Mortensen Observer for a class of variational inequalities -Lost equivalence with stochastic filtering approaches

We address the problem of deterministic sequential estimation for a nonsmooth dynamics in R governed by a variational inequality, as illustrated by the Skorokhod problem with a reflective boundary condition at 0. For smooth dynamics, Mortensen introduced an energy for the likelihood that the state variable produces-up to perturbations disturbances-a given observation in a finite time interval, while reaching a given target state at the final time. The Mortensen observer is the minimiser of this energy. For dynamics given by a variational inequality and therefore not reversible in time, we study the definition of a Mortensen estimator. On the one hand, we address this problem by relaxing the boundary constraint of the synthetic variable and then proposing an approximated variant of the Mortensen estimator that uses the resulting nonlinear smooth dynamics. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. We prove a stability result that allows us to interpret the limiting solution as the value function associated with a control problem rather than an estimation problem. In contrast to the case of smooth dynamics, here the zero-noise limit of the robust form of the Zakai equation cannot be understood from the Bellman equation of the value function arising in Mortensen's deterministic estimation. This may unveil a violation of equivalence for non-reversible dynamics between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics

Thermodynamical framework for modeling chemo-mechanical coupling in muscle contraction – Formulation and preliminary results

International audienceWe propose a multiscale model of cardiac contraction in which the molecular motors at the origin of the contractile process are considered as multistable mechanical entities endowed with internal degrees of freedom of both mechanical and chemical nature. This model provides a thermodynamical basis for modeling the complex interplay of chemical and mechanical phenomena at the sub-cellular level. Important motivations for this work include the ability to represent the experimentally observed physiological characteristics of the contractile apparatus such as (i) the passive quick force recovery mechanism, (ii) the relation between the contraction velocity and the applied force and (iii) the so called Lymn-Taylor cycle describing the metabolism.Nous proposons un modèle multi-échelle de la contraction cardiaque dans lequel les moteurs moléculaires à l'origine du processus contractile sont représentés par des élé-ments mécaniques multistables paramétrés à la fois par des degrés de liberté géométriques et par des états chimiques. Ce modèle permet de poser les fondements thermody-namiques permettant de décrire l'interaction complexe entre les phénomènes mécaniques et chimiques a l'échelle sub-cellulaire. Ce travail a pour objet de représenter les car-actéristiques physiologiques du dispositif contractile observées expérimentalement et en particulier (i) le mécanisme passif de récupération rapide de force, (ii) la relation entre la vitesse de contraction et la charge appliquée et (iii) le cycle dit de Lymn-Taylor décrivant l'activité métabolique. Abstract : We propose a multiscale model of cardiac contraction in which the molecular motors at the origin of the contractile process are considered as multistable mechanical entities endowed with internal degrees of freedom of both mechanical and chemical nature. This model provides a thermodynamical basis for modeling the complex interplay of chemical and mechanical phenomena at the sub-cellular level. Important motivations for this work include the ability to represent the experimentally observed physiological characteristics of the contractile apparatus such as (i) the passive quick force recovery mechanism, (ii) the relation between the contraction velocity and the applied force and (iii) the so called Lymn-Taylor cycle describing the metabolism

Image-driven Stochastic Identification of Boundary Conditions for Predictive Simulation

International audienceIn computer-aided interventions, biomechanical models reconstructed from the pre-operative data are used via augmented reality to facilitate the intra-operative navigation. The predictive power of such models highly depends on the knowledge of boundary conditions. However , in the context of patient-specific modeling, neither the pre-operative nor the intra-operative modalities provide a reliable information about the location and mechanical properties of the organ attachments. We present a novel image-driven method for fast identification of boundary conditions which are modelled as stochastic parameters. The method employs the reduced-order unscented Kalman filter to transform in real-time the probability distributions of the parameters, given observations extracted from intra-operative images. The method is evaluated using synthetic, phantom and real data acquired in vivo on a porcine liver. A quantitative assessment is presented and it is shown that the method significantly increases the predictive power of the biomechanical model

Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator

We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results

Combining data assimilation and machine learning to build data-driven models for unknown long time dynamics—Applications in cardiovascular modeling

We propose a method to discover differential equations describing the long-term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast-scale. Our methodology is based on a synergetic combination of data assimilation (DA), used to estimate the parameters associated with the known fast-scale dynamics, and machine learning (ML), used to infer the laws underlying the slow-scale dynamics. Specifically, by exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden. Then, we propose a ML algorithm that learns a parametric mathematical model from a collection of time series coming from the phenomenon to be modeled. Moreover, we study the interpretability of the data-driven models obtained within the black-box learning framework proposed in this paper. In particular, we show that every model can be rewritten in infinitely many different equivalent ways, thus making intrinsically ill-posed the problem of learning a parametric differential equation starting from time series. Hence, we propose a strategy that allows to select a unique representative model in each equivalence class, thus enhancing the interpretability of the results. We demonstrate the effectiveness and noise-robustness of the proposed methods through several test cases, in which we reconstruct several differential models starting from time series generated through the models themselves. Finally, we show the results obtained for a test case in the cardiovascular modeling context, which sheds light on a promising field of application of the proposed methods