A Leslie matrix approach to an age-structured epidemic

We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples of the population of susceptibles, infectives, and removals for different birth rates and two different infection rates. Finally, when the infection rate depe

A novel method to lecture and to provide assessment feedback to mathematics students

BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedback. Feedback is critical to inform students if they are on track to achieve the course learning goals. According to Sadler (1989) useful feedback should provide evidence of learning that fills a gap between what is understood and what is aimed to be understood. In mathematics, the most effective form of assessment is weekly assignments to learn how to write mathematics. This is a difficult skill that requires a lot of practice and guidance. It is not sufficient to write down the correct answer or to give a list of calculations without adequate explanation. Students must employ precise use of words, formulae, symbols and punctuation. It is essential for students to receive feedback not only on the correctness of their answers, but also on their writing abilities in mathematics. AIMS The main aim of this project is to improve student learning of mathematics by promoting changes in mathematics assessment and teaching methods through the adoption of electronic pens and tablets. This strategy allows (i) the online submission and marking of assignments which gives students constructive feedback on their assessment. (ii) lecturers to work out proofs and examples in class on a step-by-step fashion that is also very effective for lecture recording. This is aimed at facilitating students’ flexible learning making it easier to study mathematics in their own time wherever they are: at home, in a library or on public transport. APPROACH We have endeavored to leverage changes in mathematics assessment and teaching methods by adopting electronic pens and tablets to allow online submission, electronic assignments’ marking and delivery of lectures. We have purchased a number of Samsung tablets equipped with a Wacom e-pen for the teaching staff of our first year mathematics service and honours courses (total enrolment of 450 students). CONCLUSIONS Lecturers have found that the use of e-pens on tablet is a very powerful method to deliver lectures that also allows for easy recording. However, lecturers and tutors have found that whilst providing written feedback to students once the assignments are marked results in an excellent learning outcome, marking assignments on tablets is still not as easy as marking them on paper, probably due to limitations in the technology. This has resulted in delays in returning marked assignments, and thus feedback on learning, to students. REFERENCES Sadler, D.R. (1989). Formative assessment and the design of instructional systems. Instructional Science,18,144

Adaptive discrete thin plate spline smoother

The discrete thin plate spline smoother fits smooth surfaces to large data sets efficiently. It combines the favourable properties of the finite element surface fitting and thin plate splines. The efficiency of its finite element grid is improved by adaptive refinement, which adapts the precision of the solution. It reduces computational costs by refining only in sensitive regions, which are identified using error indicators. While many error indicators have been developed for the finite element method, they may not work for the discrete smoother. In this article we show three error indicators adapted from the finite element method for the discrete smoother. A numerical experiment is provided to evaluate their performance in producing efficient finite element grids. References F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pat. Anal. Mach. Int. 11.6 (1989), pp. 567–585. doi: 10.1109/34.24792. C. Chen and Y. Li. A robust method of thin plate spline and its application to DEM construction. Comput. Geosci. 48 (2012), pp. 9–16. doi: 10.1016/j.cageo.2012.05.018. L. Fang. Error estimation and adaptive refinement of finite element thin plate spline. PhD thesis. The Australian National University. http://hdl.handle.net/1885/237742. L. Fang. Error indicators and adaptive refinement of the discrete thin plate spline smoother. ANZIAM J. 60 (2018), pp. 33–51. doi: 10.21914/anziamj.v60i0.14061. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun. Stat. Simul. Comput. 19.2 (1990), pp. 433–450. doi: 10.1080/0361091900881286. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15.4 (1989), pp. 326–347. doi: 10.1145/76909.76912. R. F. Reiniger and C. K. Ross. A method of interpolation with application to oceanographic data. Deep Sea Res. Oceanographic Abs. 15.2 (1968), pp. 185–193. doi: 10.1016/0011-7471(68)90040-5. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal. 41.1 (2003), pp. 208–234. doi: 10.1137/S0036142901383296. D. Ruprecht and H. Muller. Image warping with scattered data interpolation. IEEE Comput. Graphics Appl. 15.2 (1995), pp. 37–43. doi: 10.1109/38.365004. E. G. Sewell. Analysis of a finite element method. Springer, 2012. doi: 10.1007/978-1-4684-6331-6. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput. 63.2 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x. O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24.2 (1987), pp. 337–357. doi: 10.1002/nme.1620240206

Efficient solution techniques for a finite element thin plate spline formulation

We present a new technique for solving the saddle point problem arising from a finite element based thin plate spline formulation. The solver uses the Sherman–Morrison–Woodbury formula to divide the domain into different regions depending on the properties of the data projection matrix. We analyse the conditioning of the resulting system on certain data distributions and use the results to develop effective preconditioners. We show our approach is efficient for a wide range of parameters by testing it on a number of different examples. Numerical results are given in one, two and three dimensions

Algorithm-based fault recovery of adaptively refined parallel multilevel grids

On future extreme scale computers, it is expected that faults will become an increasingly serious problem as the number of individual components grows and failures become more frequent. This is driving the interest in designing algorithms with built-in fault tolerance that can continue to operate and that can replace data even if part of the computation is lost in a failure. For fault-free computations, the use of adaptive refinement techniques in combination with finite element methods is well established. Furthermore, iterative solution techniques that incorporate information about the grid structure, such as the parallel geometric multigrid method, have been shown to be an efficient approach to solving various types of partial different equations. In this article, we present an advanced parallel adaptive multigrid method that uses dynamic data structures to store a nested sequence of meshes and the iteratively evolving solution. After a fail-stop fault, the data residing on the faulty processor will be lost. However, with suitably designed data structures, the neighbouring processors contain enough information so that a consistent mesh can be reconstructed in the faulty domain with the goal of resuming the computation without having to restart from scratch. This recovery is based on a set of carefully designed distributed algorithms that build on the existing parallel adaptive refinement routines, but which must be carefully augmented and extended

A study of the Hasegawa--Wakatani equations using an implicit explicit backward differentiation formula

The Hasegawa--Wakatani system of equations may be used to predict and study the behaviour of plasma flow. A recent analytical study of the use of linear multistep methods to solve the Hasegawa--Wakatani equations showed the backward differentiation formulas to be the most stable. Because the backward differentiation formulas require a solution of a large dense system of equations, so we implemented an implicit explicit version of the formula. We study the performance of the implicit explicit backward differentiation formula on some example problems where the behaviour of the Hasegawa--Wakatani equation is predictable. These results suggest that the implicit explicit method is appropriate to use with the Hasegawa--Wakatani equations. References S. J. Camargo, D. Biskamp, and B. D. Scott, Resistive drift-wave turbulence, Phys. Plasmas, 1, 1995, 48--62. http://www.ldeo.columbia.edu/ suzana/papers/camargo_biskamp_scott95.pdf S. J. Camargo, M. K. Tippett, and I. L. Caldas, Nonmodal energetics of resistive drift waves, Phys. Rev. E, 58, 1998, 3693--3704. doi:10.1103/PhysRevE.58.3693 G. Dahlquist, On the relation of G-stability to other stability concepts for linear multistep methods, in Topics In Numerical Analysis {III}, J. H. Miller, ed., pages 67--80. Academic Press, London, 1977. G. Dahlquist, G-stability is equivalent to A-stability, BIT, 18, 1978, 384--401. doi:10.1007/BF01932018 T. Geveci, On the rate of convergence of the Fourier spectral method for the Navier--Stokes equations, Calcolo, 26, 1989, 185--195. doi:10.1007/BF02575728 A. Hasegawa and M. Wakatani, Plasma edge turbulence, Phys. Rev. Lett., 50, 1983, 682--686. doi:10.1103/PhysRevLett.50.682 A. T. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34, 1997, 119--142. doi:10.1137/S0036142994270971 R. LeVeque, G. Dahlquist, and D. Andree, Computations related to G-stability of linear multistep methods, Tech. Rep. STAN-CS-79-738, Stanford University, Computer Science Department, School of Humanities and Sciences, May 1979. ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/79/738/CS-TR-79-738.pdf R. Numata, R. Ball, and R. L. Dewar, Bifurcation in electrostatic resistive drift wave turbulence, Phys. Plasmas, 14, 102312, 2007, 8 Pages. http://arxiv.org/abs/0708.4317 T. S. Pedersen, P. K. Michelsen, and J. J. Rasmussen, Resistive coupling in drift wave turbulence, Plasma Phys. Control. Fusion, 38, 1996, 2143--2154. doi:10.1088/0741-3335/38/12/008 L. Stals, R. Numata, and R. Ball, Stability analysis of time stepping for prolonged plasma fluid simulations. Accepted for publication in SIAM Journal of Scientific Computing