Publications

2019

• Local structure of multi-dimensional martingale optimal transport
  
  • de March Hadrien
  , 2019. This paper analyzes the support of the conditional distribution of optimal martingale transport plans in higher dimension. In the context of a distance coupling in dimension larger than 2, previous results established by Ghoussoub, Kim & Lim show that this conditional transport is concentrated on its own Choquet boundary. Moreover, when the target measure is atomic, they prove that the support is concentrated on d+1 points, and conjecture that this result is valid for arbitrary target measure. We provide a structure result of the support of the conditional optimal transport for general Lipschitz couplings. Using tools from algebraic geometry, we provide sufficient conditions for finiteness of this conditional support, together with (optimal) lower bounds on the maximal cardinality for a given coupling function. More results are obtained for specific examples of coupling functions based on distance functions. In particular, we show that the above conjecture of Ghoussoub, Kim & Lim is not valid beyond the context of atomic target distributions.
• BUILDING ARBITRAGE-FREE IMPLIED VOLATILITY: SINKHORN'S ALGORITHM AND VARIANTS
  
  • de March Hadrien
  • Henry-Labordere Pierre
  , 2019. We consider the classical problem of building an arbitrage-free implied volatility surface from bid-ask quotes. We design a fast numerical procedure, for which we prove the convergence, based on the Sinkhorn algorithm that has been recently used to solve efficiently (martingale) optimal transport problems.
• A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators
  
  • Prandi Dario
  • Rizzi Luca
  • Seri Marcello
  Journal of Differential Geometry, International Press, 2019, 111 (2), pp.339-379. In this paper we prove a sub-Riemannian version of the classical Santal\'o formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g.\ CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a ``reduction procedure'' that allows to consider only a simple subset of sub-Riemann\-ian geodesics. As an application, we derive isoperimetric-type and ($p$-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $\lambda_1(M)$ of the sub-Laplacian, \begin{equation} \lambda_1(M) \geq \frac{k \pi^2}{L^2}, \end{equation} in terms of the rank $k$ of the distribution and the length $L$ of the longest reduced sub-Riemannian geodesic contained in $M$. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations: \begin{equation} \mathbb{S}^1\hookrightarrow \mathbb{S}^{2d+1} \xrightarrow{p} \mathbb{CP}^d, \qquad \mathbb{S}^3\hookrightarrow \mathbb{S}^{4d+3} \xrightarrow{p} \mathbb{HP}^d, \qquad d \geq 1, \end{equation} where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = \pi$ and $k=2d$ or $4d$, respectively. (10.4310/jdg/1549422105)
  DOI : 10.4310/jdg/1549422105
• Simulations of ultrasonic wave propagation in concrete based on a two-dimensional numerical model validated analytically and experimentally
  
  • Yu Ting
  • Chaix Jean François
  • Audibert Lorenzo
  • Komatitsch Dimitri
  • Garnier Vincent
  • Henault Jean-Marie
  Ultrasonics, Elsevier, 2019, 92, pp.21-34. Several non-destructive evaluation techniques to characterize concrete structures are based on ultrasonic wave propagation. The interpretation of the results is often limited by the scattering phenomena between the ultrasonic wave and the high concentration aggregates contained in the cement matrix. Numerical simulations allow for further insights. This study aims to build a two-dimensional numerical model in order to reproduce and interpret ultrasonic wave propagations in concrete. The model is built in a spectral-element software package called SPECFEM2D. The validation of the numerical tool is based on the use of resin samples containing different amount of aluminum rods from low (5%) to high concentration (40%), the last one being representative of aggregate concentration in concrete. These samples are characterized using an ultrasonic testing bench (ultrasonic water tank) from 150 kHz to 370 kHz. The measured results are analyzed in terms of phase velocity and attenuation which are the main parameters of coherent waves. As homogenization models such as the Waterman-Truell or Conoir-Norris models are usually used to model coherent waves in two-phase systems, we also compare the experimental and numerical results against them. We confirm that the use of these homogenization models is limited to low concentration of scattering phase, which is not adapted to applications to concrete. Finally, we use our numerical tool to carry out a parametric study on scatterer concentration, shape, orientation and size distribution of aggregates in concrete. We show that aggregate orientation has an influence on coherent wave parameters, but aggregate shape has not. (10.1016/j.ultras.2018.07.018)
  DOI : 10.1016/j.ultras.2018.07.018
• An iterative approach to monochromatic phaseless inverse scattering
  
  • Agaltsov Alexey D.
  • Hohage Thorsten
  • Novikov Roman G.
  Inverse Problems, IOP Publishing, 2019, 35 (2), pp.024001. This paper is concerned with the inverse problem to recover a compactly supported Schrödinger potential given the differential scattering cross section, i.e. the modulus, but not the phase of the scattering amplitude. To compensate for the missing phase information we assume additional measurements of the differential cross section in the presence of known background objects. We propose an iterative scheme for the numerical solution of this problem and prove that it converges globally of arbitrarily high order depending on the smoothness of the unknown potential as the energy tends to infinity. At fixed energy, however, the proposed iteration does not converge to the true solution even for exact data. Nevertheless, numerical experiments show that it yields remarkably accurate approximations with small computational effort even for moderate energies. At small noise levels it may be worth to improve these approximations by a few steps of a locally convergent iterative regularization method, and we demonstrate to which extent this reduces the reconstruction error. (10.1088/1361-6420/aaf097)
  DOI : 10.1088/1361-6420/aaf097
• Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations
  
  • Akian Marianne
  • Fodjo Eric
  , 2019, 29, pp.183–209. We consider fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. In previous works (Akian, Fodjo, 2016 and 2017), we introduced a lower complexity probabilistic numerical algorithm for such equations by combining max-plus and numerical probabilistic approaches. The max-plus approach is in the spirit of the one of McEneaney, Kaise and Han (2011), and is based on the distributivity of monotone operators with respect to suprema. The numerical probabilistic approach is in the spirit of the one proposed by Fahim, Touzi and Warin (2011). A difficulty of the latter algorithm was in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we present new probabilistic schemes which are monotone under rather weak assumptions, and show error estimates for these schemes. These estimates will be used in further works to study the probabilistic max-plus method.
• Impact of demography on extinction/fixation events
  
  • Coron Camille
  • Méléard Sylvie
  • Villemonais Denis
  Journal of Mathematical Biology, Springer, 2019, 78 (3), pp.549-577. In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform. (10.1007/s00285-018-1283-1)
  DOI : 10.1007/s00285-018-1283-1
• Diffusion MRI simulation in thin-layer and thin-tube media using a discretization on manifolds
  
  • Nguyen Van-Dang
  • Jansson Johan
  • Tran Hoang Trong An
  • Hoffman Johan
  • Li Jing-Rebecca
  Journal of Magnetic Resonance, Elsevier, 2019, 299, pp.176-187. The Bloch-Torrey partial differential equation can be used to describe the evolution of the transverse magnetization of the imaged sample under the influence of diffusion-encoding magnetic field gradients inside the MRI scanner. The integral of the magnetization inside a voxel gives the simulated diffusion MRI signal. This paper proposes a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The variable thickness of the three-dimensional domains is included in the weak formulation established on the manifolds. We conducted a numerical study of the proposed approach by simulating the diffusion MRI signals from the extracellular space (a thin layer medium) and from neurons (a thin tube medium), comparing the results with the reference signals obtained using a standard three-dimensional finite element discretization. We show good agreements between the simulated signals using our proposed method and the reference signals for a wide range of diffusion MRI parameters. The approximation becomes better as the diffusion time increases. The method helps to significantly reduce the required simulation time, computational memory, and difficulties associated with mesh generation, thus opening the possibilities to simulating complicated structures at low cost for a better understanding of diffusion MRI in the brain. (10.1016/j.jmr.2019.01.002)
  DOI : 10.1016/j.jmr.2019.01.002
• Spectral theory for the weak decay of muons in a uniform magnetic field.
  
