Publications

2012

• A closed-form extension to the Black-Cox model
  
  • Alfonsi Aurélien
  • Lelong Jérôme
  International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2012, 15 (8), pp.1250053:1-30. In the Black-Cox model, a firm defaults when its value hits an exponential barrier. Here, we propose an hybrid model that generalizes this framework. The default intensity can take two different values and switches when the firm value crosses a barrier. Of course, the intensity level is higher below the barrier. We get an analytic formula for the Laplace transform of the default time. This result can be also extended to multiple barriers and intensity levels. Then, we explain how this model can be calibrated to Credit Default Swap prices and show its tractability on different kinds of data. We also present numerical methods to numerically recover the default time distribution. (10.1142/S0219024912500537)
  DOI : 10.1142/S0219024912500537
• Constitutive Equation Gap
  
  • Pagano Stéphane
  • Bonnet Marc
  , 2012, pp.26p.. In this chapter we examine the concept of constitutive equation gap (CEG) as a tool for the identification of parameters associated with behavior models for solid materials. The concept of CEG was initially proposed for error estimation in the finite element method. It then turned out to be also a powerful tool for identification, especially with many applications in model updating. Essentially equivalent concepts have been proposed in other contexts for solving inversion problems, such as the elec- trostatic energy functionals of Kohn and Vogelius. Two important characteristics of CEG functionals are (i) their strong and clear physical meaning, and (ii) their additive character with respect to the structure, allowing the definition of local error indicators over substructures.
• Modeling and simulation of a grand piano.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  , 2012, pp.29. The purpose of this study is the time domain modeling and numerical simulation of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependant damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from horizontal. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model that will be discretized is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependant damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties. On the one hand, numerical stability of the discrete scheme can be compromised by nonlinear and coupling terms. A very efficient way to guarantee this stability is to construct a numerical scheme which ensures the conservation (or dissipation) of a discrete equivalent of the continuous energy, across time steps. A major contribution of this work has been to develop energy preserving schemes for a class of nonlinear systems of equations, in which enters the string model. On the other hand, numerical efficiency and computation time reduction require that the unknowns of each problem's part, for which time discretization is specific, hence different, be updated separately. To achieve this artificial decoupling, adapted Schur complements are performed after Lagrange multipliers are introduced. The potential of this time domain piano modeling is emphasized by realistic numerical simulations. Beyond greatly replicating the measurements, the program allows us to investigate the influence of physical phenomena (string stiffness or nonlinearity), geometry or materials on the general vibratory behavior of the piano, sound included. Spectral enrichment, ''phantom partials'' and nonlinear precursors are clearly revealed when large playing amplitudes are involved, highlighting how this approach can help better understand how a piano works.
• Optimisation de Lois de Gestion Énergétiques des Véhicules Hybrides
  
  • Granato Giovanni
  , 2012. The purpose of the this work is to apply optimal control techniques to enhance the performance of the power management of hybrid vehicles. More precisely, the techniques concerned are viscosity solutions of Hamilton-Jacobi equations, level set methods in reachability analysis, stochastic dynamic programming, stochastic dual dynamic programming and chance constrained optimal control. This document starts by presenting the necessary technical background and models for the study of optimal power management of hybrid vehicles. The synthesis of efficient power management strategies for hybrid vehicles accounting for uncertainty in the vehicle speed is studied next. This is done via a stochastic dynamic algorithm, at a first time, and then by a stochastic dual dynamic programming algorithm. In addition, we introduce a chance constrained optimal control problem that can be used to synthesize more flexible optimal control strategies. We detail a dynamic programming principle in a form that can be readily used for the numerical synthesis of optimal feedback using a dynamic programming algorithm. Later, theoretical results regarding the reachability analysis of hybrid systems are obtained. The reachability set of a continuous-time hybrid system is characterized by a value function via a level set approach. Furthermore, we show that the value function of a hybrid optimal control problem is the unique solution of a system of quasi-variational inequalities in the viscosity sense. Then, we prove the convergence of a class of numerical schemes for the computation of the value function. As a further step in the reachability analysis, we study of the discrete-time dynamical system and the discrete-time optimal control problem for the reachability analysis of hybrid systems. Here, the focus is on a discrete-time modeling of the hybrid system, which leads to dynamic programming principle, which can be used to characterize the value function. Lastly, we describe the construction of a stochastic model of the speed profile for electric vehicles.
• Two methods of pruning Benders' cuts and their application to the management of a gas portfolio
  
  • Pfeiffer Laurent
  • Apparigliato Romain
  • Auchapt Sophie
  , 2012, pp.23. In this article, we describe a gas portfolio management problem, which is solved with the SDDP (Stochastic Dual Dynamic Programming) algorithm. We present some improvements of this algorithm and focus on methods of pruning Benders' cuts, that is to say, methods of picking out the most relevant cuts among those which have been computed. Our territory algorithm allows a quick selection and a great reduction of the number of cuts. Our second method only deletes cuts which do not contribute to the approximation of the value function, thanks to a test of usefulness. Numerical results are presented.
• Remarks on the stability of Cartesian PMLs in corners
  
  • Bécache Eliane
  • Prieto Andrés
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2012, 62 (11), pp.1639-1653. This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure–velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results. (10.1016/j.apnum.2012.05.003)
  DOI : 10.1016/j.apnum.2012.05.003
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  , 2012, pp.32. We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.
• Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case
  
  • Aronna Maria Soledad
  • Rampazzo Franco
  , 2012. In this article we study control problems with systems that are governed by ordinary differential equations whose vector fields depend linearly in the time derivatives of some components of the control. The remaining components are considered as classical controls. This kind of system is called 'impulsive system'. We assume that the vector fields multiplying the derivatives of each component of the control are commutative. We derive new necessary conditions in terms of the adjoint state and the Lie brackets of the data functions.
• Commande optimale en temps minimal d’un procédé biologique d’épuration de l’eau
  
  • Bouhafs Walid
  • Abdellatif Nahla
  • Jean Frédéric
  • Harmand Jérôme
  , 2012, pp.np. Dans ce travail, on considère un problème de contrôle d’un procédé biologique séquentiel discontinu pour le traitement de la pollution. Ce modèle fait intervenir deux réactions biologiques, l’une étant aérobie et l’autre anoxique. On s’intéresse, dans un premier temps, à un problème de contrôle optimal en temps minimal puis en temps et en énergie. On prouve l’existence de trajectoires optimales et on calcule, dans chaque cas, les contrôles optimaux correspondants. In this work, we consider an optimal control problem of a biological sequencing batch reactor for the treatment of pollutants. This model is formed of two biological reactions, one being aerobic and the other, anoxic. We are interested in a problem of optimal control in time and then, in both time and energy. The existence of the optimal trajectories is proven and the corresponding optimal controls are derived in each case.
• Étude de quelques problèmes de transmission avec changement de signe. Application aux métamatériaux.
  
  • Chesnel Lucas
  , 2012. Dans cette thèse, nous étudions quelques opérateurs présentant un changement de signe dans leur partie principale. Ces opérateurs apparaissent notamment en électromagnétisme lorsqu'on s'intéresse à la propagation des ondes dans des structures constituées de matériaux usuels et de matériaux négatifs en régime harmonique. Ici, nous appelons matériau négatif un matériau modélisé par une permittivité diélectrique et/ou une perméabilité magnétique négative(s). En raison du changement de signe des coefficients physiques, on ne peut utiliser les outils classiques pour étudier ce problème. Dans la première partie de ce mémoire, nous nous concentrons sur le problème de transmission scalaire auquel on peut réduire les équations de Maxwell lorsque la géométrie et les données présentent une invariance dans une direction. Avec la technique de la T-coercivité, basée sur des arguments géométriques, nous établissons des conditions nécessaires et suffisantes pour prouver le caractère bien posé de ce problème en domaine borné dans H^1. Nous montrons également comment on peut utiliser cette approche pour justifier la convergence des méthodes usuelles d'approximation par éléments finis. Dans un deuxième temps, au moyen de techniques différentes, issues de l'étude des équations elliptiques dans des domaines à géométrie singulière, nous définissons un nouveau cadre fonctionnel pour recouvrer le caractère Fredholm lorsque celui-ci est perdu dans H^1. Il apparaît alors un phénomène surprenant de trou noir. Tout se passe comme si des ondes étaient aspirées en un point. Nous réalisons ensuite une étude asymptotique par rapport à une petite perturbation de l'interface entre le matériau positif et le matériau négatif dans ce cadre fonctionnel. Au cours de notre analyse, nous mettons en évidence un curieux phénomène de valeur propre clignotante. La troisième partie de ce document est consacrée à l'étude des équations de Maxwell. Nous travaillons d'abord sur les équations de Maxwell 2D en exploitant les résultats obtenus pour le problème scalaire. Puis, nous nous intéressons aux équations de Maxwell 3D. Nous montrons qu'elles sont bien posées dès lors que les problèmes scalaires associés sont bien posés. Enfin, dans une quatrième partie, nous étudions le problème de transmission intérieur apparaissant en théorie de la diffraction. L'opérateur pour ce problème présente également un changement de signe dans sa partie principale. Nous abordons son étude en utilisant l'analogie existant avec le problème de transmission entre un matériau positif et un matériau négatif. Certaines configurations pour ce problème de transmission intérieur conduisent à considérer un problème de transmission du quatrième ordre avec changement de signe. Nous prouvons que cet opérateur présente des propriétés étonnamment différentes de celles de l'opérateur scalaire du second ordre.
• Time domain simulation of a piano. Part 1 : model description.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  , 2012. The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependant damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependant damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties.
• Semi-Lagrangian discontinuous Galerkin schemes for some first and second order partial differential equations
  
  • Bokanowski Olivier
  • Simarmata Giorevinus
  Journal of Scientific Computing, Springer Verlag, 2012, pp.DOI 10.1007/s10915-012-9648-x. Explicit, unconditionally stable, high order schemes for the approximation of some first and second order linear, time-dependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin elements. It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010) and of Qiu and Shu (2011), for first order equations, based on exact integration, quadrature rules, and splitting techniques. In particular we obtain high order schemes, unconditionally stable and convergent, in the case of linear second order PDEs with constant coefficients. In the case of non-constant coefficients, we construct "almost" unconditionally stable second order schemes and give precise convergence results. The schemes are tested on several academic examples, including the Black and Scholes PDE in finance.
• Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems
  
  • Alvarez Felipe
  • Bolte Jérôme
  • Bonnans J. Frederic
  • Silva Francisco
  Mathematical Programming, Series A, Springer, 2012, 135 (1-2), pp.473-507. We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If the control of the initial problem satisfies strict complementarity conditions for its Hamiltonian except for a finite number of times, the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem. Our results provide several types of efficiency measures of the penalization technique: error estimations of the control for $L^s$ norms ($s$ in $[1,+\infty]$), error estimations of the state and the adjoint state in Sobolev spaces $W^{1,s}$ ($s$ in $[1,+\infty)$) and also error estimates for the value function. For the $L^1$ norm and the logarithmic penalty, the optimal results are given. In this case we indeed establish that the penalized control and the value function errors are of order $O(\eps|\log\eps|)$.
• Quadratic conditions for bang-singular extremals
  
  • Aronna Maria Soledad
  • Bonnans J. Frederic
  • Dmitruk Andrei V.
  • Lotito Pablo
  Numerical Algebra, Control and Optimization, American Institute for Mathematical Sciences, 2012, 2 (3), pp.511-546. This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case. (10.3934/naco.2012.2.511)
  DOI : 10.3934/naco.2012.2.511
• A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces
  
  • Ghezzi Roberta
  • Jean Frédéric
  Communications on Pure and Applied Analysis, AIMS American Institute of Mathematical Sciences, 2012, 12 (2), pp.881-898. The main motivation of this paper arises from the study of Carnot--Carathéodory spaces, where the class of $1$-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $({\cal H}^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot--Carathéodory spaces. (10.3934/cpaa.2013.12.881)
  DOI : 10.3934/cpaa.2013.12.881
• Bocop - A collection of examples
  
  • Bonnans Frédéric J.
  • Martinon Pierre
  • Grélard Vincent
  , 2012. In this document we present a collection of classical optimal control problems which have been implemented and solved with Bocop. We recall the main features of the problems and of their solutions, and describe the numerical results obtained.
• A Local Ordered Upwind Method for Hamilton-Jacobi and Isaacs Equations
  
  • Cacace Simone
  • Cristiani Emiliano
  • Falcone Maurizio
  , 2012, 18. We present a generalization of the Fast Marching (FM) method for the numerical solution of a class of Hamilton-Jacobi equations, including Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. The method is able to compute an approximation of the viscosity solution concentrating the computations only in a small evolving trial region, as the original FM method. The main novelty is that the size of the trial region does not depend on the dynamics. We compare the new method with the standard iterative algorithm and the FM method, in terms of accuracy and order of computations on the grid nodes. (10.3182/20110828-6-IT-1002.02473)
  DOI : 10.3182/20110828-6-IT-1002.02473
• Collision analysis for an UAV
  
  • Desilles Anna
  • Zidani Hasnaa
  • Crück Eva
  , 2012, pp.AIAA 2012-4526. The Sense and Avoid capacity of Unmanned Aerial Vehicles (UAV) is one of the key elements to open the access to airspace for UAVs. In order to replace a pilot's See and Avoid capacity such a system has to be certified "as safe as a human pilot on-board". The problem is to prove that an unmanned aircraft equipped with a S and A system can comply with the actual air transportation regulations. This paper aims to provide mathematical and numerical tools to link together the safety objectives and sensors specifications. Our approach starts with the natural idea of a specified "safety volume" around the aircraft: the safety objective is to guarantee that no other aircraft can penetrate this volume. We use a general reachability and viability concepts to define nested sets which are meaningful to allocate sensor performances and manoeuvring capabilities necessary to protect the safety volume. Using the general framework of HJB equations for the optimal control and differential games, we give a rigorous mathematical characterization of these sets. Our approach allows also to take into account some uncertainties in the measures of the parameters of the incoming traffic. We also provide numerical tools to compute the defined sets, so that the technical specifications of a S and A system can be derived in accordance with a small set of intuitive parameters. We consider several dynamical models corresponding to the different choices of maneuvers (lateral, longitudinal and mixed). Our numerical simulations show clearly that the nature of used maneuvers is an important factor in the specifications of sensor's performances. (10.2514/6.2012-4526)
  DOI : 10.2514/6.2012-4526
• Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions
  
  • Graber Philip Jameson
  • Said-Houari Belkacem
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2012, 66, pp.81-122. The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time. (10.1007/s00245-012-9165-1)
  DOI : 10.1007/s00245-012-9165-1
• Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models
  
  • Al Haj Mohammad
  • Forcadel Nicolas
  • Monneau Régis
  , 2012. In this article, we study the existence and the uniqueness of traveling waves for a discrete reaction-diffusion equation with bistable non-linearity, namely a generalization of the fully overdamped Frenkel-Kontorova model. This model consists in a system of ODE's which describes the dynamics of crystal defects in a lattice solids. Under very poor assumptions, we prove the existence of a traveling wave solution and the uniqueness of the velocity of propagation of this traveling wave. The question of the uniqueness of the profile is also studied by proving Strong Maximum Principle or some weak asymptotics on the profile at infinity.
• Hamilton-Jacobi-Bellman approach for the climbing problem for heavy launchers
  
  • Bokanowski Olivier
  • Cristiani Emiliano
  • Laurent-Varin Julien
  • Zidani Hasnaa
  , 2012. In this paper we investigate the Hamilton-Jacobi-Bellman (HJB) approach for solving a complex real-world optimal control problem in high dimension. We consider the climbing problem for the European launcher Ariane V: The launcher has to reach the Geostationary Transfer Orbit with minimal propellant consumption under state/control constraints. In order to circumvent the well-known curse of dimensionality, we reduce the number of variables in the model exploiting the specific features concerning the dynamics of the mass. This generates a non-standard optimal control problem formulation. We show that the joint employment of the most advanced mathematical techniques for the numerical solution of HJB equations allows one to achieve practicable results in reasonable time.
• Reachability of Delayed Hybrid Systems Using Level-set Methods
  
  • Granato Giovanni
  , 2012. This study proposes an algorithm to synthesize controllers for the power management on board hybrid vehicles that allows the vehicle to reach its maximum range along a given route. The algorithm stems from a level-set approach that computes the reachable set of the system, i.e., the collection of states reachable from a certain initial condition via the computation of the value function of an optimal control problem. The discrete-time vehicle model is one of a particular class of hybrid vehicles, namely, range extender electric vehicles (REEV). This kind of hybridization departures from a full electric vehicle that has an additional module -- the range extender (RE) -- as an extra energy source in addition to its main energy source -- a high voltage battery. As an important feature, our model allows for the switching on and off of the range extender and includes a decision lag constraint, i.e., imposes two consecutive switches to be separated by a positive time interval. The approach consists in the introduction of an adequate optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a dynamic programming algorithm. This algorithm is implemented and some numerical examples are presented.
• FaIMS: A fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equation
  
  • Chaillat Stéphanie
  • Biros George
  Journal of Computational Physics, Elsevier, 2012, 231 (12), pp.4403-4421. We propose an algorithm to compute an approximate singular value decomposition of least squares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency time-harmonic wave eqation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, non-destructive evaluation, and optical tomography). (10.1016/j.jcp.2012.02.006)
  DOI : 10.1016/j.jcp.2012.02.006
• Radial orbit instability: review and perspectives
  
  • Maréchal Lionel
  • Perez Jérôme
  Transport Theory and Statistical Physics, Taylor & Francis, 2012, 40 (6), pp.p.425-439. This paper presents elements about the radial orbit instability, which occurs in spherical self-gravitating systems with a strong anisotropy in the radial velocity direction. It contains an overview on the history of radial orbit instability. We also present the symplectic method we use to explore stability of equilibrium states, directly related to the dissipation induced instability mechanism well known in theoretical mechanics and plasma physics.
• Numerical Algorithms for a Variational Problem of the Spatial Segregation of Reaction-Diffusion Systems
  
  • Arakelyan Avetik
  • Bozorgnia Farid
  , 2012. In this paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support, which are governed by a minimization problem. We use quantitative properties of both solutions and free boundaries to derive our scheme. Furthermore, the proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions. We discuss numerical implementations of the resulting approach and present computational tests.