Publications

2016

• FUNCTIONAL ITÔ VERSUS BANACH SPACE STOCHASTIC CALCULUS AND STRICT SOLUTIONS OF SEMILINEAR PATH-DEPENDENT EQUATIONS
  
  • Cosso Andrea
  • Russo Francesco
  Infinite Dimensional Analysis, Quantum Probability and Related Topics, World Scientific Publishing, 2016, 19 (04), pp.1650024. Functional It\^o calculus was introduced in order to expand a functional $F(t, X_{\cdot+t}, X_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X_{\cdot+t}, X_t)$ consists in considering the path $X_{\cdot+t}=\{X_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional It\^o calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional It\^o calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an It\^o stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation. (10.1142/S0219025716500247)
  DOI : 10.1142/S0219025716500247
• Study of a Model Equation in Detonation Theory: Multidimensional Effects
  
  • Faria Luiz
  • Kasimov A.
  • Rosales R.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (3), pp.887-909. We extend the reactive Burgers equation presented in [A. R. Kasimov, L. M. Faria,and R. R. Rosales,Phys. Rev. Lett., 110 (2013), 104104], [L. M. Faria, A. R. Kasimov, and R. R.Rosales,SIAM J. Appl. Math., 74 (2014), pp. 547–570] to include multidimensional effects. Fur-thermore, we explain how the model can be rationally justified following the ideas of the asymptotictheory developed in [L. M. Faria, A. R. Kasimov, and R. R. Rosales,J. Fluid Mech., 784 (2015),pp. 163–198]. The proposed model is a forced version of the unsteady small disturbance transonicflow equations. We show that for physically reasonable choices of forcing functions, traveling wavesolutions akin to detonation waves exist. It is demonstrated that multidimensional effects play animportant role in the stability and dynamics of the traveling waves. Numerical simulations indicatethat solutions of the model tend to form multidimensional patterns analogous to cells in gaseousdetonations. (10.1137/15M1039663)
  DOI : 10.1137/15M1039663
• A numerical study of the solution of x-mode equations around the hybrid resonance
  
  • Caldini-Queiros Céline
  • Després Bruno
  • Imbert-Gérard Lise-Marie
  • Kachanovska Maryna
  ESAIM: Proceedings and Surveys, EDP Sciences, 2016, 53, pp.1-21. Hybrid resonance is a physical phenomenon that appears for example in the heating of plasma, and as such is of scientific interest in the development of the ITER project. In this paper we focus some solutions with low regularity of Maxwell equations in plasmas under strong background magnetic field. Our purpose is two-fold. On one hand we investigate the finite element approximation of the one dimensional problem written in the frequency domain, and on the other hand we investigate two different finite difference approximations of the one dimensional time dependent problem. We will also compare the results of these different methods.
• State-constrained stochastic optimal control problems via reachability approach
  
  • Bokanowski Olivier
  • Picarelli Athena
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2016, 54 (5), pp.2568–2593. This paper deals with a class of stochastic optimal control problems (SOCP) in presence of state-constraints. It is well-known that for such problems the value function is, in general, discontinuous and its characterization by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system. Here, we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions. For this end, the SOCP is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward-reachable sets describe the value function. The main advantage of our approach is that it allows to handle easily the state constraints by an exact penalization. However, the target problem involves a new state variable and a new control variable that is unbounded. (10.1137/15M1023737)
  DOI : 10.1137/15M1023737
• Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides
  
  • Baronian Vahan
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Wave Motion, Elsevier, 2016. We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite straight isotropic elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with or without an overlap between the domains. Specific transmission conditions are used, so that only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using the Fraser bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized, in particular for the extension to the anisotropic case. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.
• Uniqueness results for inverse Robin problems with bounded coefficient
  
  • Baratchart Laurent
  • Bourgeois Laurent
  • Leblond Juliette
  Journal of Functional Analysis, Elsevier, 2016. In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, $r>n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context. (10.1016/j.jfa.2016.01.011)
  DOI : 10.1016/j.jfa.2016.01.011