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Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).

In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann–Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.

International audience; We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback.

There are two major routes to address the ubiquitous family of inverse problems appearing in signal and image processing, such as denoising or deblurring. A first route relies on Bayesian modeling, where prior probabilities are used to embody models of both the distribution of the unknown variables and their statistical dependence with the observed data. The estimation process typically relies on the minimization of an expected loss (e.g. minimum mean squared error, or MMSE). The second route has received much attention in the context of sparse regularization and compressive sensing: it consists in designing (often convex) optimization problems involving the sum of a data fidelity term and a penalty term promoting certain types of unknowns (e.g., sparsity, promoted through an 1 norm). Well known relations between these two approaches have lead to some widely spread misconceptions. In particular, while the so-called Maximum A Posterori (MAP) estimate with a Gaussian noise model does lead to an optimization problem with a quadratic data-fidelity term, we disprove through explicit examples the common belief that the converse would be true. It has already been shown [7, 9] that for denoising in the presence of additive Gaussian noise, for any prior probability on the unknowns, MMSE estimation can be expressed as a penalized least squares problem, with the apparent characteristics of a MAP estimation problem with Gaussian noise and a (generally) different prior on the unknowns. In other words, the variational approach is rich enough to build all possible MMSE estimators associated to additive Gaussian noise via a well chosen penalty. We generalize these results beyond Gaussian denoising and characterize noise models for which the same phenomenon occurs. In particular, we prove that with (a variant of) Poisson noise and any prior probability on the unknowns, MMSE estimation can again be expressed as the solution of a penalized least squares optimization problem. For additive scalar denois-ing the phenomenon holds if and only if the noise distribution is log-concave. In particular, Laplacian denoising can (perhaps surprisingly) be expressed as the solution of a penalized least squares problem. In the multivariate case, the same phenomenon occurs when the noise model belongs to a particular subset of the exponential family. For multivariate additive denoising, the phenomenon holds if and only if the noise is white and Gaussian., Comment: This work and the companion paper [10] are dedicated to the memory of Mila Nikolova, who passed away prematurely in June 2018. Mila dedicated much of her energy to bring the technical content to completion during the spring of 2018. The first author did his best to finalize the papers as Mila would have wished. He should be held responsible for any possible imperfection in the final manuscript

International audience; This paper is devoted to the derivation and analysis of accurate and efficient Perfectly Matched Layers (PML) or efficient absorbing layers for solving fractional Laplacian equations within Initial Boundary Value Problems (IBVP). Two main approaches are derived: we first propose a Fourier-based pseu-dospectral method, and then present a real space method based on an efficient computation of the fractional Laplacian with PML. Some numerical experiments and analytical results are proposed along the paper to illustrate the presented methods.

In a previous paper Chevallier et al. (2018), it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u ( t , x ) of a neural field equation (NFE). The value u ( t , x ) represents the membrane potential at time t of a typical neuron located in position x , embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u ( t , x ) . Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u ( t , x ) . To the best of our knowledge, this result appears to be new in the literature.

International audience; This paper deals with the finite element approximation of the Darcy-Brinkman-Forchheimer equation, involving a porous media with spatially-varying porosity, with mixed boundary condition such as inhomogeneous Dirichlet and traction boundary conditions. We first prove that the considered problem has a unique solution if the source terms are small enough. The convergence of a Taylor-Hood finite element approximation using a finite element interpolation of the porosity is then proved under similar smallness assumptions. Some optimal error estimates are next obtained when assuming the solution to the Darcy-Brinkman-Forchheimer model are smooth enough. We end this paper by providing a fixed-point method to solve iteratively the discrete non-linear problems and with some numerical experiments to make more precise the smallness assumptions on the source terms and to illustrate the theoretical convergence results.

In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where the Euler equations have been solved solved using a semi-implicit finite volume scheme based on the flux-vector splitting method recently proposed by Toro and Vazquez-Cendon (Comput Fluids 70:1–12, 2012). In our scheme, the nonlinear convective terms are discretized explicitly, while the pressure terms are discretized implicitly. As a consequence, the time step is restricted only by a mild CFL condition based on the fluid velocity, which makes this method particularly suitable for simulations in the low Mach number regime. However, the conservative formulation of the scheme, together with the novel subcell finite volume limiter allows also the numerical simulation of high Mach number flows with shock waves. Inserting the discrete momentum equation into the discrete total energy conservation law yields a mildly nonlinear system with the scalar pressure as the only unknown. Due to the use of staggered meshes, the resulting pressure system has the most compact stencil possible and can be efficiently solved with modern iterative methods. In order to deal with shock waves or steep gradients, the new semi-implicit DG scheme proposed in this paper includes an a posteriori subcell finite volume limiting technique. This strategy was first proposed by Dumbser et al. (J Comput Phys 278:47–75, 2014) for explicit DG schemes on collocated grids and is based on the a posteriori MOOD algorithm of Clain, Loubere and Diot. Recently, this methodology was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in Ioriatti and Dumbser (Appl Numer Math 135:443–480, 2019). Within the MOOD approach, an unlimited DG scheme first produces a so-called candidate solution for the next time level $$t^{n+1}$$. Later on, the control volumes with a non-admissible candidate solution are identified by using physical and numerical detection criteria, such as the positivity of the solution, the absence of floating point errors and the satisfaction of a relaxed discrete maximum principle (DMP). Then, in the detected troubled cells a more robust first order semi-implicit finite volume (FV) method is applied on a sub-grid composed of $$2P + 1$$ subcells, where P denotes the polynomial degree used in the DG scheme. For that purpose, the nonlinear convective terms are recomputed in the troubled cells using an explicit finite volume scheme on the subcell level. Also the linear system for the pressure needs to be assembled and solved again, but where now a low order semi-implicit finite volume scheme is used on the sub-cell level in all troubled DG elements, instead of the original high order DG method. Finally, the higher order DG polynomials are reconstructed from the piecewise constant subcell finite volume averages and the scheme proceeds to the next time step. In this paper we present, discuss and test this novel family of methods and simulate a set of classical numerical benchmark problems of compressible gasdynamics. Great attention is dedicated to 1D and 2D Riemann problems and we also show that for these test cases the scheme responds appropriately in the presence of shock waves and does not produce non-physical spurious numerical oscillations.

International audience; In this paper, we consider the linear Boltzmann equation subject to uncertainties in the initial conditions and matter parameters (cross-sections/opacities). In order to solve the underlying uncertain systems, we rely on moment theory and the construction of hierarchical moment models in the framework of parametric polynomial approximations. Such model is commonly called a generalised Polynomial Chaos (gPC) reduced model. In this paper, we prove the spectral convergence of the hierarchy of reduced model parametered by P (polynomial order) obtained from the uncertain linear Boltzmann equation.

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_n)$ which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from smooth Wasserstein distances to total variation in a quantitative way. This is a well studied topic and one can consult for instance Bally and Caramellino [Electron. J. Probab. 2014], Bogachev, Kosov and Zelenov [Trans. Amer. Math. Soc. 2018], Hu, Lu and Nualart [J. Funct. Anal. 2014], Nourdin and Poly [Stoch. Proc. Appl. 2013] and the references therein for an overview of this issue. Each of the aforementioned references share the fact that some non-degeneracy is required along the whole sequence. We provide here the first result removing this costly assumption as we require only non-degeneracy at the limit. The price to pay is to control the smooth Wasserstein distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limit as their Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main findings of Nourdin, Peccati and Swan [J. Funct. Anal. 2014]. Another application concerns the approximation of the semi-group of a diffusion process by the Euler scheme in a quantitative way and under the H\"ormander condition.

International audience; This paper introduce planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green’s third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equation, can then be numerically evalu-ated by standard quadrature rules that are irrespective of the kernel singularity. Closed-formand purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples—including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique.

In a recent paper by the first two authors, existence of martingale solutions to a stochastic nonlinear Schrodinger equation driven by a Levy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and existence of strong solutions to this problem using an abstract uniqueness result of Kurtz.

International audience; This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper, we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology. Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper, we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.

International audience; In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998). In a recent paper (Chambolle and Crismale in J Eur Math Soc (JEMS), 2018) we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti et al. (Ann Inst H Poincare Anal Non Lineaire 36:455–474, 2019) and Chambolle et al. (J Math Pures Appl, 2019. https://doi.org/10.1016/j.matpur.2019.02.001 ).

International audience; In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.

International audience; This paper is concerned with the study of the embedding circulant matrix method to simulate stationary complex-valued Gaussian sequences. The method is, in particular, shown to be well-suited to generate circularly-symmetric stationary Gaussian processes. We provide simple conditions on the complex co-variance function ensuring the theoretical validity of the minimal embedding circulant matrix method. We show that these conditions are satisfied by many examples and illustrate the algorithm. In particular, we present a simulation study involving the circularly-symmetric fractional Brownian motion, a model introduced in this paper.

International audience; This papers dealswith the zero-sum gamewith a discounted reward criterionfor piecewise deterministic Markov process (PDMPs) in general Borel spaces. Thetwo players can act on the jump rate and transition measure of the process, with thedecisions being taken just after a jump of the process. The goal of this paper is toderive conditions for the existence of min–max strategies for the infinite horizon totalexpected discounted reward function, which is composed of running and boundaryparts. The basic idea is, by using the special features of the PDMPs, to re-write theproblem via an embedded discrete-time Markov chain associated to the PDMP andre-formulate the problem as a discrete-stage zero sum game problem.

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [1] , [2] . Furthermore, in two recent papers [3] , [4] , Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain R 3 . We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin,Numer. Math.137(2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

International audience; This paper deals with the analysis of the propagation of uncertainties in computational linear dynamics for linear viscoelastic composite structures in the presence of uncertainties. In the frequency domain, the generalised damping matrix and the generalised stiffness matrix of the stochastic computational reduced-order model are random frequency-dependent matrices. Due to the causality of the dynamical system, these two frequency-dependent random matrices are statistically dependent and their probabilistic model involves a Hilbert transform. In this paper, a computational analysis of the propagation of uncertainties is presented for a composite viscoelastic structure in the frequency range.

This paper is devoted to the study of a general dynamic variational–hemivariational inequality with history-dependent operators. These operators appear in a convex potential and in a locally Lipschitz superpotential. The existence and uniqueness of a solution to the inequality problem is explored through a result on a class of nonlinear evolutionary abstract inclusions involving a nonmonotone multivalued term described by the Clarke generalized gradient. The result presented in this paper is new and general. It can be applied to study various dynamic contact problems. As an illustrative example, we apply the theory on a dynamic frictional viscoelastic contact problem in which the contact is modeled by a nonmonotone Clarke subdifferential boundary condition and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the total slip.

This paper is centred on the spectral study of a Random Fourier matrix, that is an $n\times n$ matrix $A$ whose $(j, k)$ entries are $\exp(2i\pi m X_jY_k)$, with $X_j$ and $Y_k$ two i.i.d sequences of random variables and $1\leq m\leq n$ is a real number. When they are uniformly distributed on a symmetric interval, this may be seen as a random discretization of the Finite Fourier transform, whose spectrum has been extensively studied in relation with band-limited functions. Our study is two-fold. Firstly, by pushing forward concentration inequalities, we find an accurate comparison in $\ell^2$- norm between the spectrum of $A^*A$ and the one of an integral operator that can be defined in terms of the two probability laws chosen for the rows and the columns. Our study includes the one of stationary Hermitian kernel matrices and can be generalized to non stationary ones, for which the same kind of comparison with an integral operator is possible. Because of possible applications in the data science area, these last matrices have been largely studied in the literature and our results are compared with previous ones. Secondly we concentrate on uniform distributions for the laws of $X_j$'s and $Y_k$'s, for which the integral operator is the well-known Sinc-kernel operator with parameter $m.$ Our previous study allows to translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. We have for them asymptotic results for $m, n$ and $n/m$ tending to $\infty$, as well as non asymptotic bounds in the spirit of recent work on the integral operators. As an application, we give fairly good approximations of the number of degrees of freedom and the capacity of a MIMO wireless communication network approximation model. Finally, we provide the reader with some numerical examples that illustrate the theoretical results of this paper.

