Data assimilation for 2nd order hyperbolic systems. Formulation, numerical analysis and parameter identification in a unified observer approach

Data assimilation originally developed for weather forecasting has reached today new applications in life sciences where various types of measurements can be con- sidered to handle the numerous uncertainties in the modeled system. In particular, the heart mechanical system observed through medical imaging is a perfect example of the need of adapted estimation methods for diagnosis or prognosis. 
Here we focus on a more theoretical ground with a general class of 2nd order hyperbolic system - the elasticity system or more simply the wave system - where part of the solution is measured in a subdomain.  We demonstrate - both mathematically and numerically - how the use of a sequential data assimilation strategy through the design of an observer can produce simulations with better global approximation capabilities than classical numerical approximations. When coupled to a parameter identification strategy, a complete joint state and parameter observer can be formulated ultimately leading to diagnosis applications in biomedical applications such as cardiac infarct detection.