Publications

2023

• On the concave one-dimensional random assignment problem and Young integration theory
  
  • Goldman Michael
  • Trevisan Dario
  , 2023. We investigate the one-dimensional random assignment problem in the concave case, i.e., the assignment cost is a concave power function, with exponent 0 < p < 1, of the distance between n source and n target points, that are i.i.d. random variables with a common law on an interval. We prove that the limit of a suitable renormalization of the costs exists if the exponent p is different than 1/2. Our proof in the case 1/2 < p < 1 makes use of a novel version of the Kantorovich optimal transport problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite q-variation, which may be of independent interest. We also prove a similar result for the random bipartite Traveling Salesperson Problem.
• Numerical and Theoretical Study of Patterns in a Chemotaxis Model
  
  • Payan Maxime
  • Breden Maxime
  , 2023. Models of biological interactions are in full swing. They come in different flavors, one of them being systems of partial differential equations, which are then studied by mathematicians. Here, we will be interested in the establishment of stationary solutions to problems of the form \begin{equation*} \partial_t U = \Delta (\Phi U) + f(U) \, , \end{equation*} with a computer-assisted method. Starting from a known approximate solution, we get back to study a fixed point problem to solve our initial problem. This fixed point then becomes our existing and unique theoretical solution in a neighbourhood of our approximate solution. The difficulty lies in the choices we make when reducing the problem. In particular, the non-linearity of the equations is a significant obstacle. More precisely we will look at a chemotaxis model where \begin{equation*}\left\{ \begin{array}{lll} \partial_t u &= \Delta(\gamma(v)u) &+ \sigma u (1 - u) \\ \partial_t v &= \varepsilon\Delta v &+ u - v \end{array}\right. \end{equation*} In this type of model the search for patterns and the study of their stability is interesting, which justifies our numerical to theoretical approach. We will look at different results according to the chosen function $\gamma$ : rational fraction, decreasing exponential, power series, ...
• Multi-fidelity surrogate modeling for time-series outputs
  
  • Kerleguer Baptiste
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2023, 11 (2), pp.514-539. This paper considers the surrogate modeling of a complex numerical code in a multifidelity framework when the code output is a time series. Using an experimental design of the low-and high-fidelity code levels, an original Gaussian process regression method is proposed. The code output is expanded on a basis built from the experimental design. The first coefficients of the expansion of the code output are processed by a co-kriging approach. The last coefficients are collectively processed by a kriging approach with covariance tensorization. The resulting surrogate model taking into account the uncertainty in the basis construction is shown to have better performance in terms of prediction errors and uncertainty quantification than standard dimension reduction techniques. (10.1137/20M1386694)
  DOI : 10.1137/20M1386694
• A limiting model for a low Reynolds number swimmer with N passive elastic arms
  
  • Alouges François
  • Lefebvre-Lepot Aline
  • Levillain Jessie
  Mathematics in Engineering, AIMS, 2023, 5 (5), pp.1-20. We consider a low Reynolds number artificial swimmer that consists of an active arm followed by N passive springs separated by spheres. This setup generalizes an approach proposed in Montino and DeSimone, Eur. Phys. J. E, vol. 38, 2015. We further study the limit as the number of springs tends to infinity and the parameters are scaled conveniently, and provide a rigorous proof of the convergence of the discrete model to the continuous one. Several numerical experiments show the performances of the displacement in terms of the frequency or the amplitude of the oscillation of the active arm. (10.3934/mine.2023087)
  DOI : 10.3934/mine.2023087
• Approximate Lipschitz stability for phaseless inverse scattering with background information
  
  • Sivkin Vladimir
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2023, 31 (3), pp.441-454. We prove approximate Lipschitz stability for monochromatic phaseless inverse scattering with background information in dimension d ≥ 2. Moreover, these stability estimates are given in terms of non-overdetermined and incomplete data. Related results for reconstruction from phaseless Fourier transforms are also given. Prototypes of these estimates for the phased case were given in Novikov (2013 J. Inverse Ill-Posed Problems, 21, 813-823). (10.1515/jiip-2023-0001)
  DOI : 10.1515/jiip-2023-0001
• Well-posedness of wave scattering in perturbed elastic waveguides and plates: application to an inverse problem of shape defect detection
  
  • Bonnetier Eric
  • Niclas Angèle
  • Seppecher L.
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2023, 479 (2273), pp.20220646. The aim of this work is to present theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in elastic waveguides and plates. Given surface multi-frequency wavefield measurements, we use a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, we introduce a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, we prove the well-posedness of the PDE that model elastic wave propagation in two- and three-dimensional planar waveguides. We also characterize the critical frequencies for which the Lamb decomposition is not valid. By using these results, we generalize the shape reconstruction method already developed for acoustic waveguide to two-dimensional elastic waveguides and provide a stable reconstruction method based on a mode-by-mode spacial Fourier inversion given by the scattered field. (10.1098/rspa.2022.0646)
  DOI : 10.1098/rspa.2022.0646
• Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions
  
  • Touré Cheikh
  • Auger Anne
  • Hansen Nikolaus
  Journal of Global Optimization, Springer Verlag, 2023, 86 (1), pp.163-203. Evolution Strategies (ES) are stochastic derivative-free optimization algorithms whose most prominent representative, the CMA-ES algorithm, is widely used to solve difficult numerical optimization problems. We provide the first rigorous investigation of the linear convergence of step-size adaptive ES involving a population and recombination, two ingredients crucially important in practice to be robust to local irregularities or multimodality. We investigate convergence of step-size adaptive ES with weighted recombination on composites of strictly increasing functions with continuously differentiable scaling-invariant functions with a global optimum. This function class includes functions with non-convex sublevel sets and discontinuous functions. We prove the existence of a constant r such that the logarithm of the distance to the optimum divided by the number of iterations converges to r. The constant is given as an expectation with respect to the stationary distribution of a Markov chain—its sign allows to infer linear convergence or divergence of the ES and is found numerically. Our main condition for convergence is the increase of the expected log step-size on linear functions. In contrast to previous results, our condition is equivalent to the almost sure geometric divergence of the step-size on linear functions. (10.1007/s10898-022-01249-6)
  DOI : 10.1007/s10898-022-01249-6
• Validated integration of semilinear parabolic PDEs
  
  • Bouwe van den Berg Jan
  • Breden Maxime
  • Sheombarsing Ray
  , 2023. Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton-Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift-Hohenberg equation, the Ohta-Kawasaki equation and the Kuramoto-Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.
• Hamilton–Jacobi equations for controlled gradient flows: The comparison principle
  
  • Conforti Giovanni
  • Kraaij R.C.
  • Tonon D.
  Journal of Functional Analysis, Elsevier, 2023, 284 (9), pp.109853. (10.1016/j.jfa.2023.109853)
  DOI : 10.1016/j.jfa.2023.109853
• Unifying mirror descent and dual averaging
  
  • Juditsky Anatoli
  • Kwon Joon
  • Moulines Éric
  Mathematical Programming, Springer Verlag, 2023, 199, pp.793–830. We introduce and analyze a new family of first-order optimization algorithms which generalizes and unifies both mirror descent and dual averaging. Within the framework of this family, we define new algorithms for constrained optimization that combines the advantages of mirror descent and dual averaging. Our preliminary simulation study shows that these new algorithms significantly outperform available methods in some situations. (10.1007/s10107-022-01850-3)
  DOI : 10.1007/s10107-022-01850-3
• Experimental investigation of supercritical injection
  
  • Chazelle T
  • Duhem-Duvilla L
  • Lespinasse Florian
  • Idlahcen S
  • Blaisot J.-B
  • Barviau B
  • Ribert G
  • Le Calvez Y
  • Nabet F
  • Giovangigli V
  , 2023. Over a certain value of temperature and pressure, known as the critical point, pure fluids enter the supercritical domain where the separation between the liquid and the gas states becomes much less apparent. The clear discontinuity observed when crossing the saturation curve is then replaced by a continuous evolution from liquidlike to gas-like states. Such a behavior can be found in combustion applications such as rocket engines or gas turbines (including turbojets) [1] and their improvements require a better understanding and control of the mixture of fuel and oxidizer. As very few experimental data on the behavior of injection and mixing under supercritical conditions are available in the literature, the modeling of the flame under such conditions lacks data and the validation of the simulations becomes very limited [2]. The objective of this work is to fill this gap by providing experimental quantitative data on a jet of ethane in a high-pressure chamber, filled with nitrogen at rest from atmospheric up to 6 MPa [2]. For example, Fig. 1 shows a jet of ethane (300 K) into nitrogen (333 K) at 4.5 MPa, i.e. below the critical point of ethane, 4.87 MPa. Various injection regimes are studied by shadowgraphy imaging varying the temperature, the pressure as well as the fuel mass flow rate. Then, through adequate post-processing, the spreading angle of the jet is computed, and its evolution analyzed. Finally, a thorough study of the velocity evolution for a supercritical jet is achieved for a broad range of inlet conditions (50 cases) using the Image Correlation Velocimetry (ICV) technique [3].
• Uncertainty quantification and calibration of one-dimensional arterial hemodynamics
  
  • Benmahdi Meryem
  • Le Maitre Olivier
  • Congedo Pietro Marco
  , 2023.
• AskewSGD : an annealed interval-constrained optimisation method to train quantized neural networks
  
  • Leconte Louis
  • Schechtman Sholom
  • Moulines Eric
  , 2023, 206, pp.3644--3663. In this paper, we develop a new algorithm, Annealed Skewed SGD - AskewSGD - for training deep neural networks (DNNs) with quantized weights. First, we formulate the training of quantized neural networks (QNNs) as a smoothed sequence of interval-constrained optimization problems. Then, we propose a new first-order stochastic method, AskewSGD, to solve each constrained optimization subproblem. Unlike algorithms with active sets and feasible directions, AskewSGD avoids projections or optimization under the entire feasible set and allows iterates that are infeasible. The numerical complexity of AskewSGD is comparable to existing approaches for training QNNs, such as the straight-through gradient estimator used in BinaryConnect, or other state of the art methods (ProxQuant, LUQ). We establish convergence guarantees for AskewSGD (under general assumptions for the objective function). Experimental results show that the AskewSGD algorithm performs better than or on par with state of the art methods in classical benchmarks.
• On the Strategyproofness of the Geometric Median
  
  • El-Mahdi El-Mhamdi
  • Sadegh Farhadkhani
  • Rachid Guerraoui
  • Lê-Nguyên Hoang
  , 2023. The geometric median, an instrumental component of the secure machine learning toolbox, is known to be effective when robustly aggregating models (or gradients), gathered from potentially malicious (or strategic) users. What is less known is the extent to which the geometric median incentivizes dishonest behaviors. This paper addresses this fundamental question by quantifying its strategyproofness. While we observe that the geometric median is not even approximately strategyproof, we prove that it is asymptotically α-strategyproof : when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor α, which we compute as a function of the distribution followed by the users. We then generalize our results to the case where users actually care more about specific dimensions, determining how this impacts α. We also show how the skewed geometric medians can be used to improve strategyproofness.
• Multi-Resolution Analysis for two-phase flows with heat and mass transfer
  
  • Wu Gen
  • Grenier Nicolas
  • Nore Caroline
  • Massot Marc
  , 2023. Two-phase flows combined with heat and mass transfer may present configurations in which dilatable gas pockets coexist along with nearly-incompressible liquid. Finite Volume discretisation of such compressible equations are known to introduce excessive numerical diffusion and to severely restrain time steps. Adequate solver has been developed to tackle these issues (Chalons, 2016), especially in the presence of a sharp interface (Zou, 2020, 2022). To handle multi scales inherently arising in complex two-phase flows, adaptive meshing is an efficient strategy to reduce computation cost compared to uniform grids. In the present study, we use the Multi-Resolution Analysis (MRA) (Harten, 1994), based on local wavelet basis decomposition. This method provide a measure of local error to select regions to refine or derefine and garantees a precise control of the additionnal error introduced by mesh operations, independantly of conservation equations and problems. This approach was successfully applied to two-phase flows with sharp interface representation (Han, 2014) and we extend it to low-Mach flows. We focus on configurations with heat and mass transfer, to exhibit accuracy and advantages of the MRA method, with respect to other adaptive methods (mainly Adaptive Mesh Refinement (AMR) (Berger, 1989)).
• Optimal transport for the multi-model combination of sub-seasonal ensemble forecasts
  
  • Le Coz Camille
  • Tantet Alexis
  • Flamary Rémi
  • Plougonven Riwal
  , 2023. Combining ensemble forecasts from several models has been shown to improve the skill of S2S predictions. One of the most used method for such combination is the &#8220;pooled ensemble&#8221; method, i.e. the concatenation of the ensemble members from the different models. The members of the new multi-model ensemble can simply have the same weights or be given different weights based on the skills of the models. If one sees the ensemble forecasts as discrete probability distributions, then the &#8220;pooled ensemble&#8221; is their (weighted-)barycenter with respect to the L2 distance.Here, we investigate whether a different metric when computing the barycenter may help improve the skill of S2S predictions. We consider in this work a second barycenter with respect to the Wasserstein distance. This distance is defined as the cost of the optimal transport between two distributions and has interesting properties in the distribution space, such as the possibility to preserve the temporal consistency of the ensemble members.We compare the L2 and Wasserstein barycenters for the combination of two models from the S2S database, namely ECMWF and NCEP. Their performances are evaluated for the weekly 2m-temperature over seven winters in Europe (land) in terms of different scores. The weights of the models in the barycenters are estimated from the data using grid search with cross-validation. We show that the estimation of these weights is critical as it greatly impacts the score of the barycenters. Although the NCEP ensemble generally has poorer skills than the ECMWF one, the barycenter ensembles are able to improve on both single-model ensembles (although not for all scores). At the end, the best ensemble depends on the score and on the location. These results constitute a promising first step before implementing this methodology with more than two ensembles, and ensembles having less contrasting skills. (10.5194/egusphere-egu23-13445)
  DOI : 10.5194/egusphere-egu23-13445
• CONCURRENT SHAPE OPTIMIZATION OF THE PART AND SCANNING PATH FOR POWDER BED FUSION ADDITIVE MANUFACTURING
  
  • Boissier Mathilde
  • Allaire Grégoire
  • Tournier Christophe
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2023, 61 (2), pp.697-722. (10.1137/21M1461976)
  DOI : 10.1137/21M1461976
• samurai
  
  • Bellotti Thomas
  • Gouarin Loic
  • Massot Marc
  • Matalon Pierre
  , 2023. The use of mesh adaptation methods in numerical simulation allows to drastically reduce the memory footprint and the computational costs. There are different kinds of methods: AMR patch-based, AMR cell-based, multiresolution cell-based or point-based, ... Different open source software is available to the community to manage mesh adaptation: AMReX for patch-based AMR, p4est and pablo for cell-based adaptation. The strength of samurai is that it allows to implement all the above mentioned mesh adaptation methods from the same data structure. The mesh is represented as intervals and a set algebra allows to efficiently search for subsets among these intervals. Samurai also offers a flexible and pleasant interface to easily implement numerical methods.
• Tight Regret and Complexity Bounds for Thompson Sampling via Langevin Monte Carlo
  
  • Huix Tom
  • Zhang Matthew Shunshi
  • Oliviero-Durmus Alain
  , 2023.
• Reconstruction of smooth shape defects in waveguides using locally resonant frequencies
  
  • Niclas Angèle
  • Seppecher Laurent
  Inverse Problems, IOP Publishing, 2023, 39 (5), pp.055006. This article aims to present a new method to reconstruct slowly varying width defects in 2D waveguides using locally resonant frequencies. At these frequencies, locally resonant modes propagate in the waveguide under the form of Airy functions depending on a parameter called the locally resonant point. In this particular point, the local width of the waveguide is known and its location can be recovered from boundary measurements of the wavefield. Using the same process for different frequencies, we produce a good approximation of the width in all the waveguide. Given multi-frequency measurements taken at the surface of the waveguide, we provide a L ∞-stable explicit method to reconstruct the width of the waveguide. We finally validate our method on numerical data, and we discuss its applications and limits. (10.1088/1361-6420/acc7c0)
  DOI : 10.1088/1361-6420/acc7c0
• Scaling Limit for Stochastic Control Problems in Population Dynamics
  
