Computing Nodal Distributions With Quasi‐Optimal Lebesgue Constant

ABSTRACT We propose an optimization method to minimize the Lebesgue constant of high‐order interpolative nodal distributions in the ‐dimensional simplex. The main novelty is to deterministically minimize the Lebesgue constant in the high‐dimensional simplex by consistently exploiting the first‐order derivatives of the Lebesgue function in terms of the node coordinates. To drive the minimization, we consider a sequential linear programming approach. At each iteration, we compute a candidate nodal displacement by minimizing the upper bound of first‐order approximations of the Lebesgue function sampled at the previous maxima. Then, we perform a backtracking line search procedure to choose a step‐length ensuring that the new nodal configuration reduces the Lebesgue constant. To devise our method, we use an advantageous representation of the ‐simplex in the ‐sphere orthant to exploit existing conjectures regarding the distribution and structure of optimal nodal distributions. The results show numerical evidence that our quasi‐optimal Lebesgue nodes are suitable for interpolation. Specifically, we obtain the optimal Lebesgue nodes in 1D and reproduce the current best nodal distributions in 2D. In 3D and 4D, our nodal sets improve the current best interpolative nodal configurations. Moreover, we obtain competitive condition numbers of the Vandermonde matrix and observe that the series seems to converge to one. We conclude that, in combination with existing methods to obtain fair point distributions, we can devise a two‐stage approach to compute quasi‐optimal Lebesgue nodal distributions suitable for high‐order numerical interpolation in the high‐dimensional simplex.