Publications

2022

• Optimizing Noisy Complex Systems Liable to Failure
  
  • Lunz Davin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2022, 82 (1), pp.25-48. Inspired by complex systems in social and industrial contexts, we consider a family of coupled diffusion processes modeling system components, and an associated system objective. Each process is inherently noisy, driven by a controllable drift, and fails upon reaching a critical state. Interdependence is captured via the global objective and the governing dynamics (correlated noise, cascading failures). Analytical and numerical calculations reveal that the optimal strategies to steer such systems so as to maximise the objective are highly coupled, depending strongly on the state of the entire system. Strikingly, they exhibit a rich set of bifurcations, describing qualitatively different strategies throughout the parameter space. (10.1137/21M1416126)
  DOI : 10.1137/21M1416126
• On scaling limits of random trees and maps with a prescribed degree sequence
  
  • Marzouk Cyril
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2022, 5, pp.317-386. (10.5802/ahl.125)
  DOI : 10.5802/ahl.125
• Multiply Accelerated Value Iteration for Non-Symmetric Affine Fixed Point Problems and application to Markov Decision Processes
  
  • Akian Marianne
  • Gaubert Stéphane
  • Qu Zheng
  • Saadi Omar
  SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2022, 43 (1). We analyze a modified version of Nesterov accelerated gradient algorithm, which applies to affine fixed point problems with non self-adjoint matrices, such as the ones appearing in the theory of Markov decision processes with discounted or mean payoff criteria. We characterize the spectra of matrices for which this algorithm does converge with an accelerated asymptotic rate. We also introduce a $d$th-order algorithm, and show that it yields a multiply accelerated rate under more demanding conditions on the spectrum. We subsequently apply these methods to develop accelerated schemes for non-linear fixed point problems arising from Markov decision processes. This is illustrated by numerical experiments. (10.1137/20M1367192)
  DOI : 10.1137/20M1367192
• Enriched nonconforming multiscale finite element method for Stokes flows in heterogeneous media based on high-order weighting functions
  
  • Feng Qingqing
  • Allaire Grégoire
  • Omnes Pascal
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2022, 20 (1). This paper addresses an enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling \& Simulation 2015 13:4, 1146-1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389 [math.NA], 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials. (10.1137/21M141926X)
  DOI : 10.1137/21M141926X
• DAMPING OPTIMIZATION OF VISCOELASTIC CANTILEVER BEAMS AND PLATES UNDER FREE VIBRATION
  
  • Joubert A
  • Allaire G
  • Amstutz S
  • Diani J
  Computers & Structures, Elsevier, 2022. The goal of this work is to significantly enhance the damping of linear viscoelastic structures under free vibration by relying on optimal design. Homogeneous cantilever slender beams and plates satisfying, respectively, the Euler-Bernoulli and Kirchhoff-Love assumptions are considered. A sizing optimization of the beam or plate thickness is proposed, as well as a coupled optimization of the thickness and geometry of the plate applying Hadamard's boundary variation method. The isotropic linear viscoelastic material is modeled by a classical generalized Maxwell model, well suited for polymers. Gradients of the objective functions are computed by an adjoint approach. Optimization is performed by a projected gradient algorithm and the mechanical models are evaluated by the finite element method. Numerical tests indicate that the optimal designs, as well as their damping properties, strongly depend on the material parameters.
• Linear-sized independent sets in random cographs and increasing subsequences in separable permutations
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Drmota Michael
  • Feray Valentin
  • Gerin Lucas
  • Maazoun Mickaël
  • Pierrot Adeline
  Combinatorial Theory, eScholarship, 2022, 2 (3), pp.23340676. This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions. (10.5070/C62359179)
  DOI : 10.5070/C62359179
• Generating natural adversarial Remote Sensing Images
  
  • Burnel Jean-Christophe
  • Fatras Kilian
  • Flamary Rémi
  • Courty Nicolas
  IEEE Transactions on Geoscience and Remote Sensing, Institute of Electrical and Electronics Engineers, 2022, 60, pp.1-14. Over the last years, Remote Sensing Images (RSI) analysis have started resorting to using deep neural networks to solve most of the commonly faced problems, such as detection, land cover classification or segmentation. As far as critical decision making can be based upon the results of RSI analysis, it is important to clearly identify and understand potential security threats occurring in those machine learning algorithms. Notably, it has recently been found that neural networks are particularly sensitive to carefully designed attacks, generally crafted given the full knowledge of the considered deep network. In this paper, we consider the more realistic but challenging case where one wants to generate such attacks in the case of a black-box neural network. In this case, only the prediction score of the network is accessible, given a specific input. Examples that lure away the network's prediction, while being perceptually similar to real images, are called natural or unrestricted adversarial examples. We present an original method to generate such examples, based on a variant of the Wasserstein Generative Adversarial Network. We demonstrate its effectiveness on natural adversarial hyper-spectral image generation and image modification for fooling a state-of-the-art detector. Among others, we also conduct a perceptual evaluation with human annotators to better assess the effectiveness of the proposed method. (10.1109/TGRS.2021.3110601)
  DOI : 10.1109/TGRS.2021.3110601
• TIME DEPENDENT SCANNING PATH OPTIMIZATION FOR THE POWDER BED FUSION ADDITIVE MANUFACTURING PROCESS
  
  • Boissier Mathilde
  • Allaire G
  • Tournier Christophe
  Computer-Aided Design, Elsevier, 2022, 142, pp.103122. In this paper, scanning paths optimization for the powder bed fusion additive manufacturing process is investigated. The path design is a key factor of the manufacturing time and for the control of residual stresses arising during the building, since it directly impacts the temperature distribution. In the literature, the scanning paths proposed are mainly based on existing patterns, the relevance of which is not related to the part to build. In this work, we propose an optimization algorithm to determine the scanning path without a priori restrictions. Taking into account the time dependence of the source, the manufacturing time is minimized under two constraints: melting the required structure and avoiding any overheating causing thermally induced residual stresses. The results illustrate how crucial the part's shape and topology is in the path quality and point out promising leads to define path and part design constraints. (10.1016/j.cad.2021.103122)
  DOI : 10.1016/j.cad.2021.103122
• A consistent approximation of the total perimeter functional for topology optimization algorithms
  
  • Amstutz Samuel
  • Dapogny Charles
  • Ferrer Alex
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28, pp.18:1-71. This article revolves around the total perimeter functional, one particular version of the perimeter of a shape Ω contained in a fixed computational domain D measuring the total area of its boundary ∂Ω, as opposed to its relative perimeter, which only takes into account the regions of ∂Ω strictly inside D. We construct and analyze approximate versions of the total perimeter which make sense for general “density functions” u, as generalized characteristic functions of shapes. Their use in the context of density-based topology optimization is particularly convenient insofar as they do not involve the gradient of the optimized function u. Two different constructions are proposed: while the first one involves the convolution of the function u with a smooth mollifier, the second one is based on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The “consistency” of these approximations with the original notion of total perimeter is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to 0, when the considered density function u is the characteristic function of a “regular enough” shape Ω ⊂ D. Then, we focus on the Γ-convergence of the second type of approximate total perimeter functional, that based on elliptic regularization. Several numerical examples are eventually presented in two and three space dimensions to validate our theoretical findings and demonstrate the efficiency of the proposed functionals in the context of structural optimization. (10.1051/cocv/2022005)
  DOI : 10.1051/cocv/2022005
• Provably Convergent Working Set Algorithm for Non-Convex Regularized Regression
  
  • Rakotomamonjy Alain
  • Flamary Rémi
  • Gasso Gilles
  • Salmon Joseph
  , 2022. Owing to their statistical properties, non-convex sparse regularizers have attracted much interest for estimating a sparse linear model from high dimensional data. Given that the solution is sparse, for accelerating convergence, a working set strategy addresses the optimization problem through an iterative algorithm by incre-menting the number of variables to optimize until the identification of the solution support. While those methods have been well-studied and theoretically supported for convex regularizers, this paper proposes a working set algorithm for non-convex sparse regularizers with convergence guarantees. The algorithm, named FireWorks, is based on a non-convex reformulation of a recent primal-dual approach and leverages on the geometry of the residuals. Our theoretical guarantees derive from a lower bound of the objective function decrease between two inner solver iterations and shows the convergence to a stationary point of the full problem. More importantly, we also show that convergence is preserved even when the inner solver is inexact, under sufficient decay of the error across iterations. Our experimental results demonstrate high computational gain when using our working set strategy compared to the full problem solver for both block-coordinate descent or a proximal gradient solver.
• Spectral inequality for Schrödinger's equation with multipoint potential
  
  • Grinevich Piotr
  • Novikov Roman
  Russian Mathematical Surveys, Turpion, 2022, 77 (6), pp.1021–1028. Schrödinger's equation with potential that is a sum of a regular function and a finite set of point scatterers of Bethe–Peierls type is under consideration. For this equation the spectral problem with homogeneous linear boundary conditions is considered, which covers the Dirichlet, Neumann, and Robin cases. It is shown that when the energy E is an eigenvalue with multiplicity m, it remains an eigenvalue with multiplicity at least m−n after adding n<m point scatterers. As a consequence, because for the zero potential all values of the energy are transmission eigenvalues with infinite multiplicity, this property also holds for n-point potentials, as discovered originally in a recent paper by the authors. (10.4213/rm10080e)
  DOI : 10.4213/rm10080e
• Time reversal of Markov processes with jumps under a finite entropy condition
  
  • Conforti Giovanni
  • Léonard Christian
  Stochastic Processes and their Applications, Elsevier, 2022, 144, pp.85-124. Motivated by entropic optimal transport, time reversal of Markov jump processes in Rn is investigated. Relying on an abstract integration by parts formula for the carré du champ of a Markov process recently obtained by Cattiaux, Gentil and the auhors, and using an entropic improvement strategy discovered by Föllmer in the eighties, we compute the semimartingale characteristics of the time reversed process for a wide class of jump processes with possibly unbounded variation sample paths and singular intensities of jump.
• A class of short-term models for the oil industry addressing speculative storage
  
  • Achdou Yves
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  • Rostand Antoine
  • Scheinkman Jose
  Finance and Stochastics, Springer Verlag (Germany), 2022, 26 (3), pp.631-669. This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon. (10.1007/s00780-022-00481-y)
  DOI : 10.1007/s00780-022-00481-y
• Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise
  
  • Bauzet Caroline
  • Nabet Flore
  • Schmitz Kerstin
  • Zimmermann Aleksandra
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022, 57 (2), pp.745-783. We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem. (10.1051/m2an/2022087)
  DOI : 10.1051/m2an/2022087
• Gaussian Agency problems with memory and Linear Contracts
  
  • Abi Jaber Eduardo
  • Villeneuve Stéphane
  Finance and Stochastics, Springer Verlag (Germany), 2022. Can a principal still offer optimal dynamic contracts that are linear in end-of-period outcomes when the agent controls a process that exhibits memory? We provide a positive answer by considering a general Gaussian setting where the output dynamics are not necessarily semi-martingales or Markov processes. We introduce a rich class of principal-agent models that encompasses dynamic agency models with memory. From the mathematical point of view, we develop a methodology to deal with the possible non-Markovianity and non-semimartingality of the control problem, which can no longer be directly solved by means of the usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that, for one-dimensional models, this setting always allows for optimal linear contracts in end-of-period observable outcomes with a deterministic optimal level of effort. In higher dimension, we show that linear contracts are still optimal when the effort cost function is radial and we quantify the gap between linear contracts and optimal contracts for more general quadratic costs of efforts.
• An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms
  
  • Akimoto Youhei
  • Auger Anne
  • Hansen Nikolaus
  Stochastic Processes and their Applications, Elsevier, 2022, 145, pp.269-307. We consider stochastic algorithms derived from methods for solving deterministic optimization problems, especially comparison-based algorithms derived from stochastic approximation algorithms with a constant step-size. We develop a methodology for proving geometric convergence of the parameter sequence {θn}n≥0 of such algorithms. We employ the ordinary differential equation (ODE) method, which relates a stochastic algorithm to its mean ODE, along with a Lyapunov-like function Ψ such that the geometric convergence of Ψ(θn) implies -- in the case of an optimization algorithm -- the geometric convergence of the expected distance between the optimum and the search point generated by the algorithm. We provide two sufficient conditions for Ψ(θn) to decrease at a geometric rate: Ψ should decrease "exponentially" along the solution to the mean ODE, and the deviation between the stochastic algorithm and the ODE solution (measured by Ψ) should be bounded by Ψ(θn) times a constant. We also provide practical conditions under which the two sufficient conditions may be verified easily without knowing the solution of the mean ODE. Our results are any-time bounds on Ψ(θn), so we can deduce not only the asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to a comparison-based stochastic algorithm with a constant step-size for optimization on continuous domains. (10.1016/j.spa.2021.12.005)
  DOI : 10.1016/j.spa.2021.12.005
• Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions
  
  • Isaev Mikhail
  • Novikov Roman
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 163 (July), pp.318-333. We give new formulas for finding a compactly supported function v on R^d, d≥1, from its Fourier transform Fv given within the ball B_r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given. (10.1016/j.matpur.2022.05.008)
  DOI : 10.1016/j.matpur.2022.05.008
• Differentially Private Federated Learning on Heterogeneous Data
  
  • Noble Maxence
  • Bellet Aurélien
  • Dieuleveut Aymeric
  , 2022. Federated Learning (FL) is a paradigm for large-scale distributed learning which faces two key challenges: (i) training efficiently from highly heterogeneous user data, and (ii) protecting the privacy of participating users. In this work, we propose a novel FL approach (DP-SCAFFOLD) to tackle these two challenges together by incorporating Differential Privacy (DP) constraints into the popular SCAFFOLD algorithm. We focus on the challenging setting where users communicate with a "honest-but-curious" server without any trusted intermediary, which requires to ensure privacy not only towards a third party observing the final model but also towards the server itself. Using advanced results from DP theory and optimization, we establish the convergence of our algorithm for convex and non-convex objectives. Our paper clearly highlights the trade-off between utility and privacy and demonstrates the superiority of DP-SCAFFOLD over the state-ofthe-art algorithm DP-FedAvg when the number of local updates and the level of heterogeneity grows. Our numerical results confirm our analysis and show that DP-SCAFFOLD provides significant gains in practice.
• Scintillation of partially coherent light in time-varying complex media
  
  • Garnier Josselin
  • Sølna Knut
  Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2022, 39 (8), pp.1309. We present a theory for wave scintillation in the situation of a time-dependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index, and we evaluate this in various scaling regimes. The scintillation index is a fundamental quantity that is used to analyze and optimize imaging and communication schemes. Our results are useful to quantify the scintillation index under realistic propagation scenarios and to address such optimization challenges. (10.1364/JOSAA.453358)
  DOI : 10.1364/JOSAA.453358
• A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2022, 15 (5), pp.793. A closure relation for moments equation in kinetic theory was recently introduced in [38], based on the study of the geometry of the set of moments. This relation was constructed from a projection of a moment vector toward the boundary of the set of moments and corresponds to approximating the underlying kinetic distribution as a sum of a chosen equilibrium distribution plus a sum of purely anisotropic Dirac distributions. The present work generalizes this construction for kinetic equations involving unbounded velocities, i.e. to the Hamburger problem, and provides a deeper analysis of the resulting moment system. Especially, we provide representation results for moment vectors along the boundary of the moment set that implies the well-definition of the model. And the resulting moment model is shown to be weakly hyperbolic with peculiar properties of hyperbolicity and entropy of two subsystems, corresponding respectively to the equilibrium and to the purely anisotropic parts of the underlying kinetic distribution. (10.3934/krm.2022014)
  DOI : 10.3934/krm.2022014
• Entropic turnpike estimates for the kinetic Schrödinger problem
  
  • Chiarini Alberto
  • Conforti Giovanni
  • Greco Giacomo
  • Ren Zhenjie
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2022, 27 (none). (10.1214/22-EJP850)
  DOI : 10.1214/22-EJP850
• Coupled topology optimization of structure and connections for bolted mechanical systems
  
  • Rakotondrainibe Lalaina
  • Desai Jeet
  • Orval Patrick
  • Allaire Grégoire
  European Journal of Mechanics - A/Solids, Elsevier, 2022. This work introduces a new coupled topology optimization approach for a structural assembly. Considering several parts connected by bolts, the shape and topology of potentially each part, as well as the position and number of bolts are simultaneously optimized. The main ingredients of our optimization approach are the level-set method for structural optimization, a new notion of topological derivative of an idealized model of bolt in order to decide where it is advantageous to add a new bolt, coupled with a parametric gradient-based algorithm for its position optimization. Both idealized bolt and its topological derivative handle prestressed state complexity. Several 3d numerical test cases are performed to demonstrate the efficiency of the proposed strategy for mass minimization, considering Von Mises and fatigue constraints for the bolts and compliance constraint for the structure. In particular, a simplified but industrially representative example of an accessories bracket for car engines demonstrates significant benefits. Optimizing both the structure and its connections reduces the mass by 24% compared to classical "structure-only" optimization.
• Multidimensional inverse scattering for the Schrödinger equation
  
  • Novikov Roman
  , 2022, 385, pp.75-98. We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation. A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice. In this connection our considerations include reconstructions from non-overdetermined monochromatic scattering data and formulas for phase recovering from phaseless scattering data. Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles. This paper is based, in particular, on results going back to M. Born (1926), L. Faddeev (1956, 1974), S. Manakov (1981), R.Beals, R. Coifman (1985), G. Henkin, R. Novikov (1987), and on more recent results of R. Novikov ( 1998 - 2019), A. Agaltsov, T. Hohage, R. Novikov (2019). This paper is an extended version of the talk given at the 12th ISAAC Congress, Aveiro, Portugal, 29 July - 2 August, 2019. (10.1007/978-3-030-97127-4_3)
  DOI : 10.1007/978-3-030-97127-4_3
• QLSD: Quantised Langevin Stochastic Dynamics for Bayesian Federated Learning
  
  • Vono Maxime
  • Plassier Vincent
  • Durmus Alain
  • Dieuleveut Aymeric
  • Moulines Eric
  , 2022. The objective of Federated Learning (FL) is to perform statistical inference for data which are decentralised and stored locally on networked clients. FL raises many constraints which include privacy and data ownership, communication overhead, statistical heterogeneity, and partial client participation. In this paper, we address these problems in the framework of the Bayesian paradigm. To this end, we propose a novel federated Markov Chain Monte Carlo algorithm, referred to as Quantised Langevin Stochastic Dynamics which may be seen as an extension to the FL setting of Stochastic Gradient Langevin Dynamics, which handles the communication bottleneck using gradient compression. To improve performance, we then introduce variance reduction techniques, which lead to two improved versions coined QLSD and QLSD ++. We give both non-asymptotic and asymptotic convergence guarantees for the proposed algorithms. We illustrate their performances using various Bayesian Federated Learning benchmarks.
• Leveraging Local Variation in Data: Sampling and Weighting Schemes for Supervised Deep Learning
  
  • Novello Paul
  • Poëtte Gaël
  • Lugato David
  • Congedo Pietro Marco
  Journal of Machine Learning for Modeling and Computing, Begell House, 2022, 3 (1). In the context of supervised learning of a function by a neural network, we claim and empirically verify that the neural network yields better results when the distribution of the data set focuses on regions where the function to learn is steep. We first traduce this assumption in a mathematically workable way using Taylor expansion and emphasize a new training distribution based on the derivatives of the function to learn. Then, theoretical derivations allow construction of a methodology that we call variance based samples weighting (VBSW). VBSW uses labels' local variance to weight the training points. This methodology is general, scalable, cost-effective, and significantly increases the performances of a large class of neural networks for various classification and regression tasks on image, text, and multivariate data. We highlight its benefits with experiments involving neural networks from linear models to ResNet and BERT. (10.1615/JMachLearnModelComput.2022041819)
  DOI : 10.1615/JMachLearnModelComput.2022041819