  • Jusselin Paul
  • Mastrolia Thibaut
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2023, 88 (1), pp.14. (10.1007/s00245-023-09989-x)
  DOI : 10.1007/s00245-023-09989-x
• Moment Methods for the 3D Radiative Transfer Equation Based on phi-Divergences
  
  • Abdelmalik M.R.A.
  • Cai Zhenning
  • Pichard Teddy
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2023, 417, pp.116454. The method of moments is widely used for the reduction of kinetic equations into fluid models. It consists in extracting the moments of the kinetic equation with respect to a velocity variable, but the resulting system is a priori underdetermined and requires a closure relation. In this paper, we adapt the phi-divergence based closure, recently developed for rarefied gases i.e. with a velocity variable describing R^d, to the radiative transfer equation where velocity describes the unit sphere. This closure is analyzed and a numerical method to compute it is provided. Eventually, it provides the main desirable properties to the resulting system of moments: Similarily to the entropy minimizing closure (M_N), it dissipates an entropy and captures exactly the equilibrium distribution. However, contrarily to M_N , it remains computationnally tractable even at high order and it relies on an exact quadrature formula which preserves exactly symmetry properties, i.e. it does not trigger ray effects. The purely anisotropic regimes (beams) are not captured exactly but they can be approached as close as desired and the closures remains again tractable in this limit. (10.1016/j.cma.2023.116454)
  DOI : 10.1016/j.cma.2023.116454
• On large $3/2$-stable maps
  
  • Kammerer Emmanuel
  , 2023. We discuss asymptotics of large Boltzmann random planar maps such that every vertex of degree $k$ has weight of order $k^{-2}$. Infinite maps of that kind were studied by Budd, Curien and Marzouk. These maps can be seen as the dual of the discrete $\alpha$-stable maps studied by Le Gall and Miermont for $\alpha=3/2$ or as the gaskets of critical $O(2)$-decorated random planar maps. We compute the asymptotics of the graph distance and of the first passage percolation distance between two uniform vertices, which are respectively equivalent in probability to $(\log \ell)^2/\pi^2$ and $2(\log \ell)/(\pi^2 p_{\bf q})$ when the perimeter of the map $\ell$ goes to $\infty$, where $p_{\bf q}$ is a constant which depends on the model. We also show that the diameter is of the same order as those distances for both metrics and obtain in particular that these maps do not satisfy scaling limits in the sense of Gromov-Prokhorov or Gromov-Hausdorff for lack of tightness. To study the peeling exploration of these maps, we prove local limit and scaling limit theorems for a class of random walks with heavy tails conditioned to remain positive until they die at $-\ell$ towards processes that we call stable L\'evy processes conditioned to stay positive until they jump and die at $-1$.
• Exploring differences in second order statistics for the simulation of multi-scale atomization process
  
  • Remigi Alberto
  • Tomov Petar
  • Massot Marc
  • Goudenège Ludovic
  • Demoulin F.X.
  • Duret Benjamin
  • Reveillon Julien
  , 2023.
• Simulation of polydisperse oscillating droplets through high order numerical methods for geometric moment equations
  
  • Ait-Ameur Katia
  • Loison Arthur
  • Pichard Teddy
  • Massot Marc
  , 2023. Liquid injection modeling and simulation face new challenges related to the need for predictive simulations in many fields such as combustion, chemical engineering, rocket booster propulsion and atmospheric studies. However, building up a global multi-scale model with the capability to resolve the whole injection process requires a major breakthrough in terms of both modeling and numerical methods. A new model for polydisperse sprays with coupling capabilities to the separated phases zone is proposed in (Essadki 2019, Loison 2023) and, in the present contribution, we design specific numerical methods. The key ingredient in (Loison 2023) is a good choice of variables, which can describe both the polydisperse character of a spray as well as the geometrical dynamics of non spherical droplets. The resulting system of equations is hyperbolic but has a more complex structure; realizability conditions are satisfied at the continuous level, which imply a precise framework for numerical methods. To achieve the goals of accuracy, robustness and realizability, the kinetic finite volume schemes (Bouchut 2003) and Discontinuous Galerkin methods are promising numerical approaches (Cockburn and Shu 1998). We focus here on a two-phase simulation of a polydisperse spray with oscillating droplets and assess the ability of the model and of the related numerical methods to capture the physics of such flows.