Sparsity regularization for inverse problems with non-trivial nullspaces

We study a weighted $\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter $\alpha$ tends to zero. Furthermore, for positive values of $\alpha$, we show that the regularized inverse solution equals the true source multiplied by a scalar $\gamma$, where $\gamma = 1 - c\alpha$. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by PDE-constrained optimization problems arising in connection with ECG and EEG recordings, but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems

Reconstruction of a volumetric source domain

Inverse and ill-posed problems which consist of reconstructing the unknown support of a three-dimensional volumetric source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz equations. In the case of constant physical properties, the solutions of these elliptic PDEs are sought as linear combinations of fundamental solutions, as in the method of fundamental solutions (MFS). The unknown source domain is parametrized by the radial coordinate, as a function of the spherical angles. The resulting least-squares functional estimating the gap between the measured and the computed data is regularized and minimized using the lsqnonlin toolbox routine in Matlab. Numerical results are presented and discussed for both exact and noisy data, confirming the accuracy and stability of reconstruction

Detecting an immersed body in a fluid. Stability and reconstruction

We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the stationary Stokes or Navier Stokes equations, from a single measurement of force and velocity on a portion of the boundary. Under appropriate a priori hypotheses we obtain an estimate of stability of log-log type for both cases. We then present a numerical method for the reconstruction of the body using a boundary elements representation of the solutions, combined with the iteratively regularized Gauss-Newton method, and present some partial numerical results in this direction

On Inverse Problems for Characteristic Sources in Helmholtz Equations

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented

Geometry identification and data enhancement for distributed flow measurements

The measurement of fluid motion is an important tool for researchers in fluiddynamics. Measurements with increasing precision did expedite the development of fluid-dynamic models and their theoretical understanding. Several well-established experimental techniques provide point-wise information on the flow field. In recent years novel measurement modalities have been investigated which deliver spatially resolved three-dimensional velocity measurements. Note that for methods such as particle tracking and tomographic particle imaging optical access to the flow domain is necessary. For other methods like magnetic resonance velocimetry, CT-angiography, or x-ray velocimetry this is, however, not the case. Such a property and also the fact that those methods are able to provide three-dimensional velocity fields in a rather short acquisition time makes them in particular suited for in-vivo applications. Our work is motivated by such non-invasive velocity measurement techniques for which no optical access to the interior of the geometry is needed and also not available in many cases. Here, an additional difficulty is that the exact flow geometry is in general not known a priori. The measurement techniques we are interested in, are extensions of already available medical imaging modalities. As a prototypical example, we consider magnetic resonance velocimetry, which is also suited for the measurement of turbulent fluid motion. We will also discuss computational examples using such measurement data. General purpose. Our main goal is a suitable post-processing of the available velocity data and also to obtain additional information. The measurements available from magnetic resonance velocimetry consist of several components given on a fixed field of view. The magnitude of the MRT signal corresponds to a proton density and thus e.g. the density of water molecules. Those data typically give a clear indication of the position and size of the flow geometry. The velocity data, on the other hand, are substantially perturbed outside the flow domain. This is a typical feature of measurements stemming from magnetic resonance velocimetry. Note that the surrounding noise usually has a notably higher magnitude than the actual measurements. Thus, a first necessary step will be to somehow separate the domain containing valuable velocity data from the noise surrounding it. For this reason, we apply some kind of image segmentation where we make use of the given den-sity image. Since the velocity values are given on the same field of view the segmentation directly transfers to those data. Due to the measurement procedure also the segmented velocity data are contaminated by measurement errors. Therefore, besides segmentation, additional post-processing is necessary in order to make the flow measurements available for further usage. In a second step, we propose a problem adapted data enhancement method which is able to provide a smoothed velocity field on the one hand, and also provides additional information on the other hand, like for instance the pressure drop or an estimate for the wall shear stress. The two main steps will therefore be: (i) The identification of the flow geometry, where we make use of the available density measurements. (ii) The denoising and improvement of the segmented velocity data, by using a suitable fluid-dynamical model. Outline. In part I of this thesis, we introduce our basic approach to the geom- etry identification and velocity enhancement problems described above. Both problems are formulated as optimal control problems governed by a partial differential equation and we shortly discuss some general aspects of the analysis and the solution of such problems in section 4. In part II, we thoroughly discuss and analyze the geometry identification problem introduced in section 2. The procedure is formulated as an inverse ill-posed problem and we propose a Tikhonov regularization for its stable solution. We show that the resulting optimal control problem has a solution and discuss its numerical treatment with iterative methods. Finally, a systematic discretization can be realized using finite elements which is also demonstrated by numerical tests. The velocity enhancement problem is introduced in part III. We propose a linearized flow-model which directly incorporates the available measurements. The resulting modeling error can be quantified in terms of the data error. The reconstruction method is then formulated as an optimal control problem subject to the linearized equations. We show the existence of a unique solution and derive estimates for the reconstruction error. Additionally, a reconstruction for the pressure is obtained for which we derive similar error estimates. We discuss the systematic discretization using finite elements and show preliminary computational examples for the verification of the derived estimates. In order to verify the applicability of the proposed methods to realistic data, we consider an application using experimental data in part IV. We use measurements of a human blood vessel stemming from magnetic resonance velocimetry obtained at the University Medical Center in Freiburg. After a suitable pre-processing of the available data, we apply the geometry identification method in order to obtain a discretization of the blood vessel. Using the generated mesh, we reconstruct an enhanced velocity field and the pressure from the available velocity data

IDENTIFICATION OF A CORE FROM BOUNDARY DATA

Abstract. The problem of determining the interface separating regions of constant density from boundary data of a solution of the corresponding potential equation is considered. An equivalent formulation as a nonlinear integral equation is obtained. Fourier methods are used to analyze and implement the problem. Numerical experiments based on a regularized least-squares method are presented