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An efficient method for the anisotropic diffusion equation in magnetic fields
Get PDFWe solve the anisotropic diffusion equation in 2D, where the dominant direction of diffusion is defined by a vector field which does not conform to a Cartesian grid. Our method uses operator splitting to separate the diffusion perpendicular and parallel to the vector field. The slow time scale is solved using a provably stable finite difference formulation in the perpendicular to the vector field, and an integral operator for the diffusion parallel to it. Energy estimates are shown to for the continuous and semi-discrete cases. Numerical experiments are performed showing convergence of the method, and examples is given to demonstrate the capabilities of the method.
References
L. Chacón, D. del-Castillo-Negrete, and C. D. Hauck. An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation. J. Comput. Phys. 272 (2014), pp. 719–746. doi: 10.1016/j.jcp.2014.04.049
D. C. Del Rey Fernández, J. E. Hicken, and D. W. Zingg. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95 (2014), pp. 171–196. doi: 10.1016/j.compfluid.2014.02.01
K. Duru and K. Virta. Stable and high order accurate difference methods for the elastic wave equation in discontinuous media. J. Comput. Phys. 279 (2014), pp. 37–62. doi: 10.1016/j.jcp.2014.08.04
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S. Günter, Q. Yu, J. Krüger, and K. Lackner. Modelling of heat transport in magnetised plasmas using non-aligned coordinates. J. Comput. Phys. 209.1 (2005), pp. 354–370. doi: 10.1016/j.jcp.2005.03.021
P. Helander, S. R. Hudson, and E. J. Paul. On heat conduction in an irregular magnetic field. Part 1. J. Plasma Phys. 88.1, 905880122 (Feb. 2022). doi: 10.1017/S002237782100129X
S. R. Hudson and J. Breslau. Temperature contours and ghost surfaces for chaotic magnetic fields. Phys. Rev. Lett. 100.9, 095001 (2008). doi: 10.1103/PhysRevLett.100.095001
K. Mattsson. Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51.3 (2012), pp. 650–682. doi: 10.1007/s10915-011-9525-
K. Mattsson and J. Nordström. Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199.2 (2004), pp. 503–540. doi: 10.1016/j.jcp.2004.03.001
J. Nordström and T. Lundquist. Summation-by-parts in time: The second derivative. SIAM J. Sci. Comput. 38.3 (2016), A1561–A1586. doi: 10.1137/15M103861X
E. J. Paul, S. R. Hudson, and P. Helander. Heat conduction in an irregular magnetic field. Part 2. Heat transport as a measure of the effective non-integrable volume. J. Plasma Phys. 88.1, 905880107 (2022). doi: 10.1017/S002237782100130
P. J. Roache. Code verification by the method of manufactured solutions. J. Fluids Eng. 124.1 (2002), pp. 4–10. doi: 10.1115/1.1436090
S. Steinberg and P. J. Roache. Symbolic manipulation and computational fluid dynamics. J. Comput. Phys. 57.2 (1985), pp. 251–284. doi: 10.1016/0021-9991(85)90045-
M. Svärd and J. Nordström. Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268 (2014), pp. 17–38. doi: 10.1016/j.jcp.2014.02.03
D. del-Castillo-Negrete and L. Chacón. Local and nonlocal parallel heat transport in general magnetic fields. Phys. Rev. Lett. 106.19, 195004 (2011). doi: 10.1103/PhysRevLett.106.19500
D. del-Castillo-Negrete and L. Chacón. Parallel heat transport in integrable and chaotic magnetic fields. Phy. Plasmas 19.5, 056112 (2012). doi: 10.1063/1.36960
An IMEX-based approach for the pricing of equity warrants under fractional Brownian motion models
No full textIn this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models.
doi: 10.1017/S144618112300015
On the size, spectral radius, distance spectral radius and fractional matchings in graphs
No full texthttp://dx.doi.org/10.1017/S000497271200033
Exploratory analysis of multivariate drill core time series measurements
Get PDFDemand for mineral resources is increasing, necessitating exploitation of lower grade and more heterogeneous orebodies. The high variability inherent in such orebodies leads to an increase in the cost, complexity and environmental footprint associated with mining and mineral processing. Enhanced knowledge of orebody characteristics is thus vital for mining companies to optimize profitability. We present a pilot study to investigate prediction of geometallurgical variables from drill sensor data. A comparison is made of the performance of multilayer perceptron (MLP) and multiple linear regression models (MLR) for predicting a geometallurgical variable. This comparison is based on simulated data that are physically realistic, having been derived from models fitted to the one available drill core. The comparison is made in terms of the mean and standard deviation (over repeated samples from the population) of the mean absolute error, root mean square error, and coefficient of determination. The best performing model depends on the form of the response variable and the sample size. The standard deviation of performance measures tends to be higher for the MLP, and MLR appears to offer a more consistent performance for the test cases considered.
References
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J. B. Boisvert, M. E. Rossi, K. Ehrig, and C. V. Deutsch. Geometallurgical modeling at Olympic dam mine, South Australia”. Math. Geosci. 45 (2013), pp. 901–925. doi: 10.1007/s11004-013-9462-5
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C. Both and R. Dimitrakopoulos. Applied machine learning for geometallurgical throughput prediction—A case study using production data at the Tropicana Gold Mining Complex”. Minerals 11.11 (2021), p. 1257. doi: 10.3390/min11111257
J. Chen and G. Li. Tsallis wavelet entropy and its application in power signal analysis”. Entropy 16.6 (2014), pp. 3009–3025. doi: 10.3390/e16063009
S. Coward, J. Vann, S. Dunham, and M. Stewart. The primary-response framework for geometallurgical variables”. Seventh international mining geology conference. 2009, pp. 109–113. https://www.ausimm.com/publications/conference->url: https://www.ausimm.com/publications/conference- proceedings/seventh-international-mining-geology- conference-2009/the-primary-response-framework-for- geometallurgical-variables/
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C. Dritsaki. An empirical evaluation in GARCH volatility modeling: Evidence from the Stockholm stock exchange”. J. Math. Fin. 7.2 (2017), pp. 366–390. doi: 10.4236/jmf.2017.72020
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J. Hunt, T. Kojovic, and R. Berry. Estimating comminution indices from ore mineralogy, chemistry and drill core logging”. The Second AusIMM International Geometallurgy Conference (GeoMet) 2013. 2013, pp. 173–176. http://ecite.utas.edu.au/89773>url: http://ecite.utas.edu.au/89773 on p. C210).
R. Hyndman, Y. Kang, P. Montero-Manso, T. Talagala, E. Wang, Y. Yang, M. O’Hara-Wild, S. Ben Taieb, H. Cao, D. K. Lake, N. Laptev, and J. R. Moorman. tsfeatures: Time series feature extraction. R package version 1.0.2. 2020. https://CRAN.R-project.org/package=tsfeatures>url: https://CRAN.R-project.org/package=tsfeatures on p. C222).
C. L. Johnson, D. A. Browning, and N. E. Pendock. Hyperspectral imaging applications to geometallurgy: Utilizing blast hole mineralogy to predict Au-Cu recovery and throughput at the Phoenix mine, Nevada”. Econ. Geol. 114.8 (2019), pp. 1481–1494. doi: 10.5382/econgeo.4684
E. B. Martin and A. J. Morris. An overview of multivariate statistical process control in continuous and batch process performance monitoring”. Trans. Inst. Meas. Control 18.1 (1996), pp. 51–60. doi: 10.1177/014233129601800107
E. Sepulveda, P. A. Dowd, C. Xu, and E. Addo. Multivariate modelling of geometallurgical variables by projection pursuit”. Math. Geosci. 49.1 (2017), pp. 121–143. doi: 10.1007/s11004-016-9660-z
S. J. Webb, G. R. J. Cooper, and L. D. Ashwal. Wavelet and statistical investigation of density and susceptibility data from the Bellevue drill core and Moordkopje borehole, Bushveld Complex, South Africa”. SEG Technical Program Expanded Abstracts 2008. Society of Exploration Geophysicists, 2008, pp. 1167–1171. doi: 10.1190/1.3059129
R. Zuo. Identifying geochemical anomalies associated with Cu and Pb–Zn skarn mineralization using principal component analysis and spectrum–area fractal modeling in the Gangdese Belt, Tibet (China)”. J. Geochem. Explor. 111.1-2 (2011), pp. 13–22. doi: 10.1016/J.GEXPLO.2011.06.01
Biological self-heating in industrial compost piles: A modelling case study
No full textWe consider a simple 'toy model' for the spontaneous combustion of industrial compost stockpiles. The model is a scalar non-linear differential equation which can be analysed using techniques taught in an introductory subject on non-linear ordinary differential equations. This model was used as a case study in a third year subject. We discuss how students approached some of the questions. Could they transfer their prior knowledge about differential equations to an industrial case study? How would they cope with a problem which required both pen-and-paper and numerical calculations? Students used a variety of approaches but the worked solutions only showed one. It would be beneficial for students to see that there is not one correct method to solve such problems.
References
B. Barnes and G. R. Fulford. Mathematical modelling with case studies: Using Maple and Matlab. Third edition. CRC Press, 2015. doi: 10.1201/b17896
M. I. Nelson, E. Balakrishnan, and X. D. Chen. A Semenov model of self-heating in compost piles. In: Process Safety Environ. Protect. 81 (2003), pp. 375–383. doi: 10.1205/09575820377022441
Super Catalan numbers and Fourier summations over finite fields
No full texthttp://dx.doi.org/10.1017/S000497271200033
Data-informed uncertainty quantification for wave scattering by heterogeneous media
Get PDFWe present an efficient data-driven offline/online Bayesian algorithm for uncertainty quantification (UQ) in the induced scattered field when a time-harmonic incident wave interacts with an uncertain heterogeneous medium. The incident wave of interest need not be known in advance, and the uncertainty is informed by noisy scattering data obtained from other incident waves impinging on the medium. Our UQ algorithm is accelerated by a novel stochastic reduced order model (ROM) based on the T-matrix, and the ROM is independent of both the incident wave, and other incident waves used to generate the data. This important property allows the model to be set up offline.
References
M. Bachmayr and A. Djurdjevac. Multilevel representations of isotropic Gaussian random fields on the sphere. IMA J. Numer. Anal. 43.4 (2023), pp. 1970–2000. doi: 10.1093/imanum/drac034
C. Borges and G. Biros. Reconstruction of a compactly supported sound profile in the presence of a random background medium. Inv. Prob. 34, 115007 (2018). doi: 10.1088/1361-6420/aadbc5 on p. C101).
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M. Ganesh and S. C. Hawkins. A far-field based T-matrix method for two dimensional obstacle scattering. Proceedings of the 9th Biennial Engineering Mathematics and Applications Conference, EMAC-2009. Ed. by P. Howlett, M. Nelson, and A. J. Roberts. Vol. 51. ANZIAM J. 2010, pp. C215–C230. doi: 10.21914/anziamj.v51i0.2581
M. Ganesh and S. C. Hawkins. A numerically stable T-matrix method for acoustic scattering by nonspherical particles with large aspect ratios and size parameters. J. Acoust. Soc. Am. 151 (2022), pp. 1978–1988. doi: 10.1121/10.0009679
M. Ganesh and S. C. Hawkins. Algorithm 975: TMATROM–A T-matrix reduced order model software. ACM Trans. Math. Softw. 44, 9 (2017), pp. 1–8. doi: 10.1145/3054945
M. Ganesh, S. C. Hawkins, and R. Hiptmair. Convergence analysis with parameter estimates for a reduced basis acoustic scattering T-matrix method. IMA J. Numer. Anal. 32 (2012), pp. 1348–1374. doi: 10.1093/imanum/drr041
M. Ganesh, S. C. Hawkins, A. M. Tartakovsky, and R. Tipireddy. A stochastic domain decomposition and post-processing algorithm for epistemic uncertainty quantification. Int. J. Uncertain. Quant. 13 (2023), pp. 1–22. doi: 10.1615/Int.J.UncertaintyQuantification.2023045687
S. C. Hawkins. Algorithm 1009: MieSolver–An object-oriented Mie series software for wave scattering by cylinders. ACM Trans. Math. Softw. 46, 19 (2020), pp. 1–28. doi: 10.1145/3381537 on p. C109).
S. C. Hawkins. Noisy far-field data. Published online 12th August 2023. doi: 10.5281/zenodo.8240111
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