Sylvie Boldo, François Clément, Vincent Martin, Micaela Mayero, Houda Mouhcine
Formalization of mathematics is a major topic, that includes in particular numerical analysis, towards proofs of scientific computing programs. The present study is about the finite element method, a popular method to numerically solve partial differential equations. In the long-term goal of proving its correctness, we focus here on the formal definition of what is a finite element. Mathematically, a finite element describes what happens in a cell of a mesh. It notably includes the geometry of the cell, the polynomial approximation space, and a finite set of linear forms that computationally characterizes the polynomials. Formally, we design a finite element as a record in the Rocq proof assistant with both values (such as the vertices of the cell) and proofs of validity (such as the dimension of the approximation space). The decisive validity proof is unisolvence, that makes the previous characterization unique. We then instantiate this record with the most popular and useful, the simplicial Lagrange finite elements for evenly distributed nodes, for any dimension and any polynomial degree, including the difficult unisolvence proof. These proofs require many results (definitions, lemmas, canonical structures) about finite families, affine spaces, multivariate polynomials, in the context of finite or infinite-dimensional spaces.
Clément Francois, Hossein Nassajian Mojarrad, Duc Hiep Pham, Chun-Yen Shen
Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $Δ_{d, d}(E)=\left\{\left(\|x_1-y_1\|, \|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |Δ_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg q^{\frac{10}{3}}$, then $ |Δ_{2, 2}(E)| \gg q^2$. In this paper, we present extensions and improvements of these results.
François Clément, Stefan Steinerberger
We suppose we are given a list of points $x_1, \dots, x_n \in \mathbb{R}$, a target probability measure $μ$ and are asked to add additional points $x_{n+1}, \dots, x_{n+m}$ so that $x_1, \dots, x_{n+m}$ is as close as possible to the distribution of $μ$; additionally, we want this to be true uniformly for all $m$. We propose a simple method that achieves this goal. It selects new points in regions where the existing set is lacking points and avoids regions that are already overly crowded. If we replace $μ$ by another measure $μ_2$ in the middle of the computation, the method dynamically adjusts and allows us to keep the original sampling points. $x_{n+1}$ can be computed in $\mathcal{O}(n)$ steps and we obtain state-of-the-art results. It appears to be an interesting dynamical system in its own right; we analyze a continuous mean-field version that reflects much of the same behavior.
Hend Ben Ameur, Guy Chavent, Francois Clément, Pierre Weis
We transpose an optimal control technique to the image segmentation problem. The idea is to consider image segmentation as a parameter estimation problem. The parameter to estimate is the color of the pixels of the image. We use the adaptive parameterization technique which builds iteratively an optimal representation of the parameter into uniform regions that form a partition of the domain, hence corresponding to a segmentation of the image. We minimize an error function during the iterations, and the partition of the image into regions is optimally driven by the gradient of this error. The resulting segmentation algorithm inherits desirable properties from its optimal control origin: soundness, robustness, and flexibility.
François Clément, Vincent Martin
To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness of the approach. The finite element method is one of the popular tools for the numerical resolution of a wide range of PDEs. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs for the construction of the Lagrange finite elements of any degree on simplices in positive dimension.
Hend Ben Ameur, Guy Chavent, Cheikh Fatma, François Clément, Vincent Martin, Jean E. Roberts
Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modeled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparing fracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations.
François Clément, Stefan Steinerberger
It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting.
Carolin Benjamins, Helena Graf, Sarah Segel, Difan Deng, Tim Ruhkopf, Leona Hennig, Soham Basu, Neeratyoy Mallik, Edward Bergman, Deyao Chen, François Clément, Alexander Tornede, Matthias Feurer, Katharina Eggensperger, Frank Hutter, Carola Doerr, Marius Lindauer
Hyperparameter Optimization (HPO) is crucial to develop well-performing machine learning models. In order to ease prototyping and benchmarking of HPO methods, we propose carps, a benchmark framework for Comprehensive Automated Research Performance Studies allowing to evaluate N optimizers on M benchmark tasks. In this first release of carps, we focus on the four most important types of HPO task types: blackbox, multi-fidelity, multi-objective and multi-fidelity-multi-objective. With 3 336 tasks from 5 community benchmark collections and 28 variants of 9 optimizer families, we offer the biggest go-to library to date to evaluate and compare HPO methods. The carps framework relies on a purpose-built, lightweight interface, gluing together optimizers and benchmark tasks. It also features an analysis pipeline, facilitating the evaluation of optimizers on benchmarks. However, navigating a huge number of tasks while developing and comparing methods can be computationally infeasible. To address this, we obtain a subset of representative tasks by minimizing the star discrepancy of the subset, in the space spanned by the full set. As a result, we propose an initial subset of 10 to 30 diverse tasks for each task type, and include functionality to re-compute subsets as more benchmarks become available, enabling efficient evaluations. We also establish a first set of baseline results on these tasks as a measure for future comparisons. With carps (https://www.github.com/automl/CARP-S), we make an important step in the standardization of HPO evaluation.
Sylvie Boldo, François Clément, Jean-Christophe Filliâtre, Micaela Mayero, Guillaume Melquiond, Pierre Weis
Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization of the simplest one and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical pen-and-paper proofs. To our knowledge, this is the first time such kind of mathematical proof is machine-checked.
Sylvie Boldo, Francois Clement, Jean-Christophe Filliâtre, Micaela Mayero, Guillaume Melquiond, Pierre Weis
We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method errors, and floating-point computations lead to round-off errors. We annotate this C program to specify both method error and round-off error. We use Frama-C to generate theorems that guarantee the soundness of the code. We discharge these theorems using SMT solvers, Gappa, and Coq. This involves a large Coq development to prove the adequacy of the C program to the numerical scheme and to bound errors. To our knowledge, this is the first time such a numerical analysis program is fully machine-checked.
François Clément, Diederick Vermetten, Jacob de Nobel, Alexandre D. Jesus, Luís Paquete, Carola Doerr
The $L_{\infty}$ 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 $L_{\infty}$ 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 $L_{\infty}$ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches. In this work we compare 8 popular numerical black-box optimization algorithms on the $L_{\infty}$ star discrepancy computation problem, using a wide set of instances in dimensions 2 to 15. We show that all used optimizers perform very badly on a large majority of the instances and that in many cases random search outperforms even the more sophisticated solvers. We suspect that state-of-the-art numerical black-box optimization techniques fail to capture the global structure of the problem, an important shortcoming that may guide their future development. We also provide a parallel implementation of the best-known algorithm to compute the discrepancy.
Francois Clement, Thang Pham
In this paper, we study the Erdős-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 $Δ(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we obtain stronger statements as follows: If $|A|\gg p^{\frac{13}{22}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ If $|A|\gg p^{\frac{4}{7}}$, 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 \[|A^2+A^2|\gg \min \left\lbrace \frac{p}{K^4}, \frac{|A|^{8/3}}{K^{7/3}p^{2/3}}\right\rbrace.\] As a consequence, $|A^2+A^2|\gg p$ when $|A|\gg p^{5/8}$ and $K\sim 1$, where $A^2=\{x^2\colon x\in A\}$.
Sylvie Boldo, François Clément, Florian Faissole, Vincent Martin, Micaela Mayero
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $σ$-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou's lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $L^p$~spaces such as Banach spaces.
Sylvie Boldo, François Clément, Vincent Martin, Micaela Mayero, Houda Mouhcine
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects. Once Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is the Tonelli theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. Therefore, we need to define and prove results on product spaces, hoping that they can easily derive from the existing ones on a single space. This article describes the formal definition and proof in Coq of product $σ$-algebras, product measures and their uniqueness, the construction of iterated integrals, up to the Tonelli theorem. We also advertise the \emph{Lebesgue induction principle} provided by an inductive type for {\nonnegative} measurable functions.
Neus Ramos-Escobar, Manuel Mercier, Agnès Trébuchon-Fonséca, Antoni Rodriguez-Fornells, Clément François, Daniele Schön
Mar 11, 2022·q-bio.NC·PDF Statistical learning has been proposed as a mechanism to structure and segment the continuous flow of information in several sensory modalities. Previous studies proposed that the medial temporal lobe, and in particular the hippocampus, may be crucial to parse the stream in the visual modality. However, the involvement of the hippocampus in auditory statistical learning, and specifically in speech segmentation is less clear. To explore the role of the hippocampus in speech segmentation based on statistical learning, we exposed seven pharmaco-resistant temporal lobe epilepsy patients to a continuous stream of trisyllabic pseudowords and recorded intracranial stereotaxic electro-encephalography (sEEG). We used frequency-tagging analysis to quantify neuronal synchronization of the hippocampus and auditory regions to the temporal structure of words and syllables of the stream. Results show that while auditory regions highly respond to syllable frequency, the hippocampus responds mostly to word frequency. These findings provide direct evidence of the involvement of the hippocampus in speech segmentation process and suggest a hierarchical organization of auditory information during speech processing.
François Clément, Stefan Steinerberger
The Redheffer matrix $A_n \in \mathbb{R}^{n \times n}$ is defined by setting $A_{ij} = 1$ if $j=1$ or $i$ divides $j$ and 0 otherwise. One of its many interesting properties is that $\det(A_n) = O(n^{1/2 + \varepsilon})$ is equivalent to the Riemann hypothesis. The singular vector $v \in \mathbb{R}^n$ corresponding to the largest singular value carries a lot of information: $v_k$ is small if $k$ is prime and large if $k$ has many divisors. We prove that the vector $w$ whose $k-$th entry is the sum of the inverse divisors of $k$, $w_k = \sum_{d|k} 1/d$, is close to a singular vector in a precise quantitative sense.
François Clément, Stefan Steinerberger
The Ulam sequence, described by Stanislaw Ulam in the 1960s, starts $1,2$ and then iteratively adds the smallest integer that can be uniquely written as the sum of two distinct earlier terms: this gives $1,2,3,4,6,8,11,\dots$. Already in 1972 the great French poet Raymond Queneau wrote that it `gives an impression of great irregularity'. This irregularity appears to have a lot of structure which has inspired a great deal of work; nonetheless, very little is rigorously proven. We improve the best upper bound on its growth and show that at least some small gaps have to exist: for some $c>0$ and all $n \in \mathbb{N}$ $$ \min_{1 \leq k \leq n} \frac{a_{k+1}}{a_k} \leq 1 + c\frac{\log{n}}{n}.$$
François Clément, Nathan Kirk, Art B. Owen, T. Konstantin Rusch
Points in the unit cube with low discrepancy can be constructed using algebra or, more recently, by direct computational optimization of a criterion. The usual $L_\infty$ star discrepancy is a poor criterion for this because it is computationally expensive and lacks differentiability. Its usual replacement, the $L_2$ star discrepancy, is smooth but exhibits other pathologies shown by J. Matoušek. In an attempt to address these problems, we introduce the \textit{average squared discrepancy} which averages over $2^d$ versions of the $L_2$ star discrepancy anchored in the different vertices of $[0,1]^d$. Not only can this criterion be computed in $O(dn^2)$ time, like the $L_2$ star discrepancy, but also we show that it is equivalent to a weighted symmetric $L_2$ criterion of Hickernell's by a constant factor. We compare this criterion with a wide range of traditional discrepancy measures, and show that only the average squared discrepancy avoids the problems raised by Matoušek. Furthermore, we present a comprehensive numerical study showing in particular that optimizing for the average squared discrepancy leads to strong performance for the $L_2$ star discrepancy, whereas the converse does not hold.
Zawad Chowdhury, Francois Clement, Max Horwitz
We investigate a family of $4$-regular graphs constructed to test for the presence of combinatorial structure in a sequence of distinct real numbers. We show that the graphs constructed from the Kronecker sequence can be embedded into the torus, while the graphs constructed from the binary van der Corput sequence can be embedded into the Chamanara surface, in both cases with the possible removal of one edge. These results allude to a general theory of sequence graphs which can be embedded into particular translation surfaces coming from interval exchange transformations.
Francois Clement, Nathan Kirk, Florian Pausinger
Classical jittered sampling partitions $[0,1]^d$ into $m^d$ cubes for a positive integer $m$ and randomly places a point inside each of them, providing a point set of size $N=m^d$ with small discrepancy. The aim of this note is to provide a construction of partitions that works for arbitrary $N$ and improves straight-forward constructions. We show how to construct equivolume partitions of the $d$-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. We investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques.