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

On the use of marker data to determine the kinetics of the digestive behaviour of feeds

A model of the transport process that follows the progress of digesta successively through the small intestine of a monogastric is investigated. The process is multi-phase and multi-constituent, as described in detail by Bastianelli et al. [J. Anim. Sci., 74:1873–1887, 1996]. The model describes the movement of marker substances that are used to obtain data on the interactions between the intestinal sections and digesta with differing levels of soluble fibre. A multi-stage process is modelled by a set of coupled first order linear differential equations. Solutions of steady and initial value problems provide information on the transfer rates of the processes. Properties of the solutions as functions of system parameters are examined. References M. Renton, J. Hanan and K. Burrage, Using the canonical modelling approach to simplify the simulation of function in functional-structural plant models. New Phytologist, 166:845–857, 2005. doi:10.1111/j.1469-8137.2005.01330.x D. Bastianelli, D. Sauvant and A. Rerat, Mathematical modeling of digestion and nutrient absorption in pigs. J. Animal Science, 74:1873–1887, 1996. http://www.journalofanimalscience.org/content/74/8/1873.abstract R. G. Lentle and P. W. M. Janssen, Manipulating Digestion with Foods designed to Change the Physical Characteristics of digesta. Critical Reviews in Food Science and Nutrition, 50:130–145, 2010. doi:10.1080/10408390802248726 J. France, J. H. M. Thornley, M. S. Dhanoa and R. C. Siddons, On the mathematics of digesta flow kinetics. Journal of Theoretical Biology, 113:743–758, 1985. doi:10.1016/S0022-5193(85)80191-0 A. Mazanov and J. V. Nolan, Simulation of the dynamics of nitrogen metabolism in sheep. British Journal of Nutrition, 35:149–174, 1976. doi:10.1079/BJN19760017 A. Mazanov, Stability of Multi-pool Models with Lags. Journal of Theoretical Biology, 59:429–442, 1976. doi:10.1016/0022-5193(76)90181-

Development of mathematical pathways for vet students to articulate to related higher education courses

Australia needs more qualified professionals in the areas of engineering, education, health and other sciences. The national focus on widening participation in higher education (HE) includes strengthening pathways from vocational education and training (VET). VET students often lack the mathematics skills necessary to articulate successfully to their chosen degrees. Current approaches such as bridging and foundation mathematics programs, and university in-degree support, are fragmented and not tailored or sufficiently contextualised for VET articulants. Flexible approaches are needed that enable institutions to assess the numeracy skills of VET articulants and provide resources and support to build their mathematical skills and confidence. This project is developing a series of mathematics pathways designed to improve the readiness of VET qualified students for higher education study in the areas of engineering, education and health science. Year 1 of this project focuses on engineering and education. The main VET qualifications and HE education courses have been identified and mapping the mathematical gap in knowledge between the two is underway. Mathematical pathways will be delivered as Open Education Resources and designed to be delivered flexibly. This presentation will review the progress on the mathematical pathway development and review the gaps that exist between the two sectors

Remote Sensing of Environment: Current status of Landsat program, science, and applications

Formal planning and development of what became the first Landsat satellite commenced over 50 years ago in 1967. Now, having collected earth observation data for well over four decades since the 1972 launch of Landsat- 1, the Landsat program is increasingly complex and vibrant. Critical programmatic elements are ensuring the continuity of high quality measurements for scientific and operational investigations, including ground systems, acquisition planning, data archiving and management, and provision of analysis ready data products. Free and open access to archival and new imagery has resulted in a myriad of innovative applications and novel scientific insights. The planning of future compatible satellites in the Landsat series, which maintain continuity while incorporating technological advancements, has resulted in an increased operational use of Landsat data. Governments and international agencies, among others, can now build an expectation of Landsat data into a given operational data stream. International programs and conventions (e.g., deforestation monitoring, climate change mitigation) are empowered by access to systematically collected and calibrated data with expected future continuity further contributing to the existing multi-decadal record. The increased breadth and depth of Landsat science and applications have accelerated following the launch of Landsat-8, with significant improvements in data quality. Herein, we describe the programmatic developments and institutional context for the Landsat program and the unique ability of Landsat to meet the needs of national and international programs. We then present the key trends in Landsat science that underpin many of the recent scientific and application developments and followup with more detailed thematically organized summaries. The historical context offered by archival imagery combined with new imagery allows for the development of time series algorithms that can produce information on trends and dynamics. Landsat-8 has figured prominently in these recent developments, as has the improved understanding and calibration of historical data. Following the communication of the state of Landsat science, an outlook for future launches and envisioned programmatic developments are presented. Increased linkages between satellite programs are also made possible through an expectation of future mission continuity, such as developing a virtual constellation with Sentinel-2. Successful science and applications developments create a positive feedback loop—justifying and encouraging current and future programmatic support for Landsat

Boundary treatment for virtual leaf surfaces

When working on fitting leaf surfaces for use with virtual plant models [Room et al. 1996] we encountered the unsatisfactory situation of receiving a smooth surface model that is bounded by a piecewise linear curve. To smooth the boundary, we fit a parametric piecewise cubic curve through the boundary data points and extend the surface to the new boundary curve. This method will aid us in the representation of leaf surfaces with arbitrary boundaries in future

Chebyshev Series Approximations for the Bessel Function Y n (z) of Complex Argument

We employ the truncated Chebyshev series to approximate the Bessel function of the second kind Yn (z) for jzj 8. Detailed manipulations and discussions for Y 0 (z) and Y 1 (z) are given. Results of numerical experiments are presented to demonstrate the computed accuracy by using the Chebyshev series approximation. Advantages and disadvantages of the Chebyshev series approximation compared with other polynomial approximation methods, e.g., the tau-method approximations, are discussed. Key words: Complex Bessel functions, polynomial approximations, Chebyshev series approximations, tau-method approximations. 1 Introduction Bessel functions of the first kind Jn (z) and the second kind Yn (z) of integer order play an important role in mathematical physics and engineering sciences. Numerical methods for efficiently computing these functions, based on polynomial or rational approximations, on the real line as well as in the complex plane are therefore of interests to computational physicist..

Computational strategies for surface fitting using thin plate spline finite element methods

Thin plate spline finite element methods are used to fit a surface to an irregularly scattered dataset [S. Roberts, M. Hegland, and I. Altas. Approximation of a Thin Plate Spline Smoother using Continuous Piecewise Polynomial Functions. SIAM, 1:208--234, 2003]. The computational bottleneck for this algorithm is the solution of large, ill-conditioned systems of linear equations at each step of a generalised cross validation algorithm. Preconditioning techniques are investigated to accelerate the convergence of the solution of these systems using Krylov subspace methods. The preconditioners under consideration are block diagonal, block triangular and constraint preconditioners [M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1--137, 2005]. The effectiveness of each of these preconditioners is examined on a sample dataset taken from a known surface. From our numerical investigation, constraint preconditioners appear to provide improved convergence for this surface fitting problem compared to block preconditioners

Modelling water droplet movement on a leaf surface

Modelling droplet movement on leaf surfaces is an important component in understanding how water, pesticide or nutrient is absorbed through the leaf surface. A simple mathematical model is proposed in this paper for generating a realistic, or natural looking trajectory of a water droplet traversing a virtual leaf surface. The virtual surface is comprised of a triangular mesh structure over which a hybrid Clough-Tocher seamed element interpolant is constructed from real-life scattered data captured by a laser scanner. The motion of the droplet is assumed to be affected by gravitational, frictional and surface resistance forces and the innovation of our approach is the use of thin-film theory to develop a stopping criterion for the droplet as it moves on the surface. The droplet model is verified and calibrated using experimental measurement; the results are promising and appear to capture reality quite well

On derivative estimation and the solution of least squares problems

Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation. In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding