Publications

2020

• Approximate and exact controllability of linear difference equations
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2020, 7, pp.93--142. In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum_{j=1}^N A_j x(t - \Lambda_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $\Lambda_1, \dotsc, \Lambda_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays. (10.5802/jep.112)
  DOI : 10.5802/jep.112
• 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
• 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.
• 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
• A second order analysis of McKean-Vlasov semigroups
  
  • Arnaudon Marc
  • del Moral Pierre
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2020. We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities. (10.1214/20-AAP1568)
  DOI : 10.1214/20-AAP1568
• Model Reduction for Large-Scale Earthquake Simulation in an Uncertain 3D Medium
  
  • Sochala Pierre
  • de Martin Florent
  • Le Maitre Olivier
  International Journal for Uncertainty Quantification, Begell House Publishers, 2020, 10 (2), pp.101-127. In this paper, we are interested in the seismic wave propagation into an uncertain medium. To this end, we performed an ensemble of 400 large-scale simulations that requires 4 million core-hours of CPU time. In addition to the large computational load of these simulations, solving the uncertainty propagation problem requires dedicated procedures to handle the complexities inherent to large data set size and the low number of samples. We focus on the peak ground motion at the free surface of the 3D domain, and our analysis utilizes a surrogate model combining two key ingredients for complexity mitigation: i) a dimension reduction technique using empirical orthogonal basis functions and ii) a functional approximation of the uncertain reduced coordinates by polynomial chaos expansions. We carefully validate the resulting surrogate model by estimating its predictive error using bootstrap, truncation, and cross-validation procedures. The surrogate model allows us to compute various statistical information of the uncertain prediction, including marginal and joint probability distributions, interval probability maps, and 2D fields of global sensitivity indices. (10.1615/Int.J.UncertaintyQuantification.2020031165)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020031165
• Problem and Non-Problem Gamblers: A Cross-Sectional Clustering Study by Gambling Characteristics
  
  • Guillou-Landréat Morgane
  • Chereau Boudet Isabelle
  • Perrot Bastien
  • Romo Lucia
  • Codina Irene
  • Magalon David
  • Fatseas Melina
  • Luquiens Amandine
  • Brousse Georges
  • Challet-Bouju Gaëlle
  • Grall-Bronnec Marie
  BMJ Open, BMJ Publishing Group, 2020, 10 (2), pp.e030424. OBJECTIVES: Gambling characteristics are factors that could influence problem gambling development. The aim of this study was to identify a typology of gamblers to frame risky behaviour based on gambling characteristics (age of initiation/of problem gambling, type of gambling: pure chance/chance with pseudoskills/chance with elements of skill, gambling online/offline, amount wagered monthly) and to investigate clinical factors associated with these different profiles in a large representative sample of gamblers. DESIGN AND SETTING: The study is a cross-sectional analysis to the baseline data of the french JEU cohort study (study protocol : Challet-Bouju et al, 2014). Recruitment (April 2009 to September 2011) involved clinicians and researchers from seven institutions that offer care for or conduct research on problem gamblers (PG). Participants were recruited in gambling places, and in care centres. Only participants who reported gambling in the previous year between 18 and 65 years old were included.Participants gave their written informed consent, it was approved by the French Research Ethics Committee. PARTICIPANTS: The participants were 628 gamblers : 256 non-problem gamblers (NPG), 169 problem gamblers without treatment (PGWT) and 203 problem gamblers seeking treatment (PGST). RESULTS: Six clustering models were tested, the one with three clusters displayed a lower classification error rate (7.92%) and was better suited to clinical interpretation : 'Early Onset and Short Course' (47.5%), 'Early Onset and Long Course' (35%) and 'Late Onset and Short Course' (17.5%). Gambling characteristics differed significantly between the three clusters. CONCLUSIONS: We defined clusters through the analysis of gambling variables, easy to identify, by psychiatrists or by physicians in primary care. Simple screening concerning these gambling characteristics could be constructed to prevent and to help PG identification. It is important to consider gambling characteristics : policy measures targeting gambling characteristics may reduce the risk of PG or minimise harm from gambling. TRIAL REGISTRATION NUMBER: NCT01207674 (ClinicalTrials.gov); Results. (10.1136/bmjopen-2019-030424)
  DOI : 10.1136/bmjopen-2019-030424
• EXISTENCE AND ASYMPTOTIC RESULTS FOR AN INTRINSIC MODEL OF LINEARIZED INCOMPATIBLE ELASTICITY
  
  • Amstutz Samuel
  • van Goethem Nicolas
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2020, 25 (10). A general model of incompatible linearized elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Elastic strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids.The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.
• Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems
  
  • Augier Nicolas
  • Boscain Ugo
  • Sigalotti Mario
  Mathematical Control and Related Fields, AIMS, 2020, 10, pp.877-911. We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian. (10.3934/mcrf.2020023)
  DOI : 10.3934/mcrf.2020023
• 3D positive lattice walks and spherical triangles
  
  • Bogosel B
  • Perrollaz V
  • Raschel K.
  • Trotignon A
  Journal of Combinatorial Theory, Series A, Elsevier, 2020, 172, pp.105189. In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision. (10.1016/j.jcta.2019.105189)
  DOI : 10.1016/j.jcta.2019.105189
• The effect of the terminal penalty in receding horizon control for a class of stabilization problems
  
  • Kunisch Karl
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2020, 58. The Receding Horizon Control (RHC) strategy consists in replacing an infinite-horizon stabilization problem by a sequence of finite-horizon optimal control problems, which are numerically more tractable. The dynamic programming principle ensures that if the finite-horizon problems are formulated with the exact value function as a terminal penalty function, then the RHC method generates an optimal control. This article deals with the case where the terminal cost function is chosen as a cutoff Taylor approximation of the value function. The main result is an error rate estimate for the control generated by such a method, when compared with the optimal control. The obtained estimate is of the same order as the employed Taylor approximation and decreases at an exponential rate with respect to the prediction horizon. To illustrate the methodology, the article focuses on a class of bilinear optimal control problems in infinite-dimensional Hilbert spaces. (10.1051/cocv/2019037)
  DOI : 10.1051/cocv/2019037
• Creation and annihilation of point-potentials using Moutard-type transform in spectral variable
  
  • Grinevich Piotr
  • Novikov Roman
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2020, 61 (9), pp.093501(9 pp.). We continue to develop the method for creation and annihilation of contour singularities in the d-bar-spectral data for the two-dimensional Schrödinger equation at fixed energy. Our method is based on the Moutard-type transforms for generalized analytic functions. In this note we show that this approach successfully works for point potentials . (10.1063/1.5143303)
  DOI : 10.1063/1.5143303
• Topology optimization of connections in mechanical systems
  
  • Rakotondrainibe Lalaina
  • Allaire Grégoire
  • Orval Patrick
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020. One of the issues for the automotive industry is weight reduction. For this purpose, topology optimization is used for mechanical parts and usually involves a single part. Its connections to other parts are assumed to be fixed. This paper deals with a coupled topology optimization of both the structure of a part and its connections (location and number) to other parts. The present work focuses on two models of connections, namely rigid support and spring that prepares work for bolt connection. Rigid supports are modeled by Dirichlet boundary conditions while bolt-like connections are idealized and simplified as a non-local interaction to be representative enough at a low computational cost. On the other hand, the structure is modeled by the linearized elasticity system and its topology is represented by a level set function. A coupled optimization of the structure and the location of rigid supports is performed to minimize the volume of an engine accessories bracket under a compliance constraint. This coupled topology optimization (shape and connections) provides more satisfactory performance of a part than the one given by classical shape optimization alone. The approach presented in this work is therefore one step closer to the optimization of assembled mechanical systems. Thereafter, the concept of topological derivative is adapted to create an idealized bolt. The main idea is to add a small idealized bolt at the best location and to test the optimality of the solution with this new connection. The topological derivative is tested with a 3d academic test case for a problem of compliance minimization.
• Scaling limits of discrete snakes with stable branching
  
  • Marzouk Cyril
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2020, 56 (1), pp.502-523. We consider so-called discrete snakes obtained from size-conditioned critical Bienaym\'e-Galton-Watson trees by assigning to each node a random spatial position in such a way that the increments along each edge are i.i.d. When the offspring distribution belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$, we give a necessary and sufficient condition on the tail distribution of the spatial increments for this spatial tree to converge, in a functional sense, towards the Brownian snake driven by the $\alpha$-stable L\'evy tree. We also study the case of heavier tails, and apply our result to study the number of inversions of a uniformly random permutation indexed by the tree. (10.1214/19-AIHP970)
  DOI : 10.1214/19-AIHP970
• Spectral inequalities for nonnegative tensors and their tropical analogues
  
  • Friedland Shmuel
  • Gaubert Stéphane
  Vietnam Journal of Mathematics, Springer, 2020, 48 (4), pp.893-928. We extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker–Varadhan, Friedland–Karlin, Karlin–Ost inequalities, to nonnegative tensors. Our approach involves a correspondence between nonnegative tensors, ergodic control and entropy maximization: we show in particular that the logarithm of the spectral radius of a tensor is given by en entropy maximization problem over a space of occupation measures. We study in particular the tropical analogue of the spectral radius, that we characterize as a limit of the classical spectral radius, and we give an explicit combinatorial formula for this tropical spectral radius. (10.1007/s10013-020-00432-0)
  DOI : 10.1007/s10013-020-00432-0
• 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method
  
  • Geoffroy-Donders Perle
  • Allaire Grégoire
  • Pantz Olivier
  Journal of Computational Physics, Elsevier, 2020, 401, pp.108994. This paper is motivated by the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Generalizing our previous work in 2-d we propose a method for topology optimization of structures made of periodically perforated material , where the microscopic periodic cell can be macroscopically modulated and oriented. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized mi-crostructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. Compared to the 2-d case where rotations are parametrized by a single angle, to which a confor-mality constraint can be applied, the 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the square periodic lattice is computed component by component. Several numerical examples are presented for compliance minimization in 3-d.
• Markovian explorations of random planar maps are roundish
  
  • Curien Nicolas
  • Marzouk Cyril
  Bulletin de la société mathématique de France, Société Mathématique de France, 2020, 148 (4), pp.709-732. The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual. (10.24033/bsmf.2821)
  DOI : 10.24033/bsmf.2821
• Second-order analysis for the time crisis problem
  
  • Bayen Térence
  • Pfeiffer Laurent
  Journal of Convex Analysis, Heldermann, 2020, 27 (1), pp.139-163. In this article, we prove second-order necessary optimality conditions for the so-called time crisis problem that comes up within the context of viability theory. It consists in minimizing the time spent by solutions of a controlled dynamics outside a given subset K of the state space. One essential feature is the discontinuity of the characteristic function involved in the cost functional. Thanks to a change of time and an augmentation of the dynamics, we relate the time crisis problem to an auxiliary Mayer control problem. This allows us to use the classical tools of optimal control for obtaining optimality conditions. Going back to the original problem, we deduce that way second order optimality conditions for the time crisis problem.
• Order antimorphisms of finite-dimensional cones
  
  • Walsh Cormac
  Selecta Mathematica (New Series), Springer Verlag, 2020, 26 (4), pp.paper number 53. We show that an order antimorphism on a finite-dimensional cone having no one-dimensional factors is homogeneous of degree −1. A consequence is that the existence of an order antimorphism on a finite-dimensional cone implies that the cone is a symmetric cone.
• Validation strategy of reduced-order two-fluid flow models based on a hierarchy of direct numerical simulations
  
  • Cordesse Pierre
  • Remigi Alberto
  • Duret Benjamin
  • Murrone Angelo
  • Ménard Thibaut
  • Demoulin François-Xavier
  • Massot Marc
  Flow, Turbulence and Combustion, Springer Verlag, 2020, 105 (4), pp.1381-1411. Whereas direct numerical simulation (DNS) have reached a high level of description in the field of atomization processes, they are not yet able to cope with industrial needs since they lack resolution and are too costly. Predictive simulations relying on reduced order modeling have become mandatory for applications ranging from cryotechnic to aeronautic combustion chamber liquid injection. Two-fluid models provide a good basis in order to conduct such simulations, even if recent advances allow to refine subscale modeling using geometrical variables in order to reach a unified model including separate phases and disperse phase descriptions based on high order moment methods. The simulation of such models has to rely on dedicated numerical methods and still lacks assessment of its predictive capabilities. The present paper constitutes a building block of the investigation of a hierarchy of test-cases designed to be amenable to DNS while close enough to industrial configurations, for which we propose a comparison of two-fluid compressible simulations with DNS data-bases. We focus in the present contribution on an air-assisted water atomization using a planar liquid sheet injector. Qualitative and quantitative comparisons with incompressible DNS allow us to identify and analyze strength and weaknesses of the reduced-order modeling and numerical approach in this specific configuration and set a framework for more refined models since they already provide a very interesting level of comparison on averaged quantities. (10.1007/s10494-020-00154-w)
  DOI : 10.1007/s10494-020-00154-w
• Maximization of the Steklov Eigenvalues with a Diameter Constraint
  
  • Al Sayed Abdelkader
  • Bogosel Beniamin
  • Henrot Antoine
  • Nacry Florent
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2020, 53 (1), pp.710-729. In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first non-trivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm. (10.1137/20M1335042)
  DOI : 10.1137/20M1335042
• High to Low pellet cladding gap heat transfer modeling methodology in an uncertainty quantification framework for a PWR Rod Ejection Accident with best estimate coupling
  
  • Delipei Gregory Kyriakos
  • Garnier Josselin
  • Le Pallec Jean-Charles
  • Normand Benoit
  EPJ N - Nuclear Sciences & Technologies, EDP Sciences, 2020, 6, pp.56. High to Low modeling approaches can alleviate the computationally expensive fuel modeling in nuclear reactor’s transient uncertainty quantification. This is especially the case for Rod Ejection Accident (REA) in Pressurized Water Reactors (PWR) were strong multi-physics interactions occur. In this work, we develop and propose a pellet cladding gap heat transfer (Hgap) High to Low modeling methodology for a PWR REA in an uncertainty quantification framework. The methodology involves the calibration of asimplified $Hgap$ model based on high fidelity simulations with the fuel-thermomechanics code ALCYONE1.The calibrated model is then introduced into the CEA developed CORPUS Best Estimate (BE) multi-physicscoupling between APOLLO3R© and FLICA4. This creates an Improved Best Estimate (IBE) coupling that is then used for an uncertainty quantification study. The results indicate that with IBE the distance to boiling crisis uncertainty is decreased from 57% to 42%. This is reflected to the decrease of the sensitivity of $Hgap$. In the BE coupling $Hgap$ was responsible for 50% of the output variance while in IBE it is close to 0. These results show the potential gain of High to Low approachez for $Hgap$ modeling in REA uncertainty analyses. (10.1051/epjn/2020018)
  DOI : 10.1051/epjn/2020018
• Optimality conditions in variational form for non-linear constrained stochastic control problems
  
  • Pfeiffer Laurent
  Mathematical Control and Related Fields, AIMS, 2020, 10 (3), pp.493-526. Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the probability distribution of the state variable at the final time. The analysis uses in an essential manner a convexity property of the set of reachable probability distributions. An augmented Lagrangian method based on the obtained optimality conditions is proposed and analyzed for solving iteratively the problem. At each iteration of the method, a standard stochastic optimal control problem is solved by dynamic programming. Two academical examples are investigated. (10.3934/mcrf.2020008)
  DOI : 10.3934/mcrf.2020008
• Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function
  
  • Jornet Marc
  • Calatayud Julia
  • Le Maitre Olivier
  • Cortés Juan Carlos
  Journal of Computational and Applied Mathematics, Elsevier, 2020, 374, pp.112770. This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, which is the focus of the paper, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a crude Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper. After that, the efficiency of the algorithm is improved by employing the control variates method. Numerical examples corroborate the variance reduction of the Monte Carlo approach. (10.1016/j.cam.2020.112770)
  DOI : 10.1016/j.cam.2020.112770
• SPIX: a new software package to reveal chemical reactions at trace amounts in very complex mixtures from high-resolution mass spectra data sets
  
  • Nicol Edith
  • Xu Yao
  • Varga Zsuzsanna
  • Kinani Said
  • Bouchonnet Stéphane
  • Lavielle Marc
  Rapid Communications in Mass Spectrometry, Wiley, 2020. Rationale: High-resolution mass spectrometry-based non-targeted screening has a huge potential for applications in environmental sciences, engineering and regulation. However, it produces big data for which full appropriate processing is a real challenge; the development of processing software is the last building-block to enable large-scale use of this approach. Methods: A new software application, SPIX, has been developed to extract relevant information from high-resolution mass-spectrum datasets. Dealing with intrinsic sample variability and reducing operator subjectivity, it opens up opportunities and promising prospects in many areas of analytical chemistry. SPIX is freely available at: http://spix.webpopix.org. Results: Two features of the software are presented in the field of environmental analysis. An example illustrates how SPIX reveals photodegradation reactions in wastewater by fitting kinetic models to significant changes in ion abundance over time. A second example shows the ability of SPIX to detect photoproducts at trace amounts in river water, through comparison of datasets from samples taken before and after irradiation. Conclusions: SPIX has shown its ability to reveal relevant modifications between two series of big data sets, allowing for instance to study the consequences of a given event on a complex substrate. Most of alland this is to our knowledge the only software currently available allowing thatit can reveal and monitor any kind of reaction in all types of mixture.