Historical development of the BFGS secant method and its characterization properties

The BFGS secant method is the preferred secant method for finite-dimensional unconstrained optimization. The first part of this research consists of recounting the historical development of secant methods in general and the BFGS secant method in particular. Many people believe that the secant method arose from Newton's method using finite difference approximations to the derivative. We compile historical evidence revealing that a special case of the secant method predated Newton's method by more than 3000 years. We trace the evolution of secant methods from 18th-century B.C. Babylonian clay tablets and the Egyptian Rhind Papyrus. Modifications to Newton's method yielding secant methods are discussed and methods we believe influenced and led to the construction of the BFGS secant method are explored. In the second part of our research, we examine the construction of several rank-two secant update classes that had not received much recognition in the literature. Our study of the underlying mathematical principles and characterizations inherent in the updates classes led to theorems and their proofs concerning secant updates. One class of symmetric rank-two updates that we investigate is the Dennis class. We demonstrate how it can be derived from the general rank-one update formula in a purely algebraic manner not utilizing Powell's method of iterated projections as Dennis did it. The literature abounds with update classes; we show how some are related and show containment when possible. We derive the general formula that could be used to represent all symmetric rank-two secant updates. From this, particular parameter choices yielding well-known updates and update classes are presented. We include two derivations of the Davidon class and prove that it is a maximal class. We detail known characterization properties of the BFGS secant method and describe new characterizations of several secant update classes known to contain the BFGS update. Included is a formal proof of the conjecture made by Schnabel in his 1977 Ph.D. thesis that the BFGS update is in some asymptotic sense the average of the DFP update and the Greenstadt update

A light-scattering study of the kinetic size distribution changes of aerosols

Imperial Users onl

Novel linear and nonlinear optical signal processing for ultra-high bandwidth communications

The thesis is articulated around the theme of ultra-wide bandwidth single channel signals. It focuses on the two main topics of transmission and processing of information by techniques compatible with high baudrates. The processing schemes introduced combine new linear and nonlinear optical platforms such as Fourier-domain programmable optical processors and chalcogenide chip waveguides, as well as the concept of neural network. Transmission of data is considered in the context of medium distance links of Optical Time Division Multiplexed (OTDM) data subject to environmental fluctuations. We experimentally demonstrate simultaneous compensation of differential group delay and multiple orders of dispersion at symbol rates of 640 Gbaud and 1.28 Tbaud. Signal processing at high bandwidth is envisaged both in the case of elementary post-transmission analog error mitigation and in the broader field of optical computing for high level operations (“optical processor”). A key innovation is the introduction of a novel four-wave mixing scheme implementing a dot-product operation between wavelength multiplexed channels. In particular, it is demonstrated for low-latency hash-key based all-optical error detection in links encoded with advanced modulation formats. Finally, the work presents groundbreaking concepts for compact implementation of an optical neural network as a programmable multi-purpose processor. The experimental architecture can implement neural networks with several nodes on a single optical nonlinear transfer function implementing functions such as analog-to-digital conversion. The particularity of the thesis is the new approaches to optical signal processing that potentially enable high level operations using simple optical hardware and limited cascading of components

Dynamical effects and disorder in ultracold bosonic matter

In this thesis, various aspects on the theoretical description of ultracold bosonic atoms in optical lattices are investigated. After giving a brief introduction to the fundamental concepts of BECs, atomic physics, interatomic interactions and experimental procedures in chapter (1), we derive the Bose-Hubbard model from first principles in chapter (2). In this chapter, we also introduce and discuss a technique to efficiently determine Wannier states, which, in contrast to current techniques, can also be extended to inhomogeneous systems. This technique is later extended to higher dimensional, non-separable lattices in chapter (5). The many-body physics and phases of the Bose-Hubbard is shortly presented in chapter (3) in conjunction with Gutzwiller mean-field theory, and the recently devised projection operator approach. We then return to the derivation of an improved microscopic many-body Hamiltonian, which contains higher band contributions in the presence of interactions in chapter (4). We then move on to many-particle theory. To demonstrate the conceptual relations required in the following chapter, we derive Bogoliubov theory in chapter (5.3.4) in three different ways and discuss the connections. Furthermore, this derivation goes beyond the usual version discussed in most textbooks and papers, as it accounts for the fact, that the quasi-particle Hamiltonian is not diagonalizable in the condensate and the eigenvectors have to be completed by additional vectors to form a basis. This leads to a qualitatively different quasi-particle Hamiltonian and more intricate transformation relations as a result. In the following two chapters (7, 8), we derive an extended quasi-particle theory, which goes beyond Bogoliubov theory and is not restricted to weak interactions or a large condensate fraction. This quasi-particle theory naturally contains additional modes, such as the amplitude mode in the strongly interacting condensate. Bragg spectroscopy, a momentum-resolved spectroscopic technique, is introduced and used for the first experimental detection of the amplitude mode at finite quasi-momentum in chapter (9). The closely related lattice modulation spectroscopy is discussed in chapter (10). The results of a time-dependent simulation agree with experimental data, suggesting that also the amplitude mode, and not the sound mode, was probed in these experiments. In chapter (11) the dynamics of strongly interacting bosons far from equilibrium in inhomogeneous potentials is explored. We introduce a procedure that, in conjunction with the collapse and revival of the condensate, can be used to create exotic condensates, while particularly focusing on the case of a quadratic trapping potential. Finally, in chapter (12), we turn towards the physics of disordered systems derive and discuss in detail the stochastic mean-field theory for the disordered Bose-Hubbard model

Development of Efficient Intensity Based Registration Techniques for Multi-modal Brain Images