  • Guillot Jean-Claude
  , 2019. In this article we consider a mathematical model for the weak decay of muons in a uniform magnetic field according to the Fermi theory of weak interactions with V-A coupling. With this model we associate a Hamil-tonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is self-adjoint and has a unique ground state. We specify the essential spectrum and prove the existence of asymptotic fields from which we determine the absolutely continuous spectrum. The coupling constant is supposed sufficiently small.
• A variational approach to nonlinear and interacting diffusions
  
  • Arnaudon Marc
  • del Moral Pierre
  , 2019. The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including non homogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter are also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.
• Clock Monte Carlo methods
  
  • Michel Manon
  • Tan Xiaojun
  • Deng Youjin
  Physical Review E, American Physical Society (APS), 2019. We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of ${\rm O}(N^\kappa)$ ($0 \! \leq \! \kappa \! \leq \! 1$) can be achieved compared to the Metropolis method, where $N$ is the system size and the $\kappa$ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O$(n)$ spin models in a wide parameter regime: for $n \! = \! 1,2,3$, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from $d \! = \! 1, 2, 3$ to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms. (10.1103/PhysRevE.99.010105)
  DOI : 10.1103/PhysRevE.99.010105
• Volatility uncertainty quantification in a stochastic control problem applied to energy
  
  • Bernal Francisco
  • Gobet Emmanuel
  • Printems Jacques
  Methodology and Computing in Applied Probability, Springer Verlag, 2019, 22 (1), pp.135-159. This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solution to a system of non-linear PDEs of the same kind. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We experiment the methodology in the context of swing contract (energy contract with flexibility in purchasing energy power), this allows to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model. (10.1007/s11009-019-09692-x)
  DOI : 10.1007/s11009-019-09692-x
• Some remarks on mean field games
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  Communications in Partial Differential Equations, Taylor & Francis, 2019, 44 (3), pp.205-227. In this article, we study three aspects of mean field games (MFG). The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of “noise” in discrete space models and the formulation of the Master Equation in this case. Finally, we show how MFG reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes. (10.1080/03605302.2018.1542438)
  DOI : 10.1080/03605302.2018.1542438
• A Kinetic Model of Adsorption on Crystal Surfaces
  
  • Aoki Kazuo
  • Giovangigli Vincent
  , 2019. A kinetic theory model describing physisorption and chemisorption of gas particles on a crystal surface is introduced. A single kinetic equation is used to model gas and physisorbed particles interacting with a crystal potential and colliding with phonons. The phonons are assumed to be at equilibrium and the physisorbate/gas equation is coupled to similar kinetic equations describing chemisorbed particles and crystal atoms on the surface. A kinetic entropy is introduced for the coupled system and the H theorem is established. Using the Chapman-Enskog method with a fluid scaling, the asymptotic structure of the adsorbate is investigated and fluid boundary conditions are derived from the kinetic model.
• On a sharp Poincaré-type inequality on the 2-sphere and its application in micromagnetics
  
  • Di Fratta Giovanni
  • Slastikov Valeriy
  • Zarnescu Arghir
  , 2019. The main aim of this note is to prove a sharp Poincaré-type inequality for vector-valued functions on the 2-sphere, that naturally emerges in the context of micromagnetics of spherical thin films.
• Optimization of an ORC supersonic nozzle under epistemic uncertainties due to turbulence models
  
  • Razaaly Nassim
  • Gori Giulio
  • Iaccarino Gianluca
  • Congedo Pietro Marco
  , 2019. Organic Rankine Cycle (ORC) turbines usually operate in thermodynamic regions characterized by high-pressure ratios and strong non-ideal gas effects in the flow expansion, complicating their aerodynamic design significantly. This study presents the shape optimization of a typical 2D ORC turbine cascade (Biere), under epistemic uncertainties due to turbulence models (RANS). A design vector of size eleven controls the blade geometry parametrized with B-splines. The EQUiPS module integrated into the SU2 CFD suite, incorporating perturbations to the eigenvalues and the eigenvectors of the modeled Reynolds stress tensor, is used to evaluate the interval estimates on the predictions of integrated Quantity of Interest (QoI), performing only five specific RANS simulations. For a given blade profile, the QoIs total loss pressure and mass flow rate, are assumed to be independent uniform random variables, defined by those estimates. A global surrogate-based method allowing to propose different designs at each optimization step is used to solve the constrained mono-objective optimization problem. To illustrate the suitability of the method, several statistics of the total pressure are considered for the minimization, under the constraint that the mean of the mass flow rate to be within a range.
• Surrogate-assisted Bounding-Box approach for optimization problems with tunable objectives fidelity
  
  • Rivier Mickael
  • Congedo Pietro Marco
  Journal of Global Optimization, Springer Verlag, 2019. In this work, we present a novel framework to perform multi-objective optimization when considering expensive objective functions computed with tunable fidelity. This case is typical in many engineering optimization problems, for example with simulators relying on Monte Carlo or on iterative solvers. The objectives can only be estimated, with an accuracy depending on the computational resources allocated by the user. We propose here a heuristic for allocating the resources efficiently to recover an accurate Pareto front at low computational cost. The approach is independent from the choice of the optimizer and overall very flexible for the user. The framework is based on the concept of Bounding-Box, where the estimation error can be regarded with the abstraction of an interval (in one-dimensional problems) or a product of intervals (in multi-dimensional problems) around the estimated value, naturally allowing the computation of an approximated Pareto front. This approach is then supplemented by the construction of a surrogate model on the estimated objective values. We first study the convergence of the approximated Pareto front toward the true continuous one under some hypotheses. Secondly, a numerical algorithm is proposed and tested on several numerical test-cases. (10.1007/s10898-019-00823-9)
  DOI : 10.1007/s10898-019-00823-9
• Space debris reentry prediction and ground risk estimation using a probabilistic breakup model
  
  • Sanson Francois
  • Bertorello Charles
  • Bouilly Jean-Marc
  • Congedo Pietro Marco
  , 2019. While the number of artificial space object re-enter the Earth atmosphere daily, predicting the reentry of a space debris remains an open problem. The reentry prediction is a multi-physics problem involving aerodynamics computations in rarefied and continuum flow, heat transfer calculations and structiral breakup predictions. Additionally, numerous uncertainties coming from unknown initial flight conditions, material properties or uncalibrated model parameters affect oru ability to make acurate predictions. In this wrk, we propose an original reentry prediction framework that associates deterministic physical solvers with a stochastic breakup model and uncertainty quantification tools to make robust reentry predictions and a statistical estimate of the impact location. Our method is able to predic breakup distributions and ground impact locations efficiently using simplified but robust models at reasonable computational cost. This framework is used to predict the reenty of Upper Stage deorbited from a GTO orbit. (10.2514/6.2019-2234)
  DOI : 10.2514/6.2019-2234
• Directed topological complexity
  
  • Goubault Eric
  • Sagnier Aurélien
  • Färber Michael
  , 2019. It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence.
• Wave Propagation and Imaging in Moving Random Media
  
  • Borcea Liliana
  • Garnier Josselin
  • Solna Knut
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2019, 17 (1), pp.31-67. (10.1137/18M119505X)
  DOI : 10.1137/18M119505X
• Breakup prediction under uncertainty: Application to upper stage controlled reentries from GTO orbit
  
  • Sanson Francois
  • Bouilly Jean-Marc
  • Bertorello Charles
  • Congedo Pietro Marco
  Aerospace Science and Technology, Elsevier, 2019. More and more human-made space objects re-enter the atmosphere, and yet the risk for human population remains often unknown because predicting their reentry trajectories is formidably complex. While falling back on Earth, the space object absorbs large amounts of thermal energy that affects its structural integrity.It undergoes strong aerodynamic forces that lead to one or several breakups. Breakup events have a critical influence on the rest of the trajectory are extremely challenging to predict and subject to uncertainties. In this work,we present an original model for robustly predicting the breakup of a reentering space object. This model is composed of a set of individual solvers that are coupled together such as each solver resolves a specific aspect of this multiphysics problem. This paper deals with two levels of uncertainties. The first level is the stochastic modelling of the breakup while the second level is the statistical characterization of the model input uncertainties. The framework provides robust estimates of the quantities of interest and quantitative sensitivity analysis. The objective is twofold: first to compute a robust estimate of the breakup distribution and secondly to identify the main uncertainties in the quantities of interest.Due to the significant computational cost, we use an efficient framework particularly suited to multiple solver predictions for the uncertainty quantification analysis. Then, we illustrate the breakup model for the controlled reentry of an upper stage deorbited from a Geo Transfer Orbit (GTO), which is a classical Ariane mission. (10.1016/j.ast.2019.02.031)
  DOI : 10.1016/j.ast.2019.02.031
• Ablative thermal protection system under uncertainties including pyrolysis gas composition
  
  • Rivier Mickael
  • Lachaud Jean
  • Congedo Pietro Marco
  Aerospace Science and Technology, Elsevier, 2019. Spacecrafts such as Stardust (NASA, 2006) are protected by an ablative Thermal Protection System (TPS) for their hypersonic atmospheric entry. A new generation of TPS material, called Phenolic Impregnated Carbon Ablator (PICA), has been introduced with the Stardust mission. This new generation of low density carbon-phenolic composites is now widely used in the aerospace industry. Complex heat and mass transfer phenomena coupled to phenolic pyrolysis and pyrolysis gas chemistry occur in the material during atmospheric entry. Computer programs, as the Porous material Analysis Toolbox based on OpenFoam (PATO) released open source by NASA, allow to study the material response. In this study, a non-intrusive Anchored Analysis of Variance (Anchored-ANOVA) method has been interfaced with PATO to perform low-cost sensitivity analysis on this problem featuring a large number of uncertain parameters. Then, a Polynomial-Chaos method has been employed in order to compute the statistics of some quantities of interest for the atmospheric entry of the Stardust capsule, by taking into account uncertainties on effective material properties and pyrolysis gas composition. This first study including pyrolysis gas composition uncertainties shows their key contribution to the variability of the quantities of interest. (10.1016/j.ast.2018.11.048)
  DOI : 10.1016/j.ast.2018.11.048
• Spreading and vanishing for a monostable reaction-diffusion equation with forced speed
  
  • Bouhours Juliette
  • Giletti Thomas
  Journal of Dynamics and Differential Equations, Springer Verlag, 2019, 35, pp.92 - 117. Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain where the reaction term is positive is shifting/contracting at a given speed c. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term, in dimension N ≥ 1. We investigate the behaviour of the solution u depending on the value of the advection constant c, which typically stands for the velocity of climate change. We find that, when the initial datum is compactly supported, there exists precisely three ranges for c leading to drastically different situations. In the lower speed range the solution always spreads, while in the upper range it always vanishes. More surprisingly, we find that that both spreading and vanishing may occur in an intermediate speed range. The threshold between those two outcomes is always sharp, both with respect to c and to the initial condition. We also briefly consider the case of an exponentially decreasing initial condition, where we relate the decreasing rate of the initial condition with the range of values of c such that spreading occurs. (10.1007/s10884-018-9643-5)
  DOI : 10.1007/s10884-018-9643-5
• Approximating the Volume of Tropical Polytopes is Difficult
  
  • Gaubert Stéphane
  • Maccaig Marie
  International Journal of Algebra and Computation, World Scientific Publishing, 2019, 29 (02), pp.357--389. We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $\alpha=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank. For tropical polytopes described by inequalities we prove that counting the number of integer points and calculating the volume are $\#$P-hard. (10.1142/S0218196718500686)
  DOI : 10.1142/S0218196718500686
• Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations
  
  • Court Sébastien
  • Kunisch Karl
  • Pfeiffer Laurent
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2019, 21 (3), pp.273-311. A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters. (10.4171/IFB/424)
  DOI : 10.4171/IFB/424