arXiv subjects

François Clément

Publications and source records attributed to François Clément.

34 records · Page 2Linked to original sources

arXiv2024

Finite element method. Detailed proofs to be formalized in Coq

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.

cs.LO↗

arXiv2024

Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem

Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain $L_{\infty}$-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose $L_{\infty}$ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.

cs.CG↗

arXiv2024

Outperforming the Best 1D Low-Discrepancy Constructions with a Greedy Algorithm

The design of uniformly spread sequences on $[0,1)$ has been extensively studied since the work of Weyl and van der Corput in the early $20^{\text{th}}$ century. The current best sequences are based on the Kronecker sequence with golden ratio and a permutation of the van der Corput sequence by Ostromoukhov. Despite extensive efforts, it is still unclear if it is possible to improve these constructions further. We show, using numerical experiments, that a radically different approach introduced by Kritzinger in seems to perform better than the existing methods. In particular, this construction is based on a \emph{greedy} approach, and yet outperforms very delicate number-theoretic constructions. Furthermore, we are also able to provide the first numerical results in dimensions 2 and 3, and show that the sequence remains highly regular in this new setting.

cs.CG↗

arXiv2024

Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection

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 we prove that the heuristic does not necessarily return an optimal solution, we obtain very promising results for all tested dimensions. For example, for moderate point set sizes $30 \leq n \leq 240$ in dimension 6, we obtain point sets with $L_{\infty}$ 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 also provide a comparison with a recent energy functional introduced by Steinerberger [J. Complexity, 2019], showing that our heuristic performs better on all tested instances.

cs.CG↗

arXiv2024

Constructing Optimal $L_{\infty}$ Star Discrepancy Sets

The $L_{\infty}$ star discrepancy is a very well-studied measure used to quantify the uniformity of a point set distribution. Constructing optimal point sets for this measure is seen as a very hard problem in the discrepancy community. Indeed, optimal point sets are, up to now, known only for $n\leq 6$ in dimension 2 and $n \leq 2$ for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low $L_{\infty}$ star discrepancy as possible. Firstly, we present two models to construct optimal sets and show that there always exist optimal sets with the property that no two points share a coordinate. Then, we provide possible extensions of our models to other measures, such as the extreme and periodic discrepancies. For the $L_{\infty}$ star discrepancy, we are able to compute optimal point sets for up to 21 points in dimension 2 and for up to 8 points in dimension 3. For $d=2$ and $n\ge 7$ points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.

cs.CG↗

arXiv2023

Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms

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.

cs.NE↗

arXiv2022

A Coq Formalization of the Bochner integral

The Bochner integral is a generalization of the Lebesgue integral, for functions taking their values in a Banach space. Therefore, both its mathematical definition and its formalization in the Coq proof assistant are more challenging as we cannot rely on the properties of real numbers. Our contributions include an original formalization of simple functions, Bochner integrability defined by a dependent type, and the construction of the proof of the integrability of measurable functions under mild hypotheses (weak separability). Then, we define the Bochner integral and prove several theorems, including dominated convergence and the equivalence with an existing formalization of Lebesgue integral for nonnegative functions.

cs.LO↗

arXiv2022

Lebesgue Induction and Tonelli's Theorem in Coq

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.

cs.LO↗

arXiv2021

A Coq Formalization of Lebesgue Integration of Nonnegative Functions

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.

cs.LO↗

arXiv2021

Lebesgue integration. Detailed proofs to be formalized in Coq

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. Sobolev spaces are the mathematical framework in which most weak formulations of partial derivative equations are stated, and where solutions are sought. These functional spaces are built on integration and measure theory. Hence, this chapter in functional analysis is a mandatory theoretical cornerstone for the definition of the finite element method. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs of the main results from integration and measure theory.

cs.LO↗

arXiv2016

First-Order Indicators for the Estimation of Discrete Fractures in Porous Media

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.

math.NA↗

arXiv2016

The Lax-Milgram Theorem. A detailed proof to be formalized in Coq

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. The purpose of this document is to provide the formal proof community with a very detailed pen-and-paper proof of the Lax-Milgram theorem.

cs.LO↗

arXiv2014

Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.

math.NA↗

arXiv2011

Formal Proof of a Wave Equation Resolution Scheme: the Method Error

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.

cs.LO↗

arXiv2011

Formal Proof of a Wave Equation Resolution Scheme: the Method Error

cs.LO↗

arXiv2008

The Multi-Dimensional Refinement Indicators Algorithm for Optimal Parameterization

The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori information by reducing the number of unknowns according to the physics of the problem. The refinement indicators algorithm provides a fruitful adaptive parameterization technique that parsimoniously opens the degrees of freedom in an iterative way. We present a new general form of the refinement indicators algorithm that is applicable to the estimation of multi-dimensional parameters in any PDE. In the linear case, we state the relationship between the refinement indicator and the decrease of the usual least-squares data misfit objective function. We give numerical results in the simple case of the identity model, and this application reveals the refinement indicators algorithm as an image segmentation technique.

math.NA↗