Beyond Relativism? Re-engaging Wittgenstein

Relativism is the view that there are as many worlds as there are ways of thinking and expressing the worlds that are expressed. That is to say, things are related to the ways in which we express them. Thus philosophers assert that the way we express our thoughts in language even affects the way we perceive the world. Relativism is a reaction against the view that there is one and only one way of describing the world. Therefore, relativists argue that the different conceptual abilities and habits are liable to result in different ways of seeing the world

Cauchy problem for Ultrasound Modulated EIT

Ultrasound modulation of electrical or optical properties of materials offers the possibility to devise hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient $\sigma(x)$ for $x\in X$, a bounded domain in \Rm^n, from knowledge of $\sigma(x)|\nabla u|^2(x)$ for $x\in X$, where $u$ is the solution to the elliptic equation $-\nabla\cdot\sigma\nabla u=0$ in $X$ with $u=f$ on $\partial X$. This inverse problem may be recast as a nonlinear equation, which formally takes the form of a 0-Laplacian. Whereas $p-$Laplacians with $p>1$ are well-studied variational elliptic non-linear equations, $p=1$ is a limiting case with a convex but not strictly convex functional, and the case $p<1$ admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the 0-Laplacian with full Cauchy data at the domain's boundary, which results in a, formally overdetermined, nonlinear hyperbolic equation. The paper presents existence, uniqueness, and stability results for the Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient $\sigma(x)$ can be stably reconstructed only on a subset of $X$ described as the domain of influence of the space-like part of the boundary $\partial X$ for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.Comment: 26 pages, 6 figure

Time-dependent angularly averaged inverse transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from knowledge of angularly averaged measurements performed at the boundary of a domain of interest. We show that the absorption coefficient and the spatial component of the scattering coefficient are uniquely determined by such measurements. We obtain stability results on the reconstruction of the absorption and scattering parameters with respect to the measured albedo operator. The stability results are obtained by a precise decomposition of the measurements into components with different singular behavior in the time domain

Photo-acoustic tomography in a rotating setting

Photo-acoustic tomography is a coupled-physics (hybrid) medical imaging modality that aims to reconstruct optical parameters in biological tissues from ultrasound measurements. As propagating light gets partially absorbed, the resulting thermal expansion generates minute ultrasonic signals (the photo-acoustic effect) that are measured at the boundary of a domain of interest. Standard inversion procedures first reconstruct the source of radiation by an inverse ultrasound (boundary) problem and second describe the optical parameters from internal information obtained in the first step. This paper considers the rotating experimental setting. Light emission and ultrasound measurements are fixed on a rotating gantry, resulting in a rotation-dependent source of ultrasound. The two-step procedure we just mentioned does not apply. Instead, we propose an inversion that directly aims to reconstruct the optical parameters quantitatively. The mapping from the unknown (absorption and diffusion) coefficients to the ultrasound measurement via the unknown ultrasound source is modeled as a composition of a pseudo-differential operator and a Fourier integral operator. We show that for appropriate choices of optical illuminations, the above composition is an elliptic Fourier integral operator. Under the assumption that the coefficients are unknown on a sufficiently small domain, we derive from this a (global) injectivity result (measurements uniquely characterize our coefficients) combined with an optimal stability estimate. The latter is the same as that obtained in the standard (non-rotating experimental) setting

Fluctuations of Parabolic Equations with Large Random Potentials

In this paper, we present a fluctuation analysis of a type of parabolic equations with large, highly oscillatory, random potentials around the homogenization limit. With a Feynman-Kac representation, the Kipnis-Varadhan's method, and a quantitative martingale central limit theorem, we derive the asymptotic distribution of the rescaled error between heterogeneous and homogenized solutions under different assumptions in dimension $d\geq 3$. The results depend highly on whether a stationary corrector exits.Comment: 44 pages; reorganized the structure and extended the results; to appear in SPDE: Analysis and Computation

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained by the authors for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence less computationally intensive, algorithms