Recent advances in medical imaging have resulted in the development of many imaging techniques that capture various aspects of the patients anatomy and metabolism. These are accomplished with image registration: the task of transforming images on a common anatomical coordinate space. Image registration is one of the important task for multi-modal brain images, which has paramount importance in clinical diagnosis, leads to treatment of brain diseases. In many other applications, image registration characterizes anatomical variability, to detect changes in disease state over time, and by mapping functional information into anatomical space. This thesis is focused to explore intensity-based registration techniques to accomplish precise information with accurate transformation for multi-modal brain images. In this view, we addressed mainly three important issues of image registration both in the rigid and non-rigid framework, i.e. i) information theoretic based similarity measure for alignment measurement, ii) free form deformation (FFD) based transformation, and iii) evolutionary technique based optimization of the cost function. Mutual information (MI) is a widely used information theoretic similarity measure criterion for multi-modal brain image registration. MI only dense the quantitative aspects of information based on the probability of events. For rustication of the information of events, qualitative aspect i.e. utility or saliency is a necessitate factor for consideration. In this work, a novel similarity measure is proposed, which incorporates the utility information into mutual Information, known as Enhanced Mutual Information(EMI).It is found that the maximum information gain using EMI is higher as compared to that of other state of arts. The utility or saliency employed in EMI is a scale invariant parameter, and hence it may fail to register in case of projective and perspective transformations. To overcome this bottleneck, salient region (SR) based Enhance Mutual Information (SR-EMI)is proposed, a new similarity measure for robust and accurate registration. The proposed SR-EMI based registration technique is robust to register the multi-modal brain images at a faster rate with better alignment

Reducing the expense of electronic structure calculations for larger molecules : optimized auxiliary basis sets, and system-specifically reparametrized semiempirical methods

Optimization approaches using several global and local algorithms (genetic algorithms, direct search, simplex and implicit filtering) in the search for a global minimum are applied to optimize auxiliary basis sets for quantum chemistry ab-initio calculations and to reparametrize semiempirical methods. We optimize auxiliary basis sets for RI-MP2 and RI-HF, by minimizing a suitable difference measure to the analogous calculations without the RI technique. It is shown that our methods of generating optimal auxiliary basis sets are more systematic and can be automatized more easily than the traditional approach. Hence, they can reasonably be expected to be faster and more reliable. At the same time, the quality of our basis sets is at least as good as that from the traditional approach. As an application, we present the first systematically optimized and complete set of mixed Poisson and density auxiliary basis sets for the atoms H, B, C, N, O and F, complementing the standard basis sets cc-pVXZ (X = D, T, Q and 5). As soon as efficient integral routines for this new basis function type become available, calculations with them will be much more efficient than with traditional basis sets. Similarly, these global and local optimization methods are also employed to reparametrize semiempirical methods for a difficult double proton transfer system. System-specific reparametrization of the well-known AM1, PM3 and PM5 methods is done by minimizing the error of the semiempirical calculations compared to ab-initio reference data at the MP2/aug-cc-pVDZ level. This is done at a small set of selected geometries, leading to one- and two-dimensional potential energy surfaces that are quantitatively in agreement with the ab-initio data over a much broader range of geometries. With this system-specific adaption, these reparametrized methods lead to results far superior to those obtainable with standard parameters. Nevertheless, the full speed advantage of the semiempirical approach is retained, offering the possibility to do direct dynamics studies with the potential energy surface calculated on the fly at ab-initio quality but at a fraction of the ab-initio cost. In both cases, our combination of genetic algorithm global search and Powell local search is the fastest and most robust choice for optimization, comparing with the other methods. Therefore, in these cases, a combination of global and local search is actually better than a purely local algorithm

Novel geometry gradient coils for MRI designed by genetic algorithm

This thesis concerns the design of gradient coils for magnetic resonance imaging systems. The method of design by genetic algorithm optimisation is applied to novel gradient geometries both by use of conventional computer facilities, and, by parallelisation of the design algorithm, on a supercomputer architecture. Geometries and regions of interests which are inaccessible to analytic solution are considered, and the criteria which are difficult to include in such algorithms, such as the robustness of the design, are also included. To exemplify this, in the first instance a two axis biplanar coil was designed and the performance of the genetic algorithm tested and evaluated. The coil was tested computationally; a working example was constructed and tested in a MRI scanner both on phantom objects and on a human knee. Consideration of the usefulness of the coil regions not optimised for linearity for image reconstruction was done. The gradient efficiencies of the final designs in the z and y directions respectively were 0.3 mTm-1A-1 and 0.4 mTm-1A-1 over a 15 cm diameter region of interest. The size of the interior of the gradient set was designed to be 40.0 cm x 24.4 cm x 40.0 cm, to fit within the confines of the bore of an existing scanner. The linearity in the primary direction over the region of optimisation was less than 5% for both coils. The algorithm was extended for operation on a Hitachi SR2201 supercomputer using parallelisation. The performance in this mode was evaluated and found to be favourable in comparison with the standard computer architecture, with an increase in speed in real time of a factor of-!llore than 40 in some configurations of the supercomputer. Various polygonal cross-section design shapes requiring the use of this improved computer performance were optimised and evaluated computationally. Such designs have previously been inaccessible to the genetic algorithm optimisation model. Tests were made between the performance of the genetic algorithm on various similar design problems, and simulated images from such gradient coils were produced. Finally an example of a transverse coaxial return path gradient coil is presented computationally. This coil had an internal diameter of 32 cm, a d external diameter of 44 cm and a length of 40 cm. It achieved a strength of 0.1 mTm- 1A -1 over a cylinder of diameter 20 cm and length 25 cm, with a deviation from linearity of less than 5% over this volume.British Heart Foundatio