Showing papers in "Anziam Journal in 2013"

A numerical method for the fractional Fitzhugh–Nagumo monodomain model

Abstract: A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach. References B. Baeumer, M. Kovaly, and M. M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bulletin of Mathematical Biology 69:2281–2297, 2007. doi:10.1007/s11538-007-9220-2 B. Baeumer, M. Kovaly, and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Computers and Mathematics with Applications 55:2212–2226, 2008. doi:10.1016/j.camwa.2007.11.012 N. Badie and N. Bursac, Novel micropatterned cardiac cell cultures with realistic ventricular microstructure, Biophys J 96:3873–3885, 2009. doi:10.1016/j.bpj.2009.02.019 A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, Technical report, University of Oxford, 2013. A. Bueno-Orovioy, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional dffusion models of electrical propagation in cardiac tissue: electrotonic effects and the modulated dispersion of repolarization, Technical report, University of Oxford, 2013. K. F. Decker, J. Heijman, J. R. Silva, T. J. Hund and Y. Rudy, Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium, Am. J. Physiol Heart Circ. Physiol. 296:H1017–H1026, 2009. doi:10.1152/ajpheart.01216.2008 J. S. Frank and G. A. Langer, The myocardial interstitium: its structure and its role in ionic exchange, J Cell Biol 60:586–601, 1974. doi:10.1083/jcb.60.3.586 A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond) , 117:500–544, 1952. http://jp.physoc.org/content/117/4/500.html R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J. , 1:445–466, 1961. doi:10.1016/S0006-3495(61)86902-6 D. Kay, I. W. Turner, N. Cusimano and K. Burrage, Reflections from a boundary: reflecting boundary conditions for space-fractional partial differential equations on bounded domains, Technical report, University of Oxford, 2013. . F. Liu, V. Anh and I. Turner, Numerical solution of space fractional Fokker-Planck equation. J. Comp. and Appl. Math. , 166:209–219, 2004. doi:10.1016/j.cam.2003.09.028 F. Liu, P. Zhuang, V. Anh and I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comp. , 191:12–20, 2007. doi:10.1016/j.amc.2006.08.162 R. Magin, O. Abdullah, D. Baleanu and X. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation, Journal of Magnetic Resonance 190:255–270, 2008. doi:10.1016/j.jmr.2007.11.007 M. M. Meerschaert, J. Mortensenb and S. W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion, Physica A , 367:181–190, 2006. doi:10.1016/j.physa.2005.11.015 L. C. McSpadden, R. D. Kirkton and N. Bursac, Electrotonic loading of anisotropic cardiac monolayers by unexcitable cells depends on connexin type and expression level, Am. J. Physiol. Cell Physiol. 297:C339–C351, 2009. doi:10.1152/ajpcell.00024.2009 J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE , 50:2061–2070, 1962. doi:10.1109/JRPROC.1962.288235 S. F. Roberts, J. G. Stinstra and C. S. Henriquez, Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model, Biophys J 95:3724–3737, 2008. doi:10.1529/biophysj.108.137349 J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K. A. Mardal and A. Tveitio, Computing the electrical activity in the heart , Springer-Verlag, 2006. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods , Clarendon Press, Oxford, 1985. F. J. Valdes-Parada, J. A. Ochoa-Tapia and J. Alvarez-Ramirez, Effective medium equations for fractional Fick law in porous media, Physica A , 373:339–353, 2007. doi:10.1016/j.physa.2006.06.007 Q. Yang, F. Liu and I. Turner, Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term, International Journal of Differential Equations , 2010:464321, 2010, doi:10.1155/2010/464321 W. Ying, A multilevel adaptive approach for computational cardiology , PhD thesis, Duke University, 2005. Q. Yu, F. Liu, I. Turner and K. Burrage, A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comp. , 219:4082–4095, 2012. doi:10.1016/j.amc.2012.10.056 Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation, the special issue of Fractional Calculus and Its Applications in-Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 371:20120150, 2013. doi:10.1098/rsta.2012.0150 Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical simulation of the fractional Bloch equations, J. Comp. Appl. Math. , 255:635–651, 2014. doi:10.1016/j.cam.2013.06.027 P. Zhuang, F. Liu, V. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Num. Anal. , 47:1760–1781, 2009. doi:10.1137/080730597