The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods non-negative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence., Comment: This is the corrected version of the published paper P. Alquier, B. Guedj, An Oracle Inequality for Quasi-Bayesian Non-negative Matrix Factorization, Mathematical Methods of Statistics, 2017, vol. 26, no. 1, pp. 55-67. Since then Arnak Dalalyan (ENSAE) found a mistake in the proofs. We fixed the mistake at the price of a slightly different logarithmic term in the bound

International audience; This paper is a brief presentation of those Mean Field Games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e. with diffusion) is also presented both in the stationary and dynamic case. The variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on Augmented Lagrangian Algorithms, and some new simulations are shown at the end of the paper.

Our main aim is to investigate the temperature patch problem for the two-dimensional incompressible Boussinesq system with partial viscosity: the initial temperature is the characteristic function of some simply connected domain with 𝒞1,𝜀 Holder regularity. Although recent results ensure that the 𝒞1 regularity of the patch persists for all time, whether higher order regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue. We also study the higher dimensional case, after prescribing an additional smallness condition involving critical Lebesgue or weak-Lebesgue norms of the data, so as to get a global-in-time statement. All our results stem from general properties of persistence of geometrical structures (of independent interest), that we establish in the first part of the paper.

International audience; The paper focuses on the mathematical analysis and modeling of hematopoietic stem cell (HSC) dynamics that lead to the production and regulation of blood cells in the bone morrow. The HSC population is seen as a continuous medium structured in age and space. Using the method of characteristics, we reduce the age structured system to a reaction–diffusion equation containing a nonlocal spatial term and a time delay. Firstly, we give some properties on the existence, uniqueness and positivity of the solution. Secondly, we obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional and we prove that if it is not globally asymptotic stable then, it is unstable. Thirdly, we give sufficient conditions for the existence and uniqueness of the positive steady state by using the sub-and super-solutions method. Finally, we prove the uniform persistence of the system when the trivial steady state is unstable. Throughout the paper, we provide some numerical simulations to illustrate our results. Mathematics Subject Classification. 34B18, 35B35, 37N25, 92C37.

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space L^p(I,X), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in L^p(I,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces., Comment: 40 pages

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non-ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Gruneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in 89,90,22, this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods.This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in 85, and proves to fit a wide range of numerical schemes in the literature, such as those presented in 19,64,15,82,84.

International audience; The framework of this paper is the improvement of direct-forcing immersed boundary methods in presence of moving obstacles. In particular, motivations for the use of the Direct Forcing (DF) method can be found in the advantage of a fixed computational mesh for fluid–structure interaction problems. Unfortunately, the direct forcing approach suffers a serious drawback in case of moving obstacles: the well known spurious force oscillations (SFOs). In this paper, we strengthen previous analyses of the origin of the SFO through a rigorous numerical evaluation based on Taylor expansions. We propose a remedy through an easy-to-implement regularization process (regularized DF). Formally, this regularization is related to the blending of the Navier–Stokes solver with the interpolation, but no modification of the numerical scheme is needed. This approach significantly cuts off the SFOs without increasing the computational cost. The accuracy and the space convergence order of the standard DF method are conserved. This is illustrated on numerical and physical validation test cases ranging from the Taylor–Couette problem to a cylinder with an imposed sinusoidal motion subjected to a cross-flow.

This is a companion paper based on our previous work on rare event simulation\ud methods. In this paper, we provide an alternative proof for the ergodicity\ud of shaking transformation in the Gaussian case and propose two variants of\ud the existing methods with comparisons of numerical performance. In numerical\ud tests, we also illustrate the idea of extreme scenario generation based on\ud the convergence of marginal distributions of the underlying Markov chains\ud and show the impact of the discretization of continuous time models on rare\ud event probability estimation.

International audience; This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

International audience; In this paper, an existence and uniqueness result of a class of second-order sweeping processes, with velocity in the moving set under perturbation in infinite- dimensional Hilbert spaces, is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state and is possibly unbounded. The assumptions on the Lipschitz continuity and the compactness of the moving set, and the linear growth boundedness of the perturbation force are weaker than the ones used in previous papers.

The pressure term which appears in the ROM (reduced order model) associated to the incompressible Navier–Stokes equations, in particular for the shear flows, plays an important role on the velocity. The aim of this paper is to propose a Proper Orthogonal Decomposition based reduced order model (POD–ROM) to obtain both the velocity and pressure fields for incompressible flows. Two PODs are performed, one for the velocity and the other for the pressure. Contrary to existing projection methods available in the literature, the temporal velocity and pressure coefficients are sought by minimizing the residual of the momentum equation only, without the need of a Poisson equation. For the numerical test cases considered in this paper, the proposed minimum residual projection enables to obtain accurately the pressure field, and in turn to slightly improve the velocity one. The method is tested on two fluid flows: a transient mixed-convection flow and a periodic flow around a circular cylinder. In this last case, the drag, lift and pressure coefficients, as well as the Strouhal number are properly recovered compared to those of the full model.

This paper is devoted to the study of food chain predator–prey model. This model is given by a reaction–diffusion system defined on a circular spatial domain, which includes three-state variables namely, prey and intermediate predator and top predator and incorporates the Holling type II and a modified Leslie–Gower functional response. The aim of this paper is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The local and global stabilities of the positive steady-state solution and the conditions that enable the occurrence of Hopf bifurcation and Turing instability in the circular spatial domain are proved. In the end, we carry out numerical simulations to illustrate how biological processes can affect spatiotemporal pattern formation in a disc spatial domain and different types of spatial patterns with respect to different time steps and diffusion coefficients are obtained.

In this paper, we are concerned with the numerical modelling of the propagation of electromagnetic waves in dispersive materials for nanophotonics applications. We focus on a generalized model that allows for the description of a wide range of dispersive media. The underlying differential equations are recast into a generic form and we establish an existence and uniqueness result. We then turn to the numerical treatment and propose an appropriate Discontinuous Galerkin Time Domain framework. We obtain the semi-discrete convergence and prove the stability (and in a larger extent, convergence) of a Runge Kutta 4 fully discrete scheme via a technique relying on energy principles. Finally, we validate our approach through two significant nanophotonics test cases. 1. Introduction. Among the numerous phenomena encountered in electromag-netics, many rely on the dispersive properties of materials, e.g. the fact that their phase velocity varies with frequency. Indeed, in specific ranges of wavelengths, biological tissues [GGC96], noble [JC72] and transition metals [JC74], but also glass [Fle78] and certain polymers [CC41] exhibit non-negligible dispersive behaviors. From the mathematical modeling point of view, this phenomenon is modeled by a frequency-dependent permittivity function ε(ω), often derived from physical considerations. Regarding nanophotonics applications, an accurate modeling of the permittivity function for metals in the visible spectrum is crucial. Indeed, the free electrons of metals are the key ingredient in the propagation of surface modes of particular interest, called surface plasmons [NH07]. The implementation of dispersion models in time-domain electromagnetics solvers can be achieved by different methods. The most common is certainly the Additional Differential Equation (ADE) technique, which consists in the addition of one or more ODEs to the Maxwell system, the coupling being made via source terms. A consequent literature on this topic exists in the context of Finite-Difference Time-Domain (FDTD) (see e.g. [VLDC11] and references therein). More recently, more papers are concerned with Finite Element or even Discontinuous Galerkin Time-Domain approaches (DGTD) (see e.g. [GYKR12] and [BKN11] and references therein), aiming at overcoming the limitations of FDTD. In this context, some works are more precisely focused on the numerical analysis. Several proofs exist for the standard dispersive media models and the most classical time and space discretization schemes (see e.g. all the papers of J. Li and co-authors such as [JL06, Li07, LCE08, Li09]). Let us also mention the approach of [WXZ10] for the integro-differential version of the classical dispersive models. The latter reference propose to analyze a semi-discrete divergence free discontinuous Galerkin framework. Finally, in a previous work [LS13], the authors analyzed, for the Debye model, a fully discrete scheme based on a centered fluxes nodal Discontinuous Galerkin formulation and Leap frog discretization in time. In this paper, we present a complete study of a generalized dispersive model that encapsulates a wide range of dispersive media, its higher efficiency being demonstrated

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data., Comment: With supplementary material

This paper considers the use of orthogonal arrays of strength two as experimental designs for fitting a surrogate model. Contrary to standard space-filling designs or Latin hypercube designs, the points of an orthogonal array of strength two are well distributed when they are projected on the two-dimensional faces of the unit cube. The aim is to determine if this property allows one to fit an accurate surrogate model when the computer response is governed by second-order interactions of some input variables. The first part of the paper is devoted to the construction of orthogonal arrays with space-filling properties. In the second part, orthogonal arrays are compared with standard designs for fitting a Gaussian process model.

International audience; In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model coupled to the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event.

The flow around a rigid obstacle is governed by compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain with a compact obstacle in its interior. The flight of the airflow is characterized by the work shape functional, to be minimized over a family of admissible obstacles. The lift of the airfoil is given in function of time and should be closed to the flight scenario. Therefore, the minimization for a given lift of the work functional with respect to the shape of obstacle in two spatial dimensions is considered. The shape optimization problems for the compressible Navier-Stokes equations with the nonhomogeneous Dirichlet conditions in a bounded domain with an obstacle are considered in the monograph (Plotnikov, Sokolowski, Springer, 2012) in the general case of three spatial dimensions. In the present paper, for the purposes of wellposednes of shape optimization problems, there is no restrictions on the regularity of admissible obstacles, which are simply connected compact subsets of a given, bounded hold-all domain. The presented results are derived in the general framework established in (Plotnikov, Sokolowski, Springer, 2012). It means that in two spatial dimensions we obtain the same optimal shape existence result for the compressible Navier-Stokes equations as it is for the Laplacian (Sverak, JMPA, 1993). However, the complete existence proof is substantially more complex compared to the Laplacian, and it is given with all details in (Plotnikov, Sokolowski, Springer, 2012) for the general case of three spatial dimensions. In the present paper the general theory of shape optimization developed in (Plotnikov, Sokolowski, Springer, 2012) is adapted to the particular case of two spatial dimensions. The shape optimization problem of work minimization over a class of admissible obstacles is introduced. The continuity of the work functional with respect to the obstacle in two spatial dimensions is shown for a wide class of admissible obstacles. The dependence of local solutions to the governing equations with respect to the boundary variations of obstacles is analyzed. The shape derivatives (Sokolowski, Zolesio, Springer, 1992) of solutions to the compressible Navier-Stokes equations are derived. The shape gradient (Sokolowski, Zolesio, Springer, 1992) of the work functional is obtained. The framework for numerical methods of shape optimization (Plotnikov, Sokolowski, Zochowski, MMAR'09, 2009) is established for nonstationary, compressible Navier-Stokes equations.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems., Comment: This paper is dedicated to J.-B. Hiriart-Urruty. Note on v3: to appear in ESAIM: COCV

International audience; When one solves PDEs modelling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. For instance, the underlying physical assumptions for the Euler equations are the positivity of both den- sity and pressure variables. We consider in this paper an unstructured vertex-based tesselation in R2 . Given a MUSCL finite volume scheme and given a reconstruction method (including a limiting process), the point is to determine whether the overall scheme ensures the positivity. The present work is issued from seminal papers from Perthame and Shu (On positivity preserving finite volume schemes for Euler equations, Numer. Math. 73 (1996) 119-130) and Berthon (Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2) (2006) 495-509). They proved in different frameworks that under assumptions on the cor- responding one-dimensional numerical flux, a suitable CFL condition guarantees that den- sity and pressure remain positive. We first analyse Berthon's method by presenting the ins and outs. We then propose a more general approach adding non geometric degrees of freedom. This approach includes an optimization procedure in order to make the CFL condition explicit and as less restric- tive as possible. The reconstruction method is handled independently by means of s- limiters and of an additional damping parameter. An algorithm is provided in order to specify the adjustments to make in a preexisting code based on a certain numerical flux. Numerical simulations are carried out to prove the accuracy of the method and its ability to deal with low densities and pressures.

International audience; In this paper, we describe a multiscale strategy that allows to couple stochastic and deterministic models. The transition condition enforced between the two models is weak, in the sense that it is based on volume coupling in space (rather than more classical boundary coupling) and on a volume/sample average in the random dimension. The paper then concentrates on the application of this weak coupling technique for the development of a new iterative method for the homogenization of random media. The technique is based on the coupling of the stochastic microstructure to a tentative homogenized medium, the parameters of which are initially chosen at will. Based on the results of the coupled simulation, for which Dirichlet or Neumann boundary conditions are posed at the boundary of the tentative homogenized medium, the parameters of the homogenized medium are then iteratively updated. An example shows the efficiency of the proposed approach compared to the classical KUBC and SUBC approaches in stochastic homogenization.

We are interested in the finite element solution of elliptic problems with a right-hand side of the single layer distribution type. Such problems arise when one aims at accounting for a physical hypersurface (or line, for bi-dimensional problem), but also in the context of fictitious domain methods, when one aims at accounting for the presence of an inclusion in a domain (in that case the support of the distribution is the boundary of the inclusion). The most popular way to handle numerically the single layer distribution in the finite element context is to spread it out by a regularization technique. An alternative approach consists in approximating the single layer distribution by a combination of Dirac masses. As the Dirac mass in the right hand side does not make sense at the continuous level, this approach raises particular issues. The object of the present paper is to give a theoretical background to this approach. We present a rigorous numerical analysis of this approximation, and we present two examples of application of the main result of this paper. The first one is a Poisson problem with a single layer distribution as a right-hand side and the second one is another Poisson problem where the single layer distribution is the Lagrange multiplier used to enforce a Dirichlet boundary condition on the boundary of an inclusion in the domain. Theoretical analysis is supplemented by numerical experiments in the last section.

This is the second paper of the authors in a series concerned with the development of a deterministic model for the transfer matrix of a MIMO system. Starting from the Maxwell equations, we have described in \cite{BCFM} the generic structure of such a deterministic transfer matrix. In the current paper we apply the results of \cite{BCFM} in order to study the (Shannon-Foschini) capacity behavior of a MIMO system as a function of the deterministic spread function of the environment, and the number of transmitting and receiving antennas. The antennas are assumed to fill in a given, fixed volume. Under some generic assumptions, we prove that the capacity grows much more slowly than linearly with the number of antennas. These results reinforce previous heuristic results obtained from statistical models of the transfer matrix, which also predict a sublinear behavior., Comment: 12 pages, to appear in Math. Meth. Appl. Sci

Accepted for Publication; International audience; In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.

This work is a continuation of the authors efforts to develop high-order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach. In a previous paper, a new method was proposed, based on the use of smooth forcing functions with identical shapes, mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of Lagrange multipliers in relation to a discrete set of boundary constraints. For one-dimensional elliptic problems, this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth-order algebraic convergence rate. In this paper, a spectrally accurate formulation is presented for multi-dimensional applications. Instead of being specified locally, the forcing function is defined as a convolution of a mollifier (smooth bump function) and a Lagrange multiplier function (the amplitude of the bump). The multiplier function is then approximated by Fourier series. Using a Fourier Galerkin approximation, the spectral accuracy is demonstrated on a two-dimensional Laplacian problem and on a Stokes flow around a periodic array of cylinders. In the latter, the numerical solution achieves the same high-order accuracy as a Stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom.

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.