Inverse problems in classical and quantum physics

The subject of this thesis is in the area of Applied Mathematics known as Inverse Problems. Inverse problems are those where a set of measured data is analysed in order to get as much information as possible on a model which is assumed to represent a system in the real world. We study two inverse problems in the fields of classical and quantum physics: QCD condensates from tau-decay data and the inverse conductivity problem. Despite a concentrated effort by physicists extending over many years, an understanding of QCD from first principles continues to be elusive. Fortunately, data continues to appear which provide a rather direct probe of the inner workings of the strong interactions. We use a functional method which allows us to extract within rather general assumptions phenomenological parameters of QCD (the condensates) from a comparison of the time-like experimental data with asymptotic space-like results from theory. The price to be paid for the generality of assumptions is relatively large errors in the values of the extracted parameters. Although we do not claim that our method is superior to other approaches, we hope that our results lend additional confidence to the numerical results obtained with the help of methods based on QCD sum rules. EIT is a technology developed to image the electrical conductivity distribution of a conductive medium. The technique works by performing simultaneous measurements of direct or alternating electric currents and voltages on the boundary of an object. These are the data used by an image reconstruction algorithm to determine the electrical conductivity distribution within the object. In this thesis, two approaches of EIT image reconstruction are proposed. The first is based on reformulating the inverse problem in terms of integral equations. This method uses only a single set of measurements for the reconstruction. The second approach is an algorithm based on linearisation which uses more then one set of measurements. A promising result is that one can qualitatively reconstruct the conductivity inside the cross-section of a human chest. Even though the human volunteer is neither two-dimensional nor circular, such reconstructions can be useful in medical applications: monitoring for lung problems such as accumulating fluid or a collapsed lung and noninvasive monitoring of heart function and blood flow.

Higher-order molecular properties and excitation energies in single-reference and multireference coupled-cluster theory

Coupled-cluster (CC) theory is one of the most successful approaches in high-accuracy quantum chemistry. The present thesis makes a number of contributions to the determination of molecular properties and excitation energies within the CC framework. The multireference CC (MRCC) method proposed by Mukherjee and coworkers (Mk-MRCC) has been benchmarked within the singles and doubles approximation (Mk-MRCCSD) for molecular equilibrium structures. It is demonstrated that Mk-MRCCSD yields reliable results for multireference cases where single-reference CC methods fail. At the same time, the present work also illustrates that Mk-MRCC still suffers from a number of theoretical problems and sometimes gives rise to results of unsatisfactory accuracy. To determine polarizability tensors and excitation spectra in the MRCC framework, the Mk-MRCC linear-response function has been derived together with the corresponding linear-response equations. Pilot applications show that Mk-MRCC linear-response theory suffers from a severe problem when applied to the calculation of dynamic properties and excitation energies: The Mk-MRCC sufficiency conditions give rise to a redundancy in the Mk-MRCC Jacobian matrix, which entails an artificial splitting of certain excited states. This finding has established a new paradigm in MRCC theory, namely that a convincing method should not only yield accurate energies, but ought to allow for the reliable calculation of dynamic properties as well. In the context of single-reference CC theory, an analytic expression for the dipole Hessian matrix, a third-order quantity relevant to infrared spectroscopy, has been derived and implemented within the CC singles and doubles approximation. The advantages of analytic derivatives over numerical differentiation schemes are demonstrated in some pilot applications.