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract: Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality. References R. C. Gonzalez, R. E. Woods, Digital Image Processing . Third Edition. Prentice Hall, New Jersey, USA, 2007. C. S. Panda, S. Patnaik, Filtering corrupted image and edge detection in restored grayscale image using derivative filters, International Journal of Image Processing , 3(3):105–119, 2009. http://www.cscjournals.org/csc/manuscript/Journals/IJIP/volume3/Issue3/IJIP-28.pdf Y. Zhang, Y. Pu, J. Zhou, Construction of Fractional differential Masks Based on Riemann–Liouville Definition, Journal of Computational Information Systems , 6(10):3191–3199, 2010. http://www.jofcis.com/publishedpapers/2010_6_10_3191_3199.pdf F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, J. Appl. Math. Comput. , 191:12–21, 2007. doi:10.1016/j.amc.2006.08.162 Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation, Phil Trans R Soc A , 371(1990):20120150, 2013. doi:10.1098/rsta.2012.0150 E. Sejdic, I. Djurovic, L. Stankovic, Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Processing , 91(6):1351–1369, 2011. doi:10.1016/j.sigpro.2010.10.008 B. Pesquet-Popescu and J. L. Vehel, Stochastic fractal models for image processing, IEEE Signal Processing Magazine , 19(5):48–62, 2002. doi:10.1109/MSP.2002.1028352 B. Mathieu, P. Melchior, A. Oustaloup, Ch. Ceyral, Fractional differentiation for edge detection, Signal Processing , 83(11):2421–2432, 2003. doi:10.1016/S0165-1684(03)00194-4 C. B. Gao, J. L. Zhou, J. R. Hu, F. N. Lang, Edge detection of colour image based on quaternion fractional differential, IET Image Processing , 5(3):261–272, 2011. http://digital-library.theiet.org/content/journals/10.1049/iet-ipr.2009.0409 C. B. Gao, J. L. Zhou, X. Q. Zheng, F. N. Lang, Image enhancement based on improved fractional differentiation, Journal of Computational Information Systems , 7(1):257–264, 2011. http://www.jofcis.com/publishedpapers/2011_7_1_257_264.pdf Y. Pu, J. Zhou and X. Yuan, Fractional differential mask: A fractional differential–based approach for multiscale texture enhancement, IEEE Transactions on Image Processing , 19(2):491–511, 2010. doi:10.1109/TIP.2009.2035980 M. D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences , 2006:48391, 2006. doi:10.1155/IJMMS/2006/48391

A transformation method for solving the hamilton-jacobi-bellman equation for a constrained dynamic stochastic optimal allocation problem

Abstract: In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method. doi:10.1017/S144618111300031X

Slip at the surface of an oscillating spheroidal particle in a micropolar fluid

Abstract: The axisymmetric rectilinear and rotary oscillations of a spheroidal particle in an incompressible micropolar fluid are considered. Basset type linear slip boundary conditions on the surface of the solid spheroidal particle are used for velocity and microrotation. Under the assumption of small amplitude oscillations, analytical expressions for the fluid velocity field and microrotation components are obtained in terms of a first order small parameter characterizing the deformation. For the rectilinear oscillations, the drag acting on the particle is evaluated and expressed in terms of two real parameters for the prolate and oblate spheroids. Also, the couple exerted on the spheroid is evaluated for the prolate and oblate spheroids for the rotary oscillations. Their variations with respect to the frequency, deformity, micropolarity and slip parameters are tabulated and displayed graphically. Well-known results are deduced and comparisons are made between the classical

