处评估的$s$维函数，其中目标是近似由积分出$(s-1)$变量产生的一维函数。这里，重点是精确地近似这种\emph{one-dimensional}函数的形状。当在一组网格点上计算函数时，近似此形状相对简单。然而，所需的网格点的数量随着$s$呈指数级增加。已知随着$s$的增加，LDS在积分$s$维函数时比网格更有效。然而，当使用$s$维LDS评估一维函数时，迄今为止还没有提出近似该函数的形状的方法。我们提出了一个近似方法解决这个问题。该方法是基于一个$s$维的积分规则与拟合多项式平滑函数。我们的状态和证明结果显示的条件下，这个多项式平滑函数将收敛到真正的一维函数。我们还证明了计算效率的新方法相比，基于网格的方法。
英文摘要：
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 \emph{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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