GSOR method for the equality constrained least squares problems and the generalized least squares problems

In a recent paper [4], Li et al. gave a generalized successive overrelaxation (GSCR) method for the least squares problems. In this paper, we show that the GSOR method can be applied to the equality constrained least squares (LSE) problems and the generalized least squares (GLS) problems

A variant of the AOR method for augmented systems

A variant of the AOR method for augmented system

A variant of the AOR method for augmented systems

First published in Mathematics of Computation in 2012, published by the American Mathematical Society

Encoding correlations in open quantum systems

The study of open quantum systems (OQS) is a vast topic consisting of diverse approaches being, or having been, carried out in divers places and times. One of the themes of this work is to demonstrate the equivalence of a number of apparently different techniques in order to travel a small way along the path of unifying the study of OQS. We present the fixed mode method (FMM) for deriving mappings between linear Hamiltonians and show how it can be used to verify the pre-eminence of the spectral density in the study of open quantum systems. We utilise the FMM to derive the collective mode method (CMM) and show that it encompasses the reaction coordinate (RC) mapping, the chain mapping, Fano diagonalisation (FD), and (when combined with a bath combination (BC) mapping) some varieties of the technique of discretisation within its scope. We demonstrate the versatility of the CMM by applying it to the study of ground states of the spin boson model (SBM) and dynamics of fermionic systems. Utilising a discretised multipolaron ansatz (MP); and exact diagonalisation and master equation techniques respectively. Another theme is that of the computation. By their very nature OQS require large scale computations, owing to the need to characterise correlations between the systems of interest (S) and the (effectively infinite) wider world. This means that many problems are now beyond the scope of humanity's (inbuilt) computational capacity and rely on the use of digital processors. As such it becomes increasingly important to utilise the intuition of humans and the speed of machines in an efficient manner. Without due care we could end up either producing unmanagable quantities of poorly understood data; or spend an eternity dealing with the minutiae of a derivation. With this in mind we attempt to present our results in a form that they might be efficiently encoded. That is, along with the methods themselves, we offer suggestions as to how such methods could be efficiently implemented and how they might be useful in pedagogical and academic environments. Throughout this work we have made use of continuum representations of particulate environments (or baths), moving to discretised versions only when necessitated by the constraints of a particular problem or methodology. We hope to demonstrate how thinking in terms of the continuum can aid intuition and calculations. Finally, wherever possible, we have attempted to provide `proof of principal' for our assertions. We have applied the techniques we derive and investigate (FMM, CMM, MP) to particular models in order to provide a detailed (but far from exhaustive) demonstration of how such methods could be used. The aim being that intuition developed from the `toy models' we investigate can be applied more widely to experimentally motivated problems within the field.Open Acces