Australian Mathematical Society (AustMS): E-Journals
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The combination technique applied to functionals
Get PDFFunctionals related to a solution of a problem, usually modelled by partial differential equations, can be important quantities used to capture features of the problem. For high dimensional problems the computational cost of the functionals can be large since the numerical solution of a high dimensional partial differential equation is usually expensive to compute. We develop a new sparse grid combination technique to reduce the computational cost of such functionals. Our method is based on error splitting models of the functionals. However, it is hard to obtain a concrete error splitting model for complicated approximations. We show the connection between the decay of the surpluses and the error splitting models. By using the connection, we can also apply our combination technique to functionals when we only know their computed surpluses. Numerical experiments are provided to illustrate our idea and test the performance of our method.
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Dispersal of hydrogen in the retina – a three-layer model
No full textTwo simple mathematical models of advection and diffusion of hydrogen within the retina are discussed. The work is motivated by the hydrogen clearance technique, which is used to estimate blood flow in the retina. The first model assumes that the retina consists of three, well-mixed layers with different thickness, and the second is a two-dimensional model consisting of three regions that represent the layers in the retina. Diffusion between the layers and leakage through the outer edges are considered. Solutions to the governing equations are obtained by employing Fourier series and finite difference methods for the two models, respectively. The effect of important parameters on the hydrogen concentration is examined and discussed. The results contribute to understanding the dispersal of hydrogen in the retina and in particular the effect of flow in the vascular retina. It is shown that the predominant features of the process are captured by the simpler model.
doi: https://doi.org/10.1017/S144618112200005
Consecutive square-free numbers in Piatetski-Shapiro sequences
No full texthttp://dx.doi.org/10.1017/S000497271200033
Conjugacy classes of maximal cyclic subgroups and nilpotence class of -groups
No full texthttp://dx.doi.org/10.1017/S000497271200033
On linearised polynomials, Sidon arrays and fast construction of Sidon sets
No full texthttp://dx.doi.org/10.1017/S000497271200033
An analytical approximation for convertible bonds
No full textThis paper looks at adapting the method of Medvedev and Scaillet for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we provide explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices.
doi:10.1017/S144618112200006
Improving the accuracy of retrieved cardiac electrical conductivities
Get PDFAccurate values for the six cardiac conductivities of the bidomain model are crucial for meaningful electrophysiological simulations of cardiac tissue and are yet to be achieved. A two-stage optimisation process is used to retrieve the cardiac conductivities from cardiac potentials measured on a multi-electrode array—the first stage simultaneously fits all six conductivities, and the second stage fits a subset of the conductivities (intracellular conductivities), while holding the remainder of the conductivities (extracellular conductivities) constant. Previous studies have shown that the intracellular conductivities are retrieved to a lesser degree of accuracy than extracellular conductivities. This study tests the proposition that there exists a relationship between the extracellular and intracellular conductivities during the second stage of the optimisation that affects the accuracy of the retrieved intracellular conductivities. A measure to quantify this relationship is developed using polynomial chaos. The results show that a significant relationship does exist, and thus any errors in the extracellular conductivities are magnified in the retrieved intracellular conductivities. Thus, it is suggested that future protocols for retrieving conductivities incorporate the uncertainty in the extracellular conductivities.
References
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B. M. Johnston, S. Coveney, E. T. Y. Chang, P. R. Johnston, and R. H. Clayton. Quantifying the effect of uncertainty in input parameters in a simplified bidomain model of partial thickness ischaemia. Med. Bio. Eng. Comput. 56.5 (2018), pp. 761–780. doi: 10.1007/s11517-017-1714-y
B. M. Johnston and P. R. Johnston. Approaches for determining cardiac bidomain conductivity values: Progress and challenges. Med. Bio. Eng. Comput. 58 (2020), pp. 2919–2935. doi: 10.1007/s11517-020-02272-z
B. M. Johnston and P. R. Johnston. Determining six cardiac conductivities from realistically large datasets. Math. Biosci. 266 (2015), pp. 15–22. doi: 10.1016/j.mbs.2015.05.008
B. M. Johnston, P. R. Johnston, and D. Kilpatrick. A new approach to the determination of cardiac potential distributions: Application to the analysis of electrode configurations. Math. Biosci. 202.2 (2006), pp. 288–309. doi: 10.1016/j.mbs.2006.04.004
A. Kamalakkannan, P. R Johnston, and B. M. Johnston. A modified approach to determine the six cardiac bidomain conductivities. In: Comput. Bio. Med. 135, 104549 (2021). doi: 10.1016/j.compbiomed.2021.104549
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Some contributions to measuring and understanding heterogeneity in meta-analysis
No full texthttp://dx.doi.org/10.1017/S000497271200033