A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

Least-squares congealing for large numbers of images

In this paper we pursue the task of aligning an ensemble of images in an unsupervised manner. This task has been commonly referred to as “congealing” in literature. A form of congealing, using a least-squares criteria, has been recently demonstrated to have desirable properties over conventional congealing. Least-squares congealing can be viewed as an extension of the Lucas & Kanade (LK)image alignment algorithm. It is well understood that the alignment performance for the LK algorithm, when aligning a single image with another, is theoretically and empirically equivalent for additive and compositional warps. In this paper we: (i) demonstrate that this equivalence does not hold for the extended case of congealing, (ii) characterize the inherent drawbacks associated with least-squares congealing when dealing with large numbers of images, and (iii) propose a novel method for circumventing these limitations through the application of an inverse-compositional strategy that maintains the attractive properties of the original method while being able to handle very large numbers of images.

A least squares based finite volume method for the Cahn-Hilliard and Cahn-Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn-Hilliard (CH) and recently proposed Cahn-Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.

Orthogonality defect and reduced search-space size for solving integer least-squares problems

In the context of ambiguity resolution (AR) of Global Navigation Satellite Systems (GNSS), decorrelation among entries of an ambiguity vector, integer ambiguity search and ambiguity validations are three standard procedures for solving integer least-squares problems. This paper contributes to AR issues from three aspects. Firstly, the orthogonality defect is introduced as a new measure of the performance of ambiguity decorrelation methods, and compared with the decorrelation number and with the condition number which are currently used as the judging criterion to measure the correlation of ambiguity variance-covariance matrix. Numerically, the orthogonality defect demonstrates slightly better performance as a measure of the correlation between decorrelation impact and computational efficiency than the condition number measure. Secondly, the paper examines the relationship of the decorrelation number, the condition number, the orthogonality defect and the size of the ambiguity search space with the ambiguity search candidates and search nodes. The size of the ambiguity search space can be properly estimated if the ambiguity matrix is decorrelated well, which is shown to be a significant parameter in the ambiguity search progress. Thirdly, a new ambiguity resolution scheme is proposed to improve ambiguity search efficiency through the control of the size of the ambiguity search space. The new AR scheme combines the LAMBDA search and validation procedures together, which results in a much smaller size of the search space and higher computational efficiency while retaining the same AR validation outcomes. In fact, the new scheme can deal with the case there are only one candidate, while the existing search methods require at least two candidates. If there are more than one candidate, the new scheme turns to the usual ratio-test procedure. Experimental results indicate that this combined method can indeed improve ambiguity search efficiency for both the single constellation and dual constellations respectively, showing the potential for processing high dimension integer parameters in multi-GNSS environment.

Uncertainty analysis of pollutant build-up modelling based on a Bayesian Weighted Least Squares approach

Reliable pollutant build-up prediction plays a critical role in the accuracy of urban stormwater quality modelling outcomes. However, water quality data collection is resource demanding compared to streamflow data monitoring, where a greater quantity of data is generally available. Consequently, available water quality data sets span only relatively short time scales unlike water quantity data. Therefore, the ability to take due consideration of the variability associated with pollutant processes and natural phenomena is constrained. This in turn gives rise to uncertainty in the modelling outcomes as research has shown that pollutant loadings on catchment surfaces and rainfall within an area can vary considerably over space and time scales. Therefore, the assessment of model uncertainty is an essential element of informed decision making in urban stormwater management. This paper presents the application of a range of regression approaches such as ordinary least squares regression, weighted least squares Regression and Bayesian Weighted Least Squares Regression for the estimation of uncertainty associated with pollutant build-up prediction using limited data sets. The study outcomes confirmed that the use of ordinary least squares regression with fixed model inputs and limited observational data may not provide realistic estimates. The stochastic nature of the dependent and independent variables need to be taken into consideration in pollutant build-up prediction. It was found that the use of the Bayesian approach along with the Monte Carlo simulation technique provides a powerful tool, which attempts to make the best use of the available knowledge in the prediction and thereby presents a practical solution to counteract the limitations which are otherwise imposed on water quality modelling.

Mathematical models for describing the shape of the in vitro unstretched human crystalline lens

We developed orthogonal least-squares techniques for fitting crystalline lens shapes, and used the bootstrap method to determine uncertainties associated with the estimated vertex radii of curvature and asphericities of five different models. Three existing models were investigated including one that uses two separate conics for the anterior and posterior surfaces, and two whole lens models based on a modulated hyperbolic cosine function and on a generalized conic function. Two new models were proposed including one that uses two interdependent conics and a polynomial based whole lens model. The models were used to describe the in vitro shape for a data set of twenty human lenses with ages 7–82 years. The two-conic-surface model (7 mm zone diameter) and the interdependent surfaces model had significantly lower merit functions than the other three models for the data set, indicating that most likely they can describe human lens shape over a wide age range better than the other models (although with the two-conic-surfaces model being unable to describe the lens equatorial region). Considerable differences were found between some models regarding estimates of radii of curvature and surface asphericities. The hyperbolic cosine model and the new polynomial based whole lens model had the best precision in determining the radii of curvature and surface asphericities across the five considered models. Most models found significant increase in anterior, but not posterior, radius of curvature with age. Most models found a wide scatter of asphericities, but with the asphericities usually being positive and not significantly related to age. As the interdependent surfaces model had lower merit function than three whole lens models, there is further scope to develop an accurate model of the complete shape of human lenses of all ages. The results highlight the continued difficulty in selecting an appropriate model for the crystalline lens shape.

Extended GNSS ambiguity resolution models with regularization criterion and constraints

This paper firstly presents an extended ambiguity resolution model that deals with an ill-posed problem and constraints among the estimated parameters. In the extended model, the regularization criterion is used instead of the traditional least squares in order to estimate the float ambiguities better. The existing models can be derived from the general model. Secondly, the paper examines the existing ambiguity searching methods from four aspects: exclusion of nuisance integer candidates based on the available integer constraints; integer rounding; integer bootstrapping and integer least squares estimations. Finally, this paper systematically addresses the similarities and differences between the generalized TCAR and decorrelation methods from both theoretical and practical aspects.