A robust combination technique

Abstract: One of the challenges for efficiently and effectively using petascale and exascale computers is the handling of run-time errors. Without such robustness, applications developed for these machines will have little chance of completing successfully. The sparse grid combination technique approximates the solution to a given problem by taking the linear combination of its solution on multiple grids. It is successful in many high performance computing applications due to its ability to tackle the curse of dimensionality. We present several approaches to fault tolerance using the combination technique. The first of these is implemented within the MapReduce model in order to utilise the existing fault tolerance of this framework. In addition, we present a method which utilises the redundancy shared by solutions on different grids. Finally, we describe a novel approach in which the solution is computed on additional grids which are used for alternative combinations if other grids experience failure. We include some results based on the solution of the 2D scalar advection PDE. References S. Balay, J. Brown, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang. PETSc Web page (2012). http://www.mcs.anl.gov/petsc . G. Bosilca, R. Delmas, J. Dongarra, and J. Langou. Algorithm-based fault tolerance applied to high performance computing. Journal of Parallel and Distributed Computing , 69(4):410--416 (2009). doi:10.1016/j.jpdc.2008.12.002 . H. J. Bungartz and M. Griebel. Sparse grids. Acta Numerica , 13:147--269 (2004). doi:10.1017/S0962492904000182 . F. Cappello. Fault Tolerance in Petascale/Exascale Systems: Current Knowledge, Challenges and Research Opportunities. International Journal of High Performance Computing Applications , 23(3):212--226, (2009). doi:10.1177/1094342009106189 . J. Dean and S. Ghemawat. MapReduce: Simplified data processing on large clusters. Communications of the ACM , 51(1):107--113 (2008). doi:10.1145/1327452.1327492 . J. Garcke. Sparse grids in a nutshell. In J. Garcke and M. Griebel, editors, Sparse grids and applications , volume 88 of Lecture Notes in Computational Science and Engineering , pages 57--80. Springer (2013). doi:10.1007/978-3-642-31703-3_3 . M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra , pages 263--281. IMACS, Elsevier, North Holland (1992). Zbl 0785.65101. M. Hegland. Adaptive sparse grids. ANZIAM Journal , 44:C335--C353 (2003). http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 . K.-H. Huang and J. A. Abraham. Algorithm-based fault tolerance for matrix operations. Computers, IEEE Transactions on , C-33(6):518--528 (1984). doi:10.1109/TC.1984.1676475 . C. Zenger. Sparse Grids. In W.Hackbusch, editor, Parrallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech. , Vieweg--Verlag, 31:241--251 (1991). Zbl 0763.65091.

Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

Abstract: A new optimal control model of the interactions between a growing tumour and the host immune system, along with an immunotherapy treatment strategy, is presented. The model is based on an ordinary differential equation model of interactions between the growing tumour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treatment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum allowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour. References Bunimovich–Mendrazitsky, S., Shochat, E. and Stone, L. Mathematical model of BCG immunotherapy in superficial bladder cancer. Bulletin of Mathematical Biology , 69:1847–1870, 2007. doi:10.1007/s11538-007-9195-z Burden, T., Ernstberger, J. and Fister, K. R. Optimal control applied to immunotherapy. Discrete and Continuous dynamical Systems-Series B , 4(1):135–146, 2004. doi:10.3934/dcdsb.2004.4.135 Cappuccio, A., Elishmereni, M. and Agur, Z. Cancer immunotherapy by interleukin-21: Potential treatment strategies evaluated in mathematical model. Cancer Res. , 66(14):7293–7300, 2006. doi:10.1158/0008-5472.CAN-06-0241 Cappuccio, A., Castiglione, F. and Piccoli, B. Determination of the optimal therapeutic protocols in cancer immunotherapy. Mathematical Biosciences, 209(1):1–13, 2007. doi:10.1016/j.mbs.2007.02.009 Castiglione, F. and Piccoli, B. Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bulletin of Mathematical Biology, 68(2):255–274, 2006. doi:10.1007/s11538-005-9014-3 de Pillis, L. G., Gu, W. and Radunskaya, A. E. Mixed immunotherapy and chemotherapy of tumours: modeling, applications and biological interpretations. Journal of Theoretical Biology , 238:841–862, 2006. doi:10.1016/j.jtbi.2005.06.037 de Pillis, L. G., Mallet, D. G. and Radunskaya, A. E. Spatial Tumor-Immune Modeling. Computational and Mathematical Methods in Medicine, 7(2-3):159–176, 2006. doi:10.1080/10273660600968978 de Pillis, L. G., Radunskaya, A. E. and Wiseman, C. L. A validated mathematical model of cell-mediated immune response to tumour growth. Cancer Research , 65:7950–7958, 2005. http://www.ncbi.nlm.nih.gov/pubmed/16140967 El-Gohary, A. Chaos and optimal control of equilibrium states of tumor system with drug, Chaos, Soliton and Fractals , 41:425–435, 2009. doi:10.1016/j.chaos.2008.02.003 Ghaffari, A. and Naserifar, N. Mathematical modeling and Lyapunov-based drug administration in cancer chemotherapy. Iranian Journal of Electrical and Electronic Engineering , 5(3):151–158, 2009. http://www.sid.ir/en/VEWSSID/J_pdf/106520090310.pdf Ghaffari, A. and Naserifar, N. Optimal therapeutic protocols in cancer immunotherapy. Computers in Biology and Medicine , 40:261–270, 2010. doi:10.1016/j.compbiomed.2009.12.001 Isaeva, O. G. and Osipov, V. A. Modelling of anti-tumour immune response: Immunocorrective effect of weak centimetre electromagnetic waves. Computational and Mathematical Methods in Medicine , 10:185–201, 2009. doi:10.1080/17486700802373540 Kirschner, D. and Panetta, J. C. Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol., 37:235–252, 1998. doi:10.1007/s002850050127 Kuznetsov, V. A., Makalkin, I. A., Taylor, M. A. and Perelson, A. S. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology , 56:295–321, 1994. doi:10.1007/BF02460644 N. Larmonier, J. Fraszack, D. Lakomy, Bonnotte, B. and Katsanis, E. Killer dendritic cells and their potential for cancer immunotherapy. Cancer Immunology Immunotherapy, 59:1–11, 2010. doi:10.1007/s00262-009-0736-1 Lenhart, S. and Workman, J. T. Optimal control applied to biological models. Chapman and Hall/CRC Mathematical and Computational Biology Series, 2007. http://www.crcpress.com/product/isbn/9781584886402 Mallet, D. G. and de Pillis, L. G. A cellular automata model of tumour-immune system interactions. Journal of Theoretical Biology , 239:334–350, 2006. doi:10.1016/j.jtbi.2005.08.002 Moretta, A. Natural killers and dendritic cells: rendezvous in abused tissues. Nat. Rev. Immunol. , 2(12):957-964, 2002. doi:10.1038/nri956 Murray, J. M. Some optimal control problems in cancer chemotherapy with a toxicity limit. Mathematical Biosciences , 100:49–67, 1990. doi:10.1016/0025-5564(90)90047-3 Swan, G. W. Role of optimal control theory in cancer chemotherapy. Mathematical Biosciences , 101:237–284, 1990. doi:10.1016/0025-5564(90)90021-P Swierniak, A., Ledzewicz, U. and Schattler, H. Optimal control for a class of compartmental models in cancer chemotherapy. Int. J. Appl. Math. Comput. Sci. , 13(3):357–368, 2003. https://www.amcs.uz.zgora.pl/?action=paper&paper=154 Wu, Y., Xia, L., Zhang, M. and Zhao, X. Immunodominance analysis through interactions of cd\(8^+\) T cells and dcs in lymph nodes. Math. Biosci. , 225(1):53–38, 2010. doi:10.1016/j.mbs.2010.01.009

A European option general first-order error formula

Abstract: We study the value of European security derivatives in the Black–Scholes model when the underlying asset \(\xi\) is approximated by random walks \(\xi^{(n)}\). We obtain an explicit error formula, up to a term of order \(\mathcal{O}(n^{3/2})\), which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks \(\xi^{(n)}\) for which option values converge at a speed of \(\mathcal{O}(n^{3/2})\). doi:10.1017/S1446181113000254

Nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs at high energy level

Abstract: This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods. doi:10.1017/S1446181113000175

Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations

Abstract: This article reports the behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations. The one-and-a-half-dimensional shallow water equations are the one-dimensional shallow water equations with a passive tracer or transverse velocity. The studied behaviour is with respect to the choice of numerical fluxes to evolve the mass, momentum, tracer-mass (transverse momentum), and entropy. When solving the one-and-a-half-dimensional shallow water equations using a finite volume method, we recommend the use of a double sided stencil flux for the mass and momentum, and in addition, a single sided stencil (upwind) flux for the tracer-mass. Having this recommended combination of fluxes, we use a double sided stencil entropy flux to compute the numerical entropy production, but this flux generates positive overshoots of the numerical entropy production. Positive overshoots of the numerical entropy production are avoided by use of a modified entropy flux, which satisfies a discrete numerical entropy inequality. References F. Bouchut. Efficient numerical finite volume schemes for shallow water models. In V. Zeitlin (editor), Nonlinear dynamics of rotating shallow water: methods and advances , Volume 2 of Edited series on advances in nonlinear science and complexity, pages 189--256. Elsevier, Amsterdam, 2007. http://dx.doi.org/10.1016/S1574-6909(06)02004-1 R. J. LeVeque. Finite-volume methods for hyperbolic problems . Cambridge University Press, Cambridge, 2002. http://dx.doi.org/10.1017/CBO9780511791253 S. Mungkasi and S. G. Roberts. Numerical entropy production for shallow water flows. ANZIAM Journal , 52(CTAC2010):C1--C17, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3786/1410 G. Puppo and M. Semplice. Numerical entropy and adaptivity for finite volume schemes. Communications in Computational Physics , 10(5):1132--1160, 2011. http://dx.doi.org/10.4208/cicp.250909.210111a J. J. Stoker. Water Waves: The Mathematical Theory with Application . Interscience Publishers, New York, 1957. http://onlinelibrary.wiley.com/book/10.1002/9781118033159

A comparison of multi-objective optimisation metaheuristics on the 2D airfoil design problem

Abstract: Variants of the multi-objective particle swarm optimisation (MOPSO) algorithm are investigated, mainly focusing on swarm topology, to optimise the well-known 2D airfoil design problem. The topologies used are global best, local best, wheel, and von Neumann. The results are compared to the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective tabu search (MOTS) algorithm, and it is found that the attainment surfaces achieved by some of the MOPSO variants completely dominate those of NSGA-II. In general, the MOPSO algorithms also significantly improve diversity of solutions compared to MOTS. The MOPSO algorithm proves its ability to exploit promising solutions in the presence of a large number of infeasible solutions, making it well suited to problems of this nature. References D. Abramson, A. Lewis, T. Peachey, and C. Fletcher. An automatic design optimization tool and its application to computational fluid dynamics. In Proceedings of the 2001 ACM/IEEE conference on Supercomputing (CDROM) , pages 25--25. ACM, 2001. doi:10.1145/582034.582059 . R. Eberhart and J. Kennedy. A new optimizer using particle swarm theory. In Micro Machine and Human Science, 1995. MHS'95., Proceedings of the Sixth International Symposium on , pages 39--43. IEEE, 1995. doi:10.1109/MHS.1995.494215 . J. Kennedy and R. Mendes. Population structure and particle swarm performance. In Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on , volume 2, pages 1671--1676. IEEE, 2002. doi:10.1109/CEC.2002.1004493 . J. Kennedy. Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance. In Evolutionary Computation, 1999. CEC'99. Proceedings of the 1999 Congress on , volume 3. IEEE, 1999. doi:10.1109/CEC.1999.785509 . K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. Evolutionary Computation, IEEE Transactions on , 6(2):182--197, 2002. doi:10.1109/4235.996017 . D. M. Jaeggi, G. T. Parks, T. Kipouros, and P. J. Clarkson. The development of a multi-objective tabu search algorithm for continuous optimisation problems. European Journal of Operational Research , 185(3):1192--1212, 2008. doi:10.1016/j.ejor.2006.06.048 . C. A. Coello Coello and M. S. Lechuga. MOPSO: A proposal for multiple objective particle swarm optimization. In Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on , volume 2, pages 1051--1056. IEEE, 2002. doi:10.1109/CEC.2002.1004388 . J. Kennedy and R. Eberhart. Particle swarm optimization. In Neural Networks, 1995. Proceedings., IEEE International Conference on , volume 4, pages 1942--1948. IEEE, 1995. doi:10.1109/ICNN.1995.488968 . E. N. Jacobs, K. E. Ward, and R. M. Pinkerton. NACA report no. 460: The characteristics of 78 related airfoil sections from test in the variable-density wind tunnel, 1933. http://www.esdu.com/cgi-bin/ps.pl?sess=unlicensed_1130622070421svf&t=doc&p=naca_tr460 . M. Drela. XFOIL: An analysis and design system for low Reynolds number airfoils. In T. J. Mueller, editor, Low Reynolds Number Aerodynamics , volume 54 of Lecture Notes in Engineering , pages 1--12. Springer Berlin Heidelberg, 1989. doi:10.1007/978-3-642-84010-4_1 . T. W. Sederberg and S. R. Parry. Free-form deformation of solid geometric models. ACM Siggraph Computer Graphics , 20(4):151--160, 1986. doi:10.1145/15886.15903 . R. C. Eberhart and Y. Shi. Particle swarm optimization: developments, applications and resources. In Evolutionary Computation, 2001. Proceedings of the 2001 Congress on , volume 1, pages 81--86. IEEE, 2001. doi:10.1109/CEC.2001.934374 . T. Kipouros, M. Mleczko, and A. M Savill. Use of parallel coordinates for post-analyses of multi-objective aerodynamic design optimisation in turbomachinery. In 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference , page 2138. AIAA, 2008. doi:10.2514/6.2008-2138 . A. Inselberg. The plane with parallel coordinates. The Visual Computer , 1(2):69--91, 1985. doi:10.1007/BF01898350 .

Fractional models for the migration of biological cells in complex spatial domains

Abstract: Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models are successfully used to model anomalous diffusion and spatial heterogeneity in different physical contexts. We develop a fractional model based on the Fisher--Kolmogoroff equation, analyse it for its wavespeed properties, and relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments. References R. J. Adler, R. E. Feldman and M. S. Taqqu. A practical guide to heavy tails: Statistical techniques and applications. Birkauser, 1998. I. J. Allan and D. F. Newgreen. The origin and differentiation of enteric neurons of the intestine of the fowl embryo. The American Journal of Anatomy , 157, 137--154, 1980. doi:10.1002/aja.1001570203 B. J. Binder, K. A. Landman, M. J. Simpson, M. Mariani and D. F. Newgreen. Modeling proliferative tissue growth: A general approach and an avian case study. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) , 78(3), 1--13, 2008. doi:10.1103/PhysRevE.78.031912 M. A. Breau, A. Dahmani, F. Broders--Bondon, J. P. Thiery and S. Dufour. $\beta 1$ integrins are required for the invasion of the caecum and proximal hindgut by enteric neural crest cells. Development , 136, 2791--2801, 2009. doi:10.1242/dev.031419 K. Burrage, N. Hale and D. Kay. An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM Journal on Scientific Computing , 34(4), A2145--A2172. doi:10.1137/110847007 N. R. Druckenbrod and M. L. Epstein. The patterns of neural crest advance in the cecum and colon. Developmental Biology , 287, 125--133, 2005. doi:10.1016/j.ydbio.2005.08.040 H. Engler. On the speed of spread for fractional reaction-diffusion equations. International Journal of Differential Equations , 2010, Article ID 315421, 2010. doi:10.1155/2010/315421 M. Ilic, F. Liu, I. Turner and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)--with nonhomogeneous boundary conditions. Fractional Calculus and Applied Analysis , 9, 333--349, 2006. http://eprints.qut.edu.au/23835/ P. K. Maini, D. L. S. McElwain and D. Leavesley. Travelling waves in a wound healing assay. Applied Mathematics Letters , 17, 575--580, 2004. doi:10.1016/S0893-9659(04)90128-0 R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini and P. M. Kulesa. Multiscale mechanisms of cell migration during development: theory and experiment. Development , 139, 2935--2944, 2012. doi:10.1242/dev.081471 J. D. Murray. Mathematical Biology I and II. Springer Verlag, 2003. M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen. Cell proliferation drives neural crest cell invasion of the intestine. Developmental Biology , 302, 553--568, 2007. doi:10.1016/j.ydbio.2006.10.017 H. M. Young, A. J. Bergner, R. B. Anderson, H. Enomoto, J. Milbrandt, D. F. Newgreen and P. M. Whitington. Dynamics of neural crest-derived cell migration in the embryonic mouse gut. Developmental Biology , 270, 455--473, 2004. doi:10.1016/j.ydbio.2004.03.015 P. Zhuang, F. Liu, V. Anh and I. Turner. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis , 47(3), 1760--1781, 2009. doi:10.1137/080730597

