However, despite its claims, this book is an excellent introduction to mathematical modelling, with a wealth of interesting case studies and lovely mathematics that will appeal to many of its target audience final year undergraduates, Masters and PhD students working in 'practical' mathematical modelling. The source of most of the case studies comes from the Study Groups with Industry and Mathematics that originated in 1968 at Oxford, organized by Alan Taylor and Leslie Fox, but is now a global activity. The book describes the type of problem, how they are approached and funded (typically a surprisingly cheap £10 000 to £15 000). The first chapter covers the mathematical preliminaries for the case studies which are all based on continuous media, and so all that is needed is diffusion and flow and deformation of solids and fluids. The next four chapters are based on elliptic partial differential equations and relate to simulation of a type of compressor, determination of the viscosity of a carbon paste used in smelting, simulation of an instrument for determining density, and modelling a form of nondestructive testing of pressure vessels. There follow six chapters devoted to a range of applications of the diffusion equation and various non-linear effects. These cover the cooking of a single cereal grain, waves of epidemics in animal populations, catalytic converters, problems with blast furnaces, the cooling of hot glass, and crystal growth. Chapter 11 looks at the load responses of filled viscoelastic materials. The last two chapters, based on hyperbolic equations, model the noise produced by a bird scare gun and the paper tension variations in a printing press.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FB9EBD3320909B8F7F968615CD8463B2/S0025557200173425a.pdf/div-class-title-span-class-italic-statistical-inference-span-2nd-edn-by-paul-h-garthwaite-ian-t-jolliffe-and-byron-jones-pp-328-40-hbk-2002-isbn-0-19-857226-3-oxford-university-press-div.pdf

Statistical inference (2nd edn), by Paul H. Garthwaite, Ian T. Jolliffe and Byron Jones. Pp.328. £40 (hbk). 2002. ISBN 0 19 857226 3 (Oxford University Press).

This paper describes a new stochastic finite difference time domain (S-FDTD) method for calculating the variance in the electromagnetic fields caused by variability or uncertainty in the electrical properties of the materials in the model. Details of the 1D derivation using truncated Taylor series approximations are given. The S-FDTD analysis is then compared to Monte Carlo analysis for a 1D bioelectromagnetic example. The accuracy of the method is controlled by the approximations for the cross correlations of the fields and electrical properties. In this paper, we are able to bound the variance using cross correlations of 1 (overestimate) and the reflection coefficient (underestimate).

Stochastic FDTD for Analysis of Statistical Variation in Electromagnetic Fields

A stochastic finite-difference time-domain (S-FDTD) algorithm is presented for electromagnetic-wave propagation in anisotropic magnetized plasma. This new algorithm efficiently calculates in a single simulation not only the mean electromagnetic field values, but also their variance as caused by the variability or uncertainty of the electron and ion content of the ionosphere. The structure of the ionosphere is often too variable and uncertain for electromagnetic-wave propagation problems to be solved using a deterministic formulation, particularly during space weather events. For these cases, the S-FDTD ionospheric plasma algorithm will serve as an important tool. For example, it could be used to determine the confidence level at which a communications or remote sensing or radar system will operate as expected under abnormal ionospheric conditions.

/pdf/a-3-d-stochastic-fdtd-model-of-electromagnetic-wave-29pbam5dxw.pdf

A 3-D Stochastic FDTD Model of Electromagnetic Wave Propagation in Magnetized Ionosphere Plasma

Rapid advances in antennas, propagation, electromagnetics, and materials are opening new and unexplored opportunities in body area sensing and stimulation. Next-generation wearables and implants are seamlessly providing round-the-clock monitoring. In turn, numerous applications are brought forward with the potential to ultimately transform healthcare, sports, consumer electronics, and beyond. This review paper provides a comprehensive overview, discusses challenges and opportunities, and indicates future directions for: (a) enabling technologies needed to make body area sensing and stimulation a reality, and (b) emerging bioelectromagnetics applications that may readily benefit from such technologies.

/pdf/next-generation-healthcare-enabling-technologies-for-m8ah5rgq.pdf

Next-Generation Healthcare: Enabling Technologies for Emerging Bioelectromagnetics Applications

This paper introduces an iterative scheme to overcome the unresolved issues presented in S-FDTD (stochastic finite-difference time-domain) for obtaining ensemble average field values recently reported by Smith and Furse in an attempt to replace the brute force multiple-realization also known as Monte-Carlo approach with a single-realization scheme. Our formulation is particularly useful for studying light interactions with biological cells and tissues having sub-wavelength scale features. Numerical results demonstrate that such a small scale variation can be effectively modeled with a random medium problem which when simulated with the proposed S-FDTD indeed produces a very accurate result.

/pdf/single-realization-stochastic-fdtd-for-weak-scattering-waves-3dmq52d56i.pdf

Single Realization Stochastic FDTD for Weak Scattering Waves in Biological Random Media

An approximate method for determining the statistical properties of an FDTD Analysis has been derived, allowing the determination of the mean and the variance of an FDTD simulation. This stochastic simulator progresses in the same fashion as an FDTD simulation progresses. The variance-wave builds and propagates with reflections occurring at dielectric boundaries. Computer overhead is reduced, i.e. memory and runtime are substantially reduced. Permitting the statistics, i.e. mean and variance to be determined without spending allot of time and computer resources for performing Monte Carlo analysis. This would allow the industry to look at statistical analysis in large 3D model spaces using FDTD, which currently is not being done.

A stochastic FDTD method for statistically varying biological tissues

A computing device simulates a test system by defining parameter values to be used to populate certain modeling formulas defining the test system The defined parameter values correspond to one of the many points defining a domain in which the test system is to be simulated The simulation iteratively solves the modeling formulas for each unit of the test system model space for each point in the domain in which the test system is simulated Results for the subjects of interest are calculated at each iteration using the populated modeling formulas A variance of the subjects of interest is also calculated at each iteration using a correlation coefficient obtained for the subjects of interest The iterations of defining the parameter values and calculating the value and variance of the subjects of interest in the test system model space continues until all points in the domain have been simulated

Incorporation of Uncertainty in Simulation Analysis

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Stochastic FDTD for Analysis of Statistical Variation in Electromagnetic Fields

A stochastic FDTD method for statistically varying biological tissues

Incorporation of Uncertainty in Simulation Analysis