Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Magic for the dead? The archaeology of magic in later medieval burials

THIS PAPER EXAMINES patterns in the placement of apotropaic objects and materials in high- to late-medieval burials in Britain (11th to 15th centuries). It develops an interdisciplinary classification to identify: (1) healing charms and protective amulets; (2) objects perceived to have occult natural power; (3) 'antique' items that were treated as possessing occult power; and (4) rare practices that may have been associated with the demonic magic of divination or sorcery. Making comparisons with amulets deposited in conversion-period graves of the 7th to 9th centuries it is argued that the placement of amulets with the dead was strategic to Christian belief, intended to transform or protect the corpse. The conclusion is that material traces of magic in later medieval graves have a connection to folk magic, performed by women in the care of their families, and drawing on knowledge of earlier traditions. This popular magic was integrated with Christian concerns and tolerated by local clergy, and was perhaps meant to heal or reconstitute the corpse, to ensure its reanimation on judgement day, and to protect the vulnerable dead on their journey through purgatory.

The Dark Monarch: magic and modernity in British art

This major curated exhibition, publication and events builds on Rowlands’ curatorial research. Working in collaboration with co-curators Martin Clark, Artistic Director, Tate St Ives and Michael Bracewell, cultural historian, the exhibition sought to explore new narratives within British art. The innovative curatorial methodology developed from a fiction found in the infamous novel, The Dark Monarch by Sven Berlin, Gallery Press 1962. The research sought specific archival and collection work that allowed thematic strands to emerge that represented influences across generations. The exhibition features two-hundred artworks, from the Tate Collection, archives and other significant British public and private collections. It examines the development of early Modernism, in the UK, as well as the reappearance of esoteric and arcane references in a significant strand of contemporary art practice. Historical works from Samuel Palmer, Graham Sutherland, Henry Moore and Paul Nash are shown alongside contemporary artists including Derek Jarman, Cerith Wyn Evans, Eva Rothschild, Linder and John Russell. The exhibition includes a key work by Damien Hirst ¬ the first time he has been shown at Tate St Ives and a number of contemporary commissions. The Dark Monarch publication extended the discourse of the research critically examining the tension between progressive modernity and romantic knowledge, the book focuses on the way that artworks are encoded with various histories - geological, mythical and magical. Essays examine magic as a counterpoint to modernity’s transparency and rational progress, but also draw out the links modernity has with notions such as fetishism, mana, totem, and the taboo. Often viewed as counter to Modernism, this collection of essays suggest that these products of illusion and delusion in fact belong to modernity. Drawing together 15 different writers commissioned to explore magic as a counterpoint of liberal understanding of modernity, drawing out links that modernity has with notions of fetish, taboo and occult philosophy. Including essays by Marina Warner, Ilsa Colsell, Philip Hoare, Chris Stephens, Jennifer Higgie and Morrissey.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

The method of least squares used to invert an orbit problem

Six parameters uniquely describe the orbit of a body about the Sun. Given these parameters, it is possible to make predictions of the body's position by solving its equation of motion. The parameters cannot be directly measured, so they must be inferred indirectly by an inversion method which uses measurements of other quantities in combination with the equation of motion. Inverse techniques are valuable tools in many applications where only noisy, incomplete, and indirect observations are available for estimating parameter values. The methodology of the approach is introduced and the Kepler problem is used as a real-world example. (C) 2003 American Association of Physics Teachers.

The method of predicate least squares refinement

Parameters to be determined in a least squares refinement calculation to fit a set of observed data may sometimes usefully be `predicated' to values obtained from some independent source, such as a theoretical calculation. An algorithm for achieving this in a least squares refinement calculation is described, which leaves the operator in full control of the weight that he may wish to attach to the predicate values of the parameters.