The question of using the available measurements to retrieve mathematical models characteristics (parameters, boundary conditions, initial conditions) is a key aspect of the modeling objective in biology or medicine. In a stochastic/statistical framework this question is seen as an estimation problems. From a deterministic point of view, we classical talk about inverse problems as we recover classical model inputs from outputs. When considering evolution problems,this question falls in the realm of data assimilation that can be seen from a deterministic of statistical point of view. Our objective in this course is to introduce the mathematical principles and numerical aspects behind data assimilation strategies with an emphasis on the deterministic formalism allowing to understand why data assimilation is a specific inverse problem. Our presentation will include considerations on finite dimensional problems but also on infinite dimensional problems such as the ones arising from PDE models. And we will illustrate the course with numerous examples coming from cardiovascular applications and biology.[-]

​We consider photoacoustic tomography in the presence of approximation and modelling errors. The inverse problem, i.e. estimation of the initial pressure from photoacoustic time-series measured on the boundary of the target, is approached in the framework of Bayesian inverse problems. The posterior distribution is examined in situations in which the forward model contains errors or uncertainties for example due to numerical approximations or uncertainties in the acoustic parameters. Modelling of these errors and its impact on the posterior distribution are investigated.
This is joint work with Teemu Sahlstrm, Jenni Tick and Aki Pulkkinen.[-]

​Multiwave aspects of thermoacoustic imaging and range verification during particle therapy will be discussed.
Thermoacoustic images are generated from acoustic pulses induced by heating due to lossy electromagnetic wave propagation. Quantitative thermoacoustic imaging is feasible when the electric field pattern can be accurately modeled throughout the imaging field of view and delivered quickly enough to ensure stress confinement.
Therapeutic ions slow from relativistic speeds to a dead stop within nanoseconds, generating extraordinarily high temperature and pressure spikes within a thermal core of nanometer diameter along their tracks.
Possibilities for utilizing these phenomena to verify the ion beam location within the patient will be considered.[-]

We consider the inverse problem of recovering an unknown parameter from a finite set of indirect measurements. We start with reviewing the formulation of the Bayesian approach to inverse problems. In this approach the data and the unknown parameter are modelled as random variables, the distribution of the data is given and the unknown is assumed to be drawn from a given prior distribution. The solution, called the posterior distribution, is the probability distribution of the unknown given the data, obtained through the Bayes rule. We will talk about the conditions under which this formulation leads to well-posedness of the inverse problem at the level of probability distributions. We then discuss the connection of the Bayesian approach to inverse problems with the variational regularization. This will also help us to study the properties of the modes of the posterior distribution as point estimators for the unknown parameter. We will also briefly talk about the Markov chain Monte Carlo methods in this context.[-]

This is a survey talk about the Boundary Control method. The method originates from the work by Belishev in 1987. He developed the method to solve the inverse boundary value problem for the acoustic wave equation with an isotropic sound speed. The method has proven to be very versatile and it has been applied to various inverse problems for hyperbolic partial differential equations. We review recent results based on the method and explain how a geometric version of method works in the case of the wave equation for the Laplace-Beltrami operator on a compact Riemannian manifold with boundary.[-]

The question of using the available measurements to retrieve mathematical models characteristics (parameters, boundary conditions, initial conditions) is a key aspect of the modeling objective in biology or medicine. In a stochastic/statistical framework this question is seen as an estimation problems. From a deterministic point of view, we classical talk about inverse problems as we recover classical model inputs from outputs. When considering evolution problems,this question falls in the realm of data assimilation that can be seen from a deterministic of statistical point of view. Our objective in this course is to introduce the mathematical principles and numerical aspects behind data assimilation strategies with an emphasis on the deterministic formalism allowing to understand why data assimilation is a specific inverse problem. Our presentation will include considerations on finite dimensional problems but also on infinite dimensional problems such as the ones arising from PDE models. And we will illustrate the course with numerous examples coming from cardiovascular applications and biology.[-]

The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. These lead to a coefficient identification problem for a quasilinear wave equation. Another characteristic property of ultrasound propagating in human tissue is frequency power law attenuation leading to fractional derivative damping models in time domain. In this talk we will first of all dwell on modeling of nonlinearity on one hand and fractional damping on the other hand. Then we will discuss the linear inverse problem of photoacoustic tomography with fractional damping. Finally some first results on nonlinearity parameter imaging are shown.[-]

The determination of the shape of an obstacle from its effects on known acoustic waves is an important problem in many technologies such as sonar, geophysical exploration and medical imaging. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because of its ill-posed and nonlinear nature. Its investigation requires the understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods. The main goal of this work is the development of an efficient procedure for retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics of the surface of the obstacle. We propose a solution methodology based on a regularized Newton-type method. The solution of the considered IOP by the proposed iterative method incurs, at each iteration, the solution of a linear system whose entries are the Fréchet derivatives of the elasto-acoustic field with respect to the shape parameters. We prove that these derivatives are solutions of the same direct elasto-acoustic scattering problem that differs only in the transmission conditions on the surface of the scatterer. Furthermore, the computational efficiency of the IOP solver depends mainly on the computational efficiency of the solution of the forward problems that arise at each Newton iteration. We propose to solve the direct scattering-type problems using a finite-element method based on discontinuous Galerkin approximations equipped with curved element boundaries. Numerical results will be presented to illustrate the salient features of this computational methodology and highlight its performance characteristics.

acoustics - shape derivative - inverse obstacle problem - Fréchet derivatives - inverse elasto-acoustic scattering problems[-]