Probabilistic Learning on Manifold for Optimization Under Uncertainties

Abstract

Abstract. This paper presents a challenging problem devoted to the probabilistic learning

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

“…In this Appendix, we summarize the smoothing algorithm [25,88] that is used for estimating J(w 0 ) and c(w 0 ) at any point w 0 in C w , by using only the additional realizations {y ar , = 1, . .…”

“…The smoothing algorithm is derived from a nonparametric statistical estimate of the probability density function p Y of Y. The construction of this algorithm is detailed in [25,88] and is summarized hereinafter.…”

“…(ii) The second ingredient is a smoothing technique [25,88] for estimating J(w 0 ) and c(w 0 ) at any point w 0 in C w , by using only the additional realizations generated from the given dataset. The points w 0 are those that are generated for the optimization algorithm.…”

“…In this paper, the probabilistic learning used for solving the optimization problem under uncertainties refers to an algorithm that is based on the use of the two following methodologies. (i) The first one is the novel methodology that is proposed in [25,88] for solving the CNOPPU, which consists in using -an optimizer that is adapted to the nature of the optimization problem under consideration, -a dataset that is made up of a small number of points generated by the optimizer for which only a small number of expensive evaluations of functions is carried out.…”