Biased random walks, partial differential equations and update schemes

Abstract: There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a Fokker-Planck equation is used to describe the evolution of a collection of noninteracting asymmetric random walkers. The resulting continuum description is highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to the macroscopic description of stochastic local movement rules in agent-based models. doi:10.1017/S1446181113000369

Multiscale modelling couples patches of wave-like simulations

Abstract: A multiscale model is proposed to significantly reduce the expensive numerical simulations of complicated waves over large spatial domains. The multiscale model is built from given microscale simulations of complicated physical processes such as sea ice or turbulent shallow water. Our long term aim is to enable macroscale simulations obtained by coupling small patches of simulations together over large physical distances. This initial work explores the coupling of patch simulations of wave-like pdes. With the line of development being to water waves we discuss the dynamics of two complementary fields called the ‘depth’ h and ‘velocity’ u. A staggered grid is used for the microscale simulation of the depth h and velocity u. We introduce a macroscale staggered grid to couple the microscale patches. Linear or quadratic interpolation provides boundary conditions on the field in each patch. Linear analysis of the whole coupled multiscale system establishes that the resultant macroscale dynamics is appropriate. Numerical simulations support the linear analysis. This multiscale method should empower the feasible computation of large scale simulations of wave-like dynamics with complicated underlying physics.

Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains

Abstract: We propose a multiscale approximation method for constructing numerical solutions to elliptic partial differential equations on a bounded domain. The approximate solution is constructed in a multi-level fashion, with each level using compactly supported radial basis functions on an increasingly fine mesh. Numerical experiments support the theoretical results. References S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods . Texts in Applied Mathematics. Springer, 3rd edition, 2008. doi:10.1007/978-1-4757-3658-8 . C. S. Chen, M. Ganesh, M. A. Golberg, and A. H.-D. Cheng. Multilevel compact radial functions based computational schemes for some elliptic problems. Computers and Mathematics with Applications , 43:359--378, 2002. doi:10.1016/S0898-1221(01)00292-9 . A. Chernih and Q. T. Le Gia. Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs. submitted, 2012. G. E. Fasshauer. Meshfree Approximation Methods with Matlab , volume 6 of Interdisciplinary Mathematical Sciences . World Scientific Publishing Co., Singapore, 2007. G. E. Fasshauer and J. G. Zhang. On choosing `optimal' shape parameters for RBF approximation. Numerical Algorithms , 45:345--368, 2007. doi:10.1007/s11075-007-9072-8 . P. Giesl and H. Wendland. Meshless collocation: Error estimates with application to dynamical systems. SIAM Journal on Numerical Analysis , 45:1723--1741, 2006. doi:10.1137/060658813 . Q. T. Le Gia, I. H. Sloan, and H. Wendland. Multiscale RBF collocation for solving PDEs on spheres. Numerische Mathematik , 121:99--125, 2012. doi:10.1007/s00211-011-0428-6 . S. Rippa. An algorithm for selecting a good value of the parameter c in radial basis function interpolation. Advance in Computational Mathematics , 11:193--210, 1999. doi:10.1023/A:1018975909870 . E. M. Stein. Singular Integrals and Differentiability Properties of Functions . Princeton University Press, Princeton, New Jersey, 1970. doi:10.1090/pspum/010/0482394 . H. Wendland. Meshless Galerkin methods using radial basis functions. Mathematics of Computation , 68(228):1521--1531, 1998. doi:10.1090/S0025-5718-99-01102-3 . H. Wendland. Numerical solution of variational problems by radial basis functions. In Charles K. Chui and Larry L. Schumaker, editors, Approximation Theory IX, Volume 2: Computational Aspects , pages 361--368. Vanderbilt University Press, 1998. H. Wendland. Scattered Data Approximation , volume 17 of Cambridge Monographs on Applied and Computational Mathematics . Cambridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511617539 .

A modified projected conjugate gradient algorithm for unconstrained optimization problems

Abstract: We propose a modified projected Polak–Ribiere–Polyak (PRP) conjugate gradient method, where a modified conjugacy condition and a method which generates sufficient descent directions are incorporated into the construction of a suitable conjugacy parameter. It is shown that the proposed method is a modification of the PRP method and generates sufficient descent directions at each iteration. With an Armijo- type line search, the theory of global convergence is established under two weak assumptions. Numerical experiments are employed to test the efficiency of the algorithm in solving some benchmark test problems available in the literature. The numerical results obtained indicate that the algorithm outperforms an existing similar algorithm in requiring fewer function evaluations and fewer iterations to find optimal solutions with the same tolerance. doi:10.1017/S1446181113000084

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Abstract: We study the new streamline diffusion finite element method for treating the three dimensional coupled nonlinear Schrodinger equation. We derive stability estimates and optimal convergence rates. Moreover, an a priori error estimate is obtained and we compare the corresponding optimal convergence rate for popular numerical methods such as conservative finite difference, semi-implicit finite difference, semi-discrete finite element and the time-splitting spectral method. We justify the advantage of the streamline diffusion method versus the some numerical methods with some examples. Test problems are presented to verify the efficiency and accuracy of the method. The results reveal that the proposed scheme is very effective, convenient and quite accurate for such considered problems rather than other methods. References R. A. Adams, Sobolev Spaces , Academic Press, New York (1975). J. Alberty, C. Carstensen, Discontinuous Galerkin Time Discretization in Elastoplasticity: Motivation, Numerical Algorithms, and Applications, Comput. Methods Appl. Mech. Engrg. , 191(2002), 43, 4949--4968. M. Asadzadeh, D. Rostamy, F. Zabihi, Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrodinger equations, Numerical Methods for Partial Differential Equations , APR 2013 doi:10.1002/num.21782 M. Asadzadeh, D. Rostamy and F. Zabihi, A posteriori error estimates for the solutions of a coupled wave system, Journal of Mathematical Sciences , Vol. 175, No. 2 (2011) pp. 228--248. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Method , Springer--Verlag , New York, (1994). E. Burman, Adaptive Finite Element Methods for Compressible Two--Phase Flow, Math. Mod. Meth. App. Sci. , 10 (2000), pp. 963--989. C. Carstensen, M. Jensen, Th. Gudi, A unifying theory of a posteriori error control for discontinuous Galerkin FEM , Numer. 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Izadi, Streamline diffusion Finite Element Method for coupling equations of nonlinear hyperbolic scalar conservation laws , MSc Thesis, (2005). M. Izadi, A posteriori error estimates for the coupling equations of scalar convervation laws, BIT Numer. Math. , 49(2009), pp. 697--720. C. Johnson and A. Szepessy, Adaptive Finite Element Methods for Conservation Laws Based on a posteriori Error Estimates, Comm. Pure. Appl. Math. , 48 (1995), pp. 199--234. C. Johnson, Numerical solutions of partial differential equations by the Finite Element Method , CUP, (1987). C. Johnson and J. Pitkaranta, An analysis of the Discontinous Galerkin method for a scalar hyperbolic equation, