11 research outputs found
Quantitative thermo-acoustic imaging: An exact reconstruction formula
This paper aims to mathematically advance the field of quantitative
thermo-acoustic imaging. Given several electromagnetic data sets, we establish
for the first time an analytical formula for reconstructing the absorption
coefficient from thermal energy measurements. Since the formula involves
derivatives of the given data up to the third order, it is unstable in the
sense that small measurement noises may cause large errors. However, in the
presence of measurement noise, the obtained formula, together with a noise
regularization technique, provides a good initial guess for the true absorption
coefficient. We finally correct the errors by deriving a reconstruction formula
based on the least square solution of an optimal control problem and prove that
this optimization step reduces the errors occurring and enhances the
resolution
New Stability Estimates for the Inverse Medium Problem with Internal Data
A major problem in solving multi-waves inverse problems is the presence of
critical points where the collected data completely vanishes. The set of these
critical points depend on the choice of the boundary conditions, and can be
directly determined from the data itself. To our knowledge, in the most
existing stability results, the boundary conditions are assumed to be close to
a set of CGO solutions where the critical points can be avoided. We establish
in the present work new weighted stability estimates for an electro-acoustic
inverse problem without assumptions on the presence of critical points. These
results show that the Lipschitz stability far from the critical points
deteriorates near these points to a logarithmic stability
Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation
The aim of this paper is to propose for the first time a reconstruction
scheme and a stability result for recovering from acoustic-optic data
absorption distributions with bounded variation. The paper extends earlier
results on smooth absorption distributions. It opens a door for a mathematical
and numerical framework for imaging, from internal data, parameter
distributions with high contrast in biological tissues
On Multiple Frequency Power Density Measurements
We shall give a priori conditions on the illuminations such that the
solutions to the Helmholtz equation in \Omega,
on , and their gradients satisfy certain non-zero
and linear independence properties inside the domain \Omega, provided that a
finite number of frequencies k are chosen in a fixed range. These conditions
are independent of the coefficients, in contrast to the illuminations
classically constructed by means of complex geometric optics solutions. This
theory finds applications in several hybrid problems, where unknown parameters
have to be imaged from internal power density measurements. As an example, we
discuss the microwave imaging by ultrasound deformation technique, for which we
prove new reconstruction formulae.Comment: 26 pages, 4 figure
Large Scale Inverse Problems
This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences