Building upon the exact methods presented in our earlier work [J. Complexity, 2022], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While the heuristic does not necessarily return an optimal solution, we obtain very promising results for all tested dimensions. For example, for moderate sizes 30≤n≤240, we obtain point sets in dimension 6 with L∞ star discrepancy up to 35% better than that of the first n points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with a recent energy functional introduced by Steinerberger [J. Complexity, 2019], showing that our heuristic performs better on all tested instances. Finally, our results and complementary experiments also give further empirical information on inverse star discrepancy conjectures.

The L∞ star discrepancy is a measure for the regularity of a finite set of points taken from [0, 1)d. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other applications. Unfortunately, computing the L∞ star discrepancy of a given point set is known to be a hard problem, with the best exact algorithms falling short for even moderate dimensions around 8. However, despite the difficulty of finding the global maximum that defines the L∞ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches.

Motivated by applications in instance selection, we introduce the star discrepancy subset selection problem, which consists of finding a subset of m out of n points that minimizes the star discrepancy. First, we show that this problem is NP-hard. Then, we introduce a mixed integer linear formulation (MILP) and a combinatorial branch-and-bound (BB) algorithm for the star discrepancy subset selection problem and we evaluate both approaches against random subset selection and a greedy construction on different use-cases in dimension two and three. Our results show that the MILP and BB are efficient in dimension two for large and small m/n ratio, respectively, and for not too large n. However, the performance of both approaches decays strongly for larger dimensions and set sizes. As a side effect of our empirical comparisons we obtain point sets of discrepancy values that are much smaller than those of common low-discrepancy sequences, random point sets, and of Latin Hypercube Sampling. This suggests that subset selection could be an interesting approach for generating point sets of small discrepancy value.

In this paper, we study the Erdös--Falconer distance problem in five dimensions for sets of Cartesian product structures. More precisely, we show that for $A\subset \mathbb{F}_p$ with $|A|\gg p^{\frac{13}{22}}$, then $\Delta(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we are able to obtain stronger conclusions as follows: łinebreak 1. If $p^{13/22}\ll |A|\ll p^{\frac{2}{3}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p$. 2. If $p^{4/7}\ll |A|\ll p^{\frac{2}{3}}$, then $(A-A)^2+(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p$. We also prove that if $p^{4/7}\ll |A-A|=K|A|\le p^{5/8}$, then $|A^2+A^2|\gg \min \{ \frac{p}{K^4}, \frac{|A|^{8/3}}{K^{7/3}p^{2/3}}\}$. As a consequence, $|A^2+A^2|\gg p$ when $|A|\sim p^{5/8}$ and $K\sim 1$, where $A^2=\{x^2: x\in A\}$.

The Finite Element Method is a widely-used method to solve numerical problems coming for instance from physics or biology. To obtain the highest confidence on the correction of numerical simulation programs implementing the Finite Element Method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. The Lax–Milgram theorem may be seen as one of those theoretical cornerstones: under some completeness and coercivity assumptions, it states existence and uniqueness of the solution to the weak formulation of some boundary value problems. This article presents the full formal proof of the Lax–Milgram theorem in Coq. It requires many results from linear algebra, geometry, functional analysis, and Hilbert spaces.

The growing availability of high throughput measurement devices in the operating room makes possible the collection of a huge amount of data about the state of the patient and the doctors’ practice during a surgical operation. This paper explores the possibility of extracting from these data relevant information and pertinent decision rules in order to support the daily anesthesia procedures. In particular we focus on machine learning strategies to design a closed-loop controller that, in a near future, could play the role of a decision support tool and, in a further perspective, the one of automatic pilot of the anesthesia procedure. Two strategies (direct and inverse) for learning a controller from observed data are assessed on the basis of a database of measurements collected in recent years by the ULB Erasme anaesthesiology group. The preliminary results of the learning approach applied to the regulation of hypnosis through the bispectral index (BIS) in a simulated framework appear to be promising and worthy of future investigation.