Search results for "Inverse problem"

Inverse problems for $p$-Laplace type equations under monotonicity assumptions

2016

We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $\sigma_1 \geq \sigma_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.

Joint Gaussian processes for inverse modeling

2017

Solving inverse problems is central in geosciences and remote sensing. Very often a mechanistic physical model of the system exists that solves the forward problem. Inverting the implied radiative transfer model (RTM) equations numerically implies, however, challenging and computationally demanding problems. Statistical models tackle the inverse problem and predict the biophysical parameter of interest from radiance data, exploiting either in situ data or simulated data from an RTM. We introduce a novel nonlinear and nonparametric statistical inversion model which incorporates both real observations and RTM-simulated data. The proposed Joint Gaussian Process (JGP) provides a solid framework…

A rigidity theorem for Lagrangian deformations

2005

We consider deformations of singular Lagrangian varieties in symplectic manifolds. We prove that a Lagrangian deformation of a Lagrangian complete intersection is analytically rigid provided that this is the case infinitesimally. This result is given as a consequence of the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations.

2020

Abstract This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schrodinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity equation is put into relation with a long jump random walk with weights.

Recent progress in electrical impedance tomography

2003

We consider the inverse problem of finding cavities within some body from electrostatic measurements on the boundary. By a cavity we understand any object with a different electrical conductivity from the background material of the body. We survey two algorithms for solving this inverse problem, namely the factorization method and a MUSIC-type algorithm. In particular, we present a number of numerical results to highlight the potential and the limitations of these two methods.

The factorization method for real elliptic problems

2006

The Factorization Method localizes inclusions inside a body from mea- surements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering sym- metric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfi…

A regularized Newton method for locating thin tubular conductivity inhomogeneities

2011

We consider the inverse problem of determining the position and shape of a thin tubular object, such as for instance a wire, a thin channel or a curve-like crack, embedded in some three-dimensional homogeneous body from a single measurement of electrostatic currents and potentials on the boundary of the body. Using an asymptotic model describing perturbations of electrostatic potentials caused by such thin objects, we reformulate the inverse problem as a nonlinear operator equation. We establish Frechet differentiability of the corresponding operator, compute its Frechet derivative and set up a regularized Newton scheme to solve the inverse problem numerically. We discuss our implementation…

The Calderón Problem for a Space-Time Fractional Parabolic Equation

2020

In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded...

Jacobian of solutions to the conductivity equation in limited view

2022

Abstract The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the bounda…

The Calderón problem for the fractional Schrödinger equation

2020

We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calder\'on problem.