## Publications

### Articles

[A17] J. Aniort, L. Chupin, N. Cîndea. Mathematical model of calcium exchange during haemodialysis using a citrate containing dialysate. Mathematical Medicine and Biology: A Journal of the IMA, Volume 35, Issue suppl_1, 16 March 2018, Pages 87–120.

Calcium has two important roles in haemodialysis. It participates in the activation of blood coagulation and calcium intakes have a major impact on patient mineral and bone metabolism. The aim of this article is to propose a mathematical model for calcium ions concentration in a dialyzer during haemodialysis using a citrate dialysate. The model is composed of two elements. The first describes the flows of blood and dialysate in a dialyzer fibre. It was obtained by asymptotic analysis and takes into account the anisotropy of the fibres forming a dialyzer. Newtonian and non-Newtonian blood rheologies were tested. The second part of the model predicts the evolution of the concentration of five chemical species present in these fluids. The fluid velocity field drives the convective part of a convection–reaction–diffusion system that models the exchange of free and complexed calcium. We performed several numerical experiments to calculate the free calcium concentration in the blood in a dialyzer using dialysates with or without citrate. The choice of blood rheology had little effect on the fluid velocity field. Our model predicts that only a citrate based dialysate without calcium can decrease free calcium concentration at the blood membrane interface low enough to inhibit blood coagulation. Moreover for a given calcium dialysate concentration, adding citrate to the dialysate decreases total calcium concentration in the blood at the dialyzer outlet. This decrease of the calcium concentration can be compensated by infusing in the dialyzed blood a quantity of calcium computed from the model.

[A16] N. Cîndea, S. Micu, I. Roventa. Boundary Controllability for Finite-Differences Semidiscretizations of a Clamped Beam Equation. SIAM Journal on Control and Optimization 2017, Vol. 55, No. 2 : pp. 785-817.

This article deals with the boundary observability and controllability properties of a space finite-differences semidiscretization of the clamped beam equation. We make a detailed spectral analysis of the system and, by combining numerical estimates with asymptotic expansions, we localize all the eigenvalues of the corresponding discrete operator depending on the mesh size $h$. Then, an Ingham's type inequality and a discrete multiplier method allow us to deduce that the uniform (with respect to $h$) observability property holds if and only if the eigenfrequencies are filtered out in the range ${\cal O}\left(1/h^4\right)$.

[A15] N. Cîndea, A. Münch. Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach. Inverse Problems 32 (11), 2016.

We introduce a direct method that makes it possible to solve numerically inverse type problems for linear hyperbolic equations posed in ${\rm{\Omega }}\times (0,T)$ − $\rm{\Omega}$, a bounded subset of ${{\mathbb{R}}}^{N}$. We consider the simultaneous reconstruction of both the state and the source term from a partial boundary observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric conditions, we show the well-posedness of this mixed formulation (in particular the inf–sup condition) and then introduce a numerical approximation based on space-time finite element discretization. We prove the strong convergence of the approximation and then discuss several examples in the one- and two-dimensional cases.

[A14] N. Cîndea, A. Münch. Inverse problems for linear hyperbolic equations using mixed formulations. Inverse Problems 31 (7), 075-001, 2015.

We introduce in this document a direct method allowing to solve nu-merically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\Omega \times (0, T)$ - $\Omega$ a bounded subset of $\mathbb{R}^n$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observa-tion to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discussed several example for $N = 1$ and $N = 2$. The problem of the reconstruction of both the state and the source term is also addressed.

[A13] N. Cîndea, S. Micu, I. Roventa, M. Tucsnak. Particle supported control of a fluid-particle system. Journal de Mathématiques Pures et Appliquées, 104 (2015) 311–353.

In this paper we study, from a control theoretic view point, a 1D model of fluid-particle interaction. More precisely, we consider a point mass moving in a pipe filled with a fluid. The fluid is modelled by the viscous Burgers equation whereas the point mass obeys Newton's second law. The control variable is a force acting on the mass point. The main result of the paper asserts that for any initial data there exist a time $T>0$ and a control such that, at the end of the control process, the particle reaches a point arbitrarily close to a given target, whereas the velocities of the fluid and of the point mass are driven exactly to zero. Therefore, within this simplified model, we can control simultaneously the fluid and the particle, by using inputs acting on the moving point only. Moreover, the main result holds without any smallness assumptions on the initial data. Alternatively, we can see our results as yielding controllability of the viscous Burgers equation by a moving internal boundary.

[A12] C. Castro, N. Cîndea, A. Münch. Controllability of the linear 1D-wave equation with inner moving forces. SIAM J. Control Optim., 52(6), 4027–4056, 2014.

This paper deals with the numerical computation of distributed null controls for the 1D wave equation. We consider supports of the controls that may vary with respect to the time variable. The goal is to compute approximations of such controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. Assuming a geometric optic condition on the support of the controls, we first prove a generalized observability inequality for the homogeneous wave equation. We then introduce and prove the well-posedness of a mixed formulation that characterizes the controls of minimal square-integrable norm. Such mixed formulation, introduced in [Cindea and Münch, A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations], and solved in the framework of the (space-time) finite element method, is particularly well-adapted to address the case of time dependent support. Several numerical experiments are discussed.

[A11] N. Cîndea, A. Imperiale, P. Moireau. Data assimilation of time under-sampled measurements using observers, the wave-like equation example. ESAIM: COCV 21 (2015) 635–669.

We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of the spatial domain and for some time samples (typically the sampling step in time is much larger than the time discretization step). To assimilate the available data, two strategies are proposed and analyzed. The first strategy consists in assimilating data only if they are available and the second one in assimilating interpolation of the available data at all the discretization times. In order to tackle the spurious high frequencies which appear when we discretize the wave equation, for both strategies, we introduce a numerical viscous term. In this case, we prove some error estimates between the exact solution and our observers. Numerical simulations illustrate the theoretical results in the case of the one dimensional wave equation.

[A10] N. Cîndea, A. Münch. A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations. Calcolo, September 2015, Volume 52, Issue 3, pp 245-288.

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea, Fernandez-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls : the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and his adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case.

[A9] N. Cîndea, E. Fernandez-Cara, A. Münch. Numerical controllability of the wave equation through primal method and Carleman estimates. ESAIM: COCV 19 (2013) 1076–1108.

This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute an approximation of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not use in this work duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and of the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.

[A8] N. Cîndea, S. Micu, J. Morais Pereira. Approximation of periodic solutions for a dissipative hyperbolic equation. Numerische Mathematik. Volume 124, Issue 3 (2013), Page 559-601.

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force f . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on f , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is addaed to the system. The same is true if the nonhomogeneous term f is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.

[A7] D. Chapelle, N. Cîndea, M. De Buhan, P. Moireau. Exponential Convergence of an Observer Based on Partial Field Measurements for the Wave Equation. Mathematical Problems in Engineering (MPE). Vol. 2012, Pages 1-12, 2012.

We analyze an observer strategy based on partial --i.e. in a subdomain -- measurements of the solution of a wave equation, in order to compensate for unknown initial conditions. We prove the exponential convergence of this observer under a non-standard observability condition, whereas using measurements of the time-derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlability. Nevertheless, we directly relate our specific condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.

[A6] D. Chapelle, N. Cîndea, P. Moireau. Improving convergence in numerical analysis using observers - The wave-like equation case. Mathematical Models and Methods in Applied Sciences (M3AS), Vol. 22, No. 12 (2012) 1250040 (35 pages).

We propose an observer-based approach to circumvent the issue of unbounded approximation errors -- with respect to the length of the time window considered -- in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addaition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability -- to be preserved through discretization. The latter property warrants exponential stability for the corrected dynamics, hence provides bounded approximation errors over time. Special care is needed to establish the required observability at the discrete level, in particular due to the fact that we focus on an original observer method adapted to measurements of the main variable, whereas measurements of the time-derivative -- admissible, of course, albeit less frequent in practical systems -- lead to a stability analysis in which existing results can be more directly applied. We also provide some detailed application examples with several such wave-like equations, and the corresponding numerical assessments illustrate the performance of our approach.

[A5] N. Cîndea, S. Micu, A. Pazoto. Periodic solutions for a weakly dissipated hybrid system. Journal of Mathematical Analysis and Applications, Volume 385, Issue 1, Pages 399-413, 2012.

We consider the motion of a stretched string coupled with a rigid body at one end and we study the existence of periodic solution when a periodic force $f$ acts on the body. The main difficulty of the study is related to the weak dissipation that characterizes this hybrid system, which does not ensure a uniform decay rate of the energy. Under addaitional regularity conditions on $f$, we use a perturbation argument in order to prove the existence of a periodic solution. In the last part of the paper we present some numerical simulations based on the theoretical results.

[A4] N. Cîndea, S. Micu, M. Tucsnak. An approximation method for exact controls of vibrating systems. SIAM J. Control Optim. 49, pp. 1283-1305, 2011.

We propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's stabilizability implies controllability" principle with the Galerkin's method. The main new feature brought in by this work consists in giving precise error estimates. In order to test the efficiency of the method, we consider two illustrative examples (with the finite element approximations of the wave and the beam equations) and we describe the corresponding simulations.

[A3] N. Cîndea, S. Micu, Roventa, M. Tucsnak. Controllability of a nonlinear hybrid system. An. Univ. Craiova Ser. Mat. Inform.. 38 (1), pp. 35-48, 2011.

In this paper we study a controllability problem for a simplified 1-d nonlinear system which models the self-propelled motion of a rigid body in a fluid located on the real axis. The control variable is the difference of the velocities of the fluid and the solid and depends only on time. The main result of the paper asserts that any final position and velocity of the rigid body can be reached by a suitable input function.

[A2] N. Cîndea, M. Tucsnak. Internal exact observability of a perturbed Euler-Bernoulli equation. Annals of the Academy of Romanian Scientists, Series on Mathematics and its Applications, Volume 2, no.2, 205-221, 2010.

In this work we prove that the exact internal observability for the Euler-Bernoulli equation is robust with respect to a class of linear perturbations. Our results yield, in particular, that for rectangular do- mains we have the exact observability in an arbitrarily small time and with an arbitrarily small observation region. The usual method of tack- ling lower order terms, using Carleman estimates, cannot be applied in this context. More precisely, it is not known if Carleman estimates hold for the evolution Euler-Bernoulli equation with arbitrarily small observation region. Therefore we use a method combining frequency domain techniques, a compactness-uniqueness argument and a Carleman estimate for elliptic problems.

[A1] N. Cîndea, M. Tucsnak. Local exact controllability for Berger plate equation. Mathematics of Control, Signals, and Systems (MCSS), Volume 21, Number 2, 93-110, 2009.

We study the exact controllability of a nonlinear plate equation by the means of a control which acts on an internal region of the plate. The main result asserts that this system is locally exactly controllable if the associated linear Euler-Bernoulli system is exactly controllable. In particular, for rectangular domains, we obtain that the Berger system is locally exactly controllable in arbitrarily small time and for every open and nonempty control region.

### Proceedings and book chapters

[P5] N. Cîndea, S. Micu, I. Roventa. Uniform Observability for a Finite Differences Discretization of a Clamped Beam Equation. 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016 — Bertinoro, Italy, 13—15 June 2016.

The aim of this paper is to prove a uniform observability inequality for a finite differences semi-discretization of a clamped beam equation. A discrete multiplier method is employed in order to obtain the uniform observability of the eigenvectors of the matrix driving the semi-discrete system, corresponding to eigenfrequencies smaller than a precise filtering threshold. This result can be generalized to the uniform observability of every filtered solution. Numerical simulations, concerning the dual controllability problem, illustrate the theoretical results.

[P4] I.F. Bugariu, N. Cîndea, S. Micu, I. Roventa. Controllability of the Space Semi-Discrete Approximation for the Beam Equation Control of partial differential equations, Proceedings of the 19th IFAC World Congress, 2014.

The aim of this work is to study of the numerical approximation of the controls for the hinged beam equation. A consequence of the numerical spurious high frequencies is the lack of the uniform controllability property of the semi-discrete model for the beam equation, in the classical setting. We solve this deficiency by adding a vanishing numerical viscosity term, which will damp out these high frequencies. An approximation algorithm based on the conjugate gradient method and some numerical experiments are presented.

[P3] N. Cîndea, S. Micu, I. Roventa, M. Tucsnak. Numerical Aspects and Controllability of a One Dimensional Fluid-Structure Model. Control of Systems Governed by Partial Differential Equations, 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations (2013).

In this paper we summarize some recent results from cite{Hib2013} concerning the controllability of a one dimensional fluid-structure model. These results are confirmed by numerical experiments in some particular cases. More precisely, we consider a simplified model for a point swimmer moving, according to Newton's law, in an one dimensional fluid which is modeled by the viscous Burgers equation. The control variable is the relative velocity of the swimmer with respect to the fluid. The main result of the paper gives the conditions in which we can drive the fluid and the velocity of the point mass to zero. A set of reachable positions of the point mass is also obtained. From the numerical point of view, we compute the $L^2$-minimal norm control for a linearized and simplified model. The method we used combine a finite elements discretization in space, a finite-difference centered scheme in time and the conjugate gradient method.

[P2] N. Cîndea, B. Fabrèges, F. de Gournay, C. Poignard. Optimal placement of electrodes in an electroporation process. ESAIM: Proceedings, Volume 30, 34-43, 2010.

Electroporation consists in increasing the permeability of a tissue by applying high voltage pulses. In this paper we discuss the question of optimal placement and optimal loading of electrodes such that electroporation holds only in a given open set of the domain. The electroporated set of the domain is where the norm of the electric field is above a given threshold value. We use a standard gradient algorithm to optimize the loading of the electrodes and shape sensitivity analysis and a gradient algorithm in order to move the electrodes. We also discuss the choice of objective functions to be chosen in the gradient algorithm.

[P1] N. Cîndea, M. Tucsnak. Fast and strongly localized observation for a perturbed plate equation. International Series of Numerical Mathematics, Vol. 158, 73-83, 2009, Birkhauser.

The aim of this work is to study the exact observability of a perturbed plate equation. A fast and strongly localized observation result was proven using a perturbation argument of an Euler-Bernoulli plate equation and a unique continuation result for bi-Laplacian.

### Articles in Magnetic Resonance in Medicine

[MR2] N. Cîndea, F. Odille, G. Bosser, J. Felblinger, P.-A. Vuissoz. Reconstruction from free-breathing cardiac MRI data using reproducing kernel Hilbert spaces. Magn Reson Med, 2010, 63 , 59-67.

This paper describes a rigorous framework for reconstructing MR images of the heart, acquired continuously over the cardiac and respiratory cycle. The framework generalizes existing techniques, commonly referred to as retrospective gating, and is based on the properties of reproducing kernel Hilbert spaces. The reconstruction problem is formulated as a moment problem in a multidimensional reproducing kernel Hilbert spaces (a two-dimensional space for cardiac and respiratory resolved imaging). Several reproducing kernel Hilbert spaces were tested and compared, including those corresponding to commonly used interpolation techniques (sinc-based and splines kernels) and a more specific kernel allowed by the framework (based on a first-order Sobolev RKHS). The Sobolev reproducing kernel Hilbert spaces was shown to allow improved reconstructions in both simulated and real data from healthy volunteers, acquired in free breathing.

[MR1] F. Odille, N. Cîndea, D. Mandry, C. Pasquier, P.-A. Vuissoz, J. Felblinger. Generalized MRI reconstruction including elastic physiological motion and coil sensitivity encoding. Magn Reson Med, 2008, 59, 1401-1411.

This article describes a general framework for multiple coil MRI reconstruction in the presence of elastic physiological motion. On the assumption that motion is known or can be predicted, it is shown that the reconstruction problem is equivalent to solving an integral equation--known in the literature as a Fredholm equation of the first kind--with a generalized kernel comprising Fourier and coil sensitivity encoding, modified by physiological motion information. Numerical solutions are found using an iterative linear system solver. The different steps in the numerical resolution are discussed, in particular it is shown how over-determination can be used to improve the conditioning of the generalized encoding operator. Practical implementation requires prior knowledge of displacement fields, so a model of patient motion is described which allows elastic displacements to be predicted from various input signals (e.g., respiratory belts, ECG, navigator echoes), after a free-breathing calibration scan. Practical implementation was demonstrated with a moving phantom setup and in two free-breathing healthy subjects, with images from the thoracic-abdominal region. Results show that the method effectively suppresses the motion blurring/ghosting artifacts, and that scan repetitions can be used as a source of over-determination to improve the reconstruction.

### Thèse de doctorat

Problèmes inverses et contrôlabilité avec applications en élasticité et IRM. Soutenue le 29 mars 2010 à IECN.

Le but de cette thèse est d'étudier, du point de vue théorique, la contrôlabilité exacte de certaines équations aux dérivées partielles qui modélisent les vibrations élastiques, et d'appliquer les résultats ainsi obtenus à la résolution des problèmes inverses provenant de l'imagerie par résonance magnétique (IRM).

Cette thèse comporte deux parties. La première partie, intitulée Contrôlabilité et observabilité de quelques équations des plaques'', discute la problématique de la contrôlabilité, respectivement de l'observabilité, de l'équation des plaques perturbées avec des termes linéaires ou non linéaires. Des résultats récents ont prouvé que l'observabilité exacte d'un système qui modélise les vibrations d'une structures élastique (équation des ondes ou des plaques) implique l'existence d'une solution du problème inverse de la récupération d'un terme source dans l'équation à partir de l'observation. Ainsi, dans le Chapitre 2 de cette thèse nous avons démontré l'observabilité interne exacte de l'équation des plaques perturbées par des termes linéaires d'ordre un et dans le Chapitre 3 la contrôlabilité exacte locale d'une équation des plaques non linéaire attribuée à Berger. Le Chapitre 4 introduit une méthode numérique pour l'approximation des contrôles exactes dans des systèmes d'ordre deux en temps.

La deuxième partie de la thèse est dédiée à l'imagerie par résonance magnétique. Plus précisément, on s'intéresse aux méthodes de reconstruction des images pour des objets en mouvement, l'exemple typique étant l'imagerie cardiaque en respiration libre. Dans le Chapitre 6, nous avons formulé la reconstruction d'images cardiaques acquises en respiration libre comme un problème des moments dans un espace de Hilbert à noyau reproductif. L'existence d'une solution pour un tel problème des moments est prouvée par des outils bien connus dans la théorie du contrôle. Nous avons validé cette méthode en utilisant des images simulées numériquement et les images de cinq volontaires sains.

La connexion entre les deux parties de la thèse est réalisée par le Chapitre 7 où l'on présente le problème inverse d'identification d'un terme source dans l'équation des ondes à partir d'une observation correspondante à un enregistrement IRM.

En conclusion, nous avons montré qu'on peut utiliser les outils de la théorie de contrôle pour des problèmes inverses provenant de l'IRM des objets en mouvement, à la condition de connaître l'équation du mouvement.

Dernière modification : 25 avril 2018.