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

Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

Correct quantum chemistry in a minimal basis from effective Hamiltonians

We describe how to create ab-initio effective Hamiltonians that qualitatively describe correct chemistry even when used with a minimal basis. The Hamiltonians are obtained by folding correlation down from a large parent basis into a small, or minimal, target basis, using the machinery of canonical transformations. We demonstrate the quality of these effective Hamiltonians to correctly capture a wide range of excited states in water, nitrogen, and ethylene, and to describe ground and excited state bond-breaking in nitrogen and the chromium dimer, all in small or minimal basis sets

Syndromics: A Bioinformatics Approach for Neurotrauma Research

Substantial scientific progress has been made in the past 50 years in delineating many of the biological mechanisms involved in the primary and secondary injuries following trauma to the spinal cord and brain. These advances have highlighted numerous potential therapeutic approaches that may help restore function after injury. Despite these advances, bench-to-bedside translation has remained elusive. Translational testing of novel therapies requires standardized measures of function for comparison across different laboratories, paradigms, and species. Although numerous functional assessments have been developed in animal models, it remains unclear how to best integrate this information to describe the complete translational “syndrome” produced by neurotrauma. The present paper describes a multivariate statistical framework for integrating diverse neurotrauma data and reviews the few papers to date that have taken an information-intensive approach for basic neurotrauma research. We argue that these papers can be described as the seminal works of a new field that we call “syndromics”, which aim to apply informatics tools to disease models to characterize the full set of mechanistic inter-relationships from multi-scale data. In the future, centralized databases of raw neurotrauma data will enable better syndromic approaches and aid future translational research, leading to more efficient testing regimens and more clinically relevant findings