Inverse problem

Summary

An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects. Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe. They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, slope stability analysis and many other fiel

Official source

https://en.wikipedia.org/wiki/Inverse_problem

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The course provides an introduction to scientific computing. Several numerical methods are presented for the computer solution of mathematical problems arising in different applications. The software MATLAB is used to solve the problems and verify the theoretical properties of the numerical methods.

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L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à appliquer cette méthode à des cas simples et à l'exploiter pour résoudre les problèmes de la pratique.

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Inverse identification of infarcted areas inside the cardiac muscle through an optimal control formulation for cardiac electrophysiology equations

Seif Ben Bader

The project deals with the finite element approximation of the monodomain equation, which models the propagation of the electrical potential in the cardiac muscle through a coupled system of PDE’s. The goal is to recover the shape of an infarcted area inside the cardiac muscle, by measuring the electrical potential on the boundary of the domain representing a portion of tissue. An optimal control formulation of the inverse problem will be analyzed, by considering infarcted areas described by few parameters (e.g width and position). To reach the description of the state problem through a monodomain model for the transmembrane potential, coupled with a system of ODEs describing the ionic model, some steps will be performed by considering first a linear time-dependent state problem; then a nonlinear time-independent state problem; finally, a nonlinear coupled state problem.

2015

LinvPy : a Python package for linear inverse problems

Guillaume François Paul Beaud

The goal of this project is to make a Python package including the tau-estimator algorithm to solve linear inverse problems. The package must be distributed, well documented, easy to use and easy to extend for future developers.

2016

Coupled inverse modeling of a controlled irrigation experiment using multiple hydro-geophysical data

Damiano Pasetto, Matteo Rossi

Geophysical surveys can provide useful, albeit indirect, information on vadose zone processes. However, the ability to provide a quantitative description of the subsurface hydrological phenomena requires to fully integrate geophysical data into hydrological modeling. Here, we describe a controlled infiltration experiment that was monitored using both electrical resistivity tomography (ERT) and ground-penetrating radar (GPR). The experimental site has a simple, well-characterized subsoil structure: the vadose zone is composed of aeolic sand with largely homogeneous and isotropic properties. In order to estimate the unknown soil hydraulic conductivity, we apply a data assimilation technique based on a sequential importance resampling (SIR) approach. The SIR approach allows a simple assimilation of either or both geophysical datasets taking into account the associated measurement uncertainties. We demonstrate that, compared to a simpler, uncoupled hydro-geophysical approach, the coupled data assimilation process provides a more reliable parameter estimation and better reproduces the evolution of the infiltrating water plume. The coupled procedure is indeed much superior to the uncoupled approach that suffers from the artifacts of the geophysical inversion step and produces severe mass balance errors. The combined assimilation of GPR and ERT data is then investigated, highlighting strengths and weaknesses of the two datasets. In the case at hand GPR energy propagates in form of a guided wave that, over time, shows different energy distribution between propagation modes as a consequence of the evolving thickness of the wet layer. We found that the GPR inversion procedure may produce estimates on the depth of the infiltrating front that are not as informative as the ERT dataset.

2015