Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and it is based on the Fr\'echet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.Comment: 3 figure

Equality and Justification: the Original Position Reconsidered

The original position has an elegance and power beyond most philosophical pictures. It has captured the attention of readers across the world through many generations of students, and is famous well beyond philosophical circles. Yet, as renowned as the original position has become, it is also typically misrepresented and misunderstood. In particular, John Rawls’ method of reasoning behind the veil of ignorance is frequently presented as drawing a conclusion mandated by rational choice theory. My aim, in this brief note, is to clarify the main purpose of the original position and to articulate its main defining features in contrast to this dominant misreading

On the reachable set for the one-dimensional heat equation

International audienceThe goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions

The Morozov's principle applied to data assimilation problems

This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation

Family and Marriage: Institutions and the Need for Social Goods

Institutions, if unjust, ought to be reformed or even abolished. This radical Rawlsian thought leads to the question of whether the family ought to be abolished, given its negative impact on the very possibility of delivering equality of life chances. In this article, we address questions regarding the justice of the family, and of marriage, and reflect on rights, equality, and the provision of social goods by institutions. There is a temptation to justify our social institutions in terms which highlight their universal accessibility and benefits. But we may best understand the claim of some of our most important institutions where we recognize that they are forms of social good which may legitimately benefit some without having to benefit all. Their abolition is unjustified where there is sufficient value in them given our collective needs that it is unreasonable for some to refuse the means to maintain and promote these goods

Stability estimates for Navier-Stokes equations and application to inverse problems

International audienceIn this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method

Faut-il abandonner la position originelle ?

La position originelle a une élégance et un attrait qui vont au-delà de la plupart des figures philosophiques. Il s’agit peut-être de l’aspect de la théorie rawlsienne qui ait le plus fasciné les lecteurs de Théorie de la justice. Bien que très étudiée, réinterprétée et critiquée, la position originelle est souvent présentée de façon erronée. En particulier, la méthode de raisonnement de John Rawls derrière le voile d’ignorance est souvent comprise comme si les conclusions qu’elle induit se situaient dans le seul cadre du choix rationnel. Il y a donc différentes interprétations de la position originelle, correspondant à des perspectives plus ou moins proches d’une perspective éthique dans le cadre du choix rationnel, que ce soit le conséquentialisme ou l’utilitarisme. Mon but dans cet article est de clarifier le but de la position originelle rawlsienne. J’explique la différence entre une interprétation qui part du choix rationnel et des préférences des individus pour définir ce qui est exigé par la justice distributive et une interprétation qui met au contraire l’accent sur la justification raisonnable. Je montre que même si nous renonçons à l’ambition d’arriver à des principes de justice dans la seule perspective du choix rationnel, nous devons retenir plutôt qu’abandonner la position originelle. Je conclus en montrant que la principale différence entre l’utilitarisme et la théorie rawlsienne réside non dans la distribution que ces deux théories recommandent, mais dans la stratégie de justification qu’elles adoptent. // The original position has an elegance and power beyond most philosophical pictures. It has captured the attention of readers across the world through many generations of students and is famous well beyond philosophical circles. Yet, as renowned as the original position has become, it has also been typically misrepresented and misunderstood. In particular, John Rawls’ method of reasoning behind the veil of ignorance is frequently presented as drawing a conclusion mandated by rational choice theory. There are, therefore, several interpretations of the original position, which correspond to perspectives closer or more removed from an ethical outlook framed through rational choice, be that consequentialism or utilitarianism. My aim in this article is to clarify the main purpose of a Rawlsian original position. I articulate the differences between on the one hand, a perspective which focuses on rational choice and on preferences of individuals to define what justice requires, and, on the other, one which stresses reasonable justification to each individual. I argue that even if we abandon the ambition to derive principles of justice from the sole perspective of rational choice, we must retain rather than relinquish the original position. I conclude by showing that the main difference between utilitarianism and Rawlsian theory are not in the distribution they recommend, but in the strategy of justification adopted