Thermoacoustic tomography with an arbitrary elliptic operator

Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stability estimates are obatined for an arbitrary variable principal part of that operator. Convergence of the Quasi-Reversibility Method to the exact solution is also established for this case. Both complete and incomplete data collection cases are considered.Comment: 16 page

The geometry of nonlinear least squares with applications to sloppy models and optimization

Parameter estimation by nonlinear least squares minimization is a common problem with an elegant geometric interpretation: the possible parameter values of a model induce a manifold in the space of data predictions. The minimization problem is then to find the point on the manifold closest to the data. We show that the model manifolds of a large class of models, known as sloppy models, have many universal features; they are characterized by a geometric series of widths, extrinsic curvatures, and parameter-effects curvatures. A number of common difficulties in optimizing least squares problems are due to this common structure. First, algorithms tend to run into the boundaries of the model manifold, causing parameters to diverge or become unphysical. We introduce the model graph as an extension of the model manifold to remedy this problem. We argue that appropriate priors can remove the boundaries and improve convergence rates. We show that typical fits will have many evaporated parameters. Second, bare model parameters are usually ill-suited to describing model behavior; cost contours in parameter space tend to form hierarchies of plateaus and canyons. Geometrically, we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter-effects curvature on the manifold. Using coordinates based on geodesic motion, these narrow canyons are transformed in many cases into a single quadratic, isotropic basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting algorithms as an Euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively. By adding a geodesic acceleration adjustment to these algorithms, we alleviate the difficulties from parameter-effects curvature, improving both efficiency and success rates at finding good fits.Comment: 40 pages, 29 Figure

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization

We address the inverse problem of cosmic large-scale structure reconstruction from a Bayesian perspective. For a linear data model, a number of known and novel reconstruction schemes, which differ in terms of the underlying signal prior, data likelihood, and numerical inverse extra-regularization schemes are derived and classified. The Bayesian methodology presented in this paper tries to unify and extend the following methods: Wiener-filtering, Tikhonov regularization, Ridge regression, Maximum Entropy, and inverse regularization techniques. The inverse techniques considered here are the asymptotic regularization, the Jacobi, Steepest Descent, Newton-Raphson, Landweber-Fridman, and both linear and non-linear Krylov methods based on Fletcher-Reeves, Polak-Ribiere, and Hestenes-Stiefel Conjugate Gradients. The structures of the up-to-date highest-performing algorithms are presented, based on an operator scheme, which permits one to exploit the power of fast Fourier transforms. Using such an implementation of the generalized Wiener-filter in the novel ARGO-software package, the different numerical schemes are benchmarked with 1-, 2-, and 3-dimensional problems including structured white and Poissonian noise, data windowing and blurring effects. A novel numerical Krylov scheme is shown to be superior in terms of performance and fidelity. These fast inverse methods ultimately will enable the application of sampling techniques to explore complex joint posterior distributions. We outline how the space of the dark-matter density field, the peculiar velocity field, and the power spectrum can jointly be investigated by a Gibbs-sampling process. Such a method can be applied for the redshift distortions correction of the observed galaxies and for time-reversal reconstructions of the initial density field.Comment: 40 pages, 11 figure

The Iteratively Regularized Gau{\ss}-Newton Method with Convex Constraints and Applications in 4Pi-Microscopy

This paper is concerned with the numerical solution of nonlinear ill-posed operator equations involving convex constraints. We study a Newton-type method which consists in applying linear Tikhonov regularization with convex constraints to the Newton equations in each iteration step. Convergence of this iterative regularization method is analyzed if both the operator and the right hand side are given with errors and all error levels tend to zero. Our study has been motivated by the joint estimation of object and phase in 4Pi microscopy, which leads to a semi-blind deconvolution problem with nonnegativity constraints. The performance of the proposed algorithm is illustrated both for simulated and for three-dimensional experimental data