Publications

2020

• On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems
  
  • Breiten Tobias
  • Pfeiffer Laurent
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (2), pp.26. Optimal control problems with a very large time horizon can be tackled with the Receding Horizon Control (RHC) method, which consists in solving a sequence of optimal control problems with small prediction horizon. The main result of this article is the proof of the exponential convergence (with respect to the prediction horizon) of the control generated by the RHC method towards the exact solution of the problem. The result is established for a class of infinite-dimensional linear-quadratic optimal control problems with time-independent dynamics and integral cost. Such problems satisfy the turnpike property: the optimal trajectory remains most of the time very close to the solution to the associated static optimization problem. Specific terminal cost functions, derived from the Lagrange multiplier associated with the static optimization problem, are employed in the implementation of the RHC method. (10.1137/18M1225811)
  DOI : 10.1137/18M1225811
• Accuracy assessment of the Non-Ideal Computational Fluid Dynamics model for siloxane MDM from the open-source SU2 suite
  
  • Gori Giulio
  • Zocca Marta
  • Cammi Giorgia
  • Spinelli Andrea
  • Congedo Pietro Marco
  • Guardone Alberto
  European Journal of Mechanics - B/Fluids, Elsevier, 2020, 79, pp.109-120. The first-ever accuracy assessment of a computational model for Non-Ideal Compressible-Fluid Dynamics (NICFD) flows is presented. The assessment relies on a comparison between numerical predictions, from the open-source suite SU2, and pressure and Mach number measurements of compressible fluid flows in the non-ideal regime. Namely, measurements regard supersonic flows of siloxane MDM (Octamethyltrisiloxane, C 8 H 24 O 2 Si 3) vapor expanding along isentropes in the close proximity of the liquid-vapor saturation curve. The model accuracy assessment takes advantage of an Uncertainty Quantification (UQ) analysis, to compute the variability of the numerical solution with respect the uncertainties affecting the test-rig operating conditions. This allows for an uncertainty-based assessment of the accuracy of numerical predictions. The test set is representative of typical operating conditions of Organic Rankine Cycle systems and it includes compressible flows expanding through a converging-diverging nozzle in mildly-to-highly non-ideal conditions. All the considered flows are well represented by the computational model. Therefore, the reliability of the numerical implementation and the predictiveness of the NICFD model are confirmed. (10.1016/j.euromechflu.2019.08.014)
  DOI : 10.1016/j.euromechflu.2019.08.014
• On Invariance and Linear Convergence of Evolution Strategies with Augmented Lagrangian Constraint Handling
  
  • Atamna Asma
  • Auger Anne
  • Hansen Nikolaus
  Theoretical Computer Science, Elsevier, 2020, 832, pp.68-97. In the context of numerical constrained optimization, we investigate stochastic algorithms, in particular evolution strategies, handling constraints via augmented Lagrangian approaches. In those approaches, the original constrained problem is turned into an unconstrained one and the function optimized is an augmented Lagrangian whose parameters are adapted during the optimization. The use of an augmented Lagrangian however breaks a central invariance property of evolution strategies, namely invariance to strictly increasing transformations of the objective function. We formalize nevertheless that an evolution strategy with augmented Lagrangian constraint handling should preserve invariance to strictly increasing affine transformations of the objective function and the scaling of the constraints—a subclass of strictly increasing transformations. We show that this invariance property is important for the linear convergence of these algorithms and show how both properties are connected. (10.1016/j.tcs.2018.10.006)
  DOI : 10.1016/j.tcs.2018.10.006
• Parametric inference for diffusions observed at stopping times
  
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2020, 14 (1). In this paper we study the problem of parametric inference for multidimensional diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject (such as [Doh87, GJ93, Gob01, AM04] among others) consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids (studied in [GL14, GS18]). Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp. (10.1214/20-EJS1708)
  DOI : 10.1214/20-EJS1708
• The boundary of random planar maps via looptrees
  
  • Kortchemski Igor
  • Richier Loïc
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2020, 29 (2), pp.391-430. (10.5802/afst.1636)
  DOI : 10.5802/afst.1636
• A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors
  
  • Akian Marianne
  • Gaubert Stéphane
  • Hochart Antoine
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2020, 40, pp.207--231. We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $\alpha, \beta > 0$, the "slice space" $\mathcal{S}_\alpha^\beta := \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid \alpha x \leq f(x) \leq \beta x \}$ is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations $g$ of $f$, all the orbits of $g$ are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of $f$ near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors. (10.3934/dcds.2020009)
  DOI : 10.3934/dcds.2020009
• Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models
  
  • Bonnet Celine
  • Chahour Keltoum
  • Clément Frédérique
  • Postel Marie
  • Yvinec Romain
  ESAIM: Proceedings and Surveys, EDP Sciences, 2020, 67, pp.72-99. In this study, we describe different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles. We put a special focus on representing the populationlevel feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We take advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We first study the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we provide detailed numerical evidence of convergence to the limit behavior. We reproduce the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging. (10.1051/proc/202067006)
  DOI : 10.1051/proc/202067006
• Variance reduction for Markov chains with application to MCMC
  
  • Belomestny D
  • Iosipoi L
  • Moulines E
  • Naumov A
  • Samsonov S
  Statistics and Computing, Springer Verlag (Germany), 2020. In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.
• Tropical planar networks
  
  • Gaubert Stéphane
  • Niv Adi
  Linear Algebra and its Applications, Elsevier, 2020, 595, pp.123-144. We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices. (10.1016/j.laa.2020.02.019)
  DOI : 10.1016/j.laa.2020.02.019
• The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities
  
  • Backhoff Julio
  • Conforti Giovanni
  • Gentil Ivan
  • Léonard Christian
  Probability Theory and Related Fields, Springer Verlag, 2020, 178, pp.475-530. (10.1007/s00440-020-00977-8)
  DOI : 10.1007/s00440-020-00977-8
• Optimal Hedging Under Fast-Varying Stochastic Volatility
  
  • Garnier Josselin
  • Sølna Knut
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2020, 11 (1), pp.274-325. In a market with a rough or Markovian mean-reverting stochastic volatility thereis no perfect hedge. Here it is shown how various delta-type hedging strategies perform and canbe evaluated in such markets in the case of European options.A precise characterization of thehedging cost, the replication cost caused by the volatilityfluctuations, is presented in an asymptoticregime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset basedhedging strategy in the considered regime is identified as the so-called “practitioners” delta hedgingscheme. It is moreover shown that the performances of the delta-type hedging schemes are essentiallyindependent of the regularity of the volatility paths in theconsidered regime and that the hedgingcosts are related to a Vega risk martingale whose magnitude is proportional to a new market riskparameter. It is also shown via numerical simulations that the proposed hedging schemes whichderive from option price approximations in the regime of rapid mean reversion, are robust: the“practitioners” delta hedging scheme that is identified as being optimal by our asymptotic analysiswhen the mean reversion time is small seems to be optimal witharbitrary mean reversion times. (10.1137/18M1221655)
  DOI : 10.1137/18M1221655
• Regression Monte Carlo methods for HJB-type equations: which approximation space?
  
  • Barrera David
  • Gobet Emmanuel
  • Lopez-Salas Jose
  • Turkedjiev Plamen
  • Vasquez Carlos
  • Zubelli Jorge
  , 2020.
• Kinetic derivation of diffuse-interface fluid models
  
  • Giovangigli Vincent
  Physical Review E, American Physical Society (APS), 2020, 102. We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, and Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a second-order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential. (10.1103/physreve.102.012110)
  DOI : 10.1103/physreve.102.012110
• A quantitative McDiarmid’s inequality for geometrically ergodic Markov chains
  
  • Havet Antoine
  • Lerasle Matthieu
  • Moulines Éric
  • Vernet Elodie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2020, 25. (10.1214/20-ECP286)
  DOI : 10.1214/20-ECP286
• Commentaires sur le rapport de surveillance de culture du MON 810 en 2018. Paris, le 25 février 2020
  
  • Du Haut Conseil Des Biotechnologies Comité Scientifique
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2020, pp.35 p.. Les analyses contenues dans le rapport de surveillance de Bayer Agriculture BVBA ne font apparaître aucun problème majeur associé à la culture de maïs MON 810 en 2018. Toutefois, le CS du HCB identifie encore certaines faiblesses et limites méthodologiques concernant la surveillance de la sensibilité des ravageurs ciblés à la toxine Cry1Ab, remettant en question les conclusions du rapport. Le HCB estime notamment que l’utilisation d’une dose diagnostic présente certaines limites pour la détection précoce de l’évolution de la résistance, tant dans son principe intrinsèque que dans sa mise en oeuvre par Bayer, et recommande une méthode alternative de type F2 screen permettant de déterminer la fréquence des allèles de résistance au sein d’une population de ravageurs cibles. Par ailleurs, le HCB formule des recommandations destinées à renforcer la mise en oeuvre des zones refuges pour prévenir ou retarder le développement de résistance à la toxine Cry1Ab chez les ravageurs ciblés. Concernant la surveillance générale, le CS du HCB relève un problème de pertinence méthodologique quant aux questions étudiées, avec des règles de décision arbitraires, des conclusions incorrectement justifiées et un possible biais associé au format d’enquête auprès du panel d’agriculteurs qui ont accepté de répondre au questionnaire. Enfin, le CS du HCB recommande que le rapport de surveillance considère la présence de téosinte dans des zones de culture du maïs MON 810 en Espagne et les risques potentiels associés à une éventuelle introgression de gènes de maïs MON 810 chez le téosinte.
• Intensity fluctuations in random waveguides
  
  • Garnier Josselin
  Communications in Mathematical Sciences, International Press, 2020, 18 (4), pp.947-971. (10.4310/CMS.2020.v18.n4.a3)
  DOI : 10.4310/CMS.2020.v18.n4.a3
• Wave Propagation in Randomly Perturbed Weakly Coupled Waveguides
  
  • Borcea Liliana
  • Garnier Josselin
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2020, 18 (1), pp.44-78. We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation and evanescent modes with amplitudes that are random fields, due to scattering. We obtain a detailed characterization of these amplitudes and thus quantify the transfer of power between the two waveguides in terms of their separation distance. The results show that, no matter how small the fluctuations of the interfacesare, they have significant effect at sufficiently large distance of propagation, which manifests in two ways: The first effect is well known and consists of power leakage from the guided modes to the radiation ones. The second effect consists of blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power. Its quantification is the main practical result ofthe paper. (10.1137/18M1230591)
  DOI : 10.1137/18M1230591
• High-Resolution Interferometric Synthetic Aperture Imaging in Scattering Media
  
  • Borcea Liliana
  • Garnier Josselin
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2020, 13 (1), pp.291-316. The goal of synthetic aperture imaging is to estimate the reflectivity of a remoteregion of interest by processing data gathered with a moving sensor which emits periodically a signaland records the backscattered wave. We introduce and analyze a high-resolution interferometric method for synthetic aperture imaging through an unknown scattering medium which distorts thewave. The method builds on the coherent interferometric (CINT) approach which uses empiricalcross-correlations of the measurements to mitigate the distortion, at the expense of a loss of resolutionof the image. The new method shows that, while mitigating the wave distortion, it is possible toobtain a robust and sharp estimate of the modulus of the Fourier transform of the reflectivity function.A high-resolution image can then be obtained by a phase retrieval algorithm. (10.1137/19M1272470)
  DOI : 10.1137/19M1272470
• The tropicalization of the entropic barrier
  
  • Allamigeon Xavier
  • Aznag Abdellah
  • Gaubert Stéphane
  • Hamdi Yassine
  , 2020. The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. (SIAM J. Applied Alg. Geom., 2018). One consequence is that the number of linear pieces in the tropical entropic central path can be exponential in the dimension and the number of inequalities defining the linear program.
• Medical innovations to maintain the function in patients with chronic PJI for whom explantation is not desirable: a pathophysiology-, multidisciplinary-, and experience-based approach
  
  • Ferry Tristan
  • Batailler Cécile
  • Brosset Sophie
  • Kolenda Camille
  • Goutelle Sylvain
  • Sappey-Marinier Elliot
  • Josse Jérôme
  • Laurent Frédéric
  • Lustig Sébastien
  SICOT-J, EDP Open, 2020, 6, pp.26. Introduction: PJI is the most dramatic complication after joint arthroplasty. In patients with chronic infection, prosthesis exchange is in theory the rule. However, this surgical approach is sometimes not desirable especially in elderly patients with multiple comorbidities, as it could be associated with a dramatic loss of function, reduction of the bone stock, fracture, or peroperative death. We propose here to report different approaches that can help to maintain the function in such patients based on a pathophysiology-, multidisciplinary-, and an experience-based approach. Methods: We describe the different points that are needed to treat such patients: (i) the multidisciplinary care management; (ii) understanding the mechanism of bacterial persistence; (iii) optimization of the conservative surgical approach; (iv) use of suppressive antimicrobial therapy (SAT); (v) implementation of innovative agents that could be used locally to target the biofilm. Results: In France, a nation-wide network called CRIOAc has been created and funded by the French Health ministry to manage complex bone and joint infection. Based on the understanding of the complex pathophysiology of PJI, it seems to be feasible to propose conservative surgical treatment such as “debridement antibiotics and implant retention” (with or without soft-tissue coverage) followed by SAT to control the disease progression. Finally, there is a rational for the use of particular agents that have the ability to target the bacteria embedded in biofilm such as bacteriophages and phage lysins. Discussion: This multistep approach is probably a key determinant to propose innovative management in patients with complex PJI, to improve the outcome. Conclusion: Conservative treatment has a high potential in patients with chronic PJI for whom explantation is not desirable. The next step will be to evaluate such practices in nation-wide clinical trials. (10.1051/sicotj/2020021)
  DOI : 10.1051/sicotj/2020021
• How a moving passive observer can perceive its environment ? The Unruh effect revisited
  
  • Fink Mathias
  • Garnier Josselin
  Wave Motion, Elsevier, 2020, 93, pp.102462. (10.1016/j.wavemoti.2019.102462)
  DOI : 10.1016/j.wavemoti.2019.102462
• Multipoint formulas for scattered far field in multidimensions
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2020, 36 (9), pp.095001(12 pp.). We give asymptotic formulas for finding the scattering amplitude at fixed frequency and angles (scattered far field) from the scattering wave function given at $n$ points in dimension $d\geq 2$. These formulas are explicit and their precision is proportional to $n$. To our knowledge these formulas are new already for $n\geq 2$. (10.1088/1361-6420/aba891)
  DOI : 10.1088/1361-6420/aba891
• Support optimization in additive manufacturing for geometric and thermo-mechanical constraints
  
  • Allaire Grégoire
  • Bihr Martin
  • Bogosel Beniamin
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020, 61, pp.2377-2399. Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.
• Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2020, 8 (3), pp.1061–1089. The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $f^*$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\theta$. We aim at deriving the function $f^*$, as well as the probabilistic distribution of $f^*(\theta)$ given a probability distribution $\pi$ for $\theta$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\theta \mapsto f^*(\theta)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{f^*_K}(\cdot)$ of $f^*$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{f^*_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{f^*_K}(\theta)$ when $\theta$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically. (10.1137/18M1178517)
  DOI : 10.1137/18M1178517
• A MOMENT CLOSURE BASED ON A PROJECTION ON THE BOUNDARY OF THE REALIZABILITY DOMAIN: 1D CASE
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2020, 13 (6), pp.1243-1280. This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function. Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry. (10.3934/xx.xx.xx.xx)
  DOI : 10.3934/xx.xx.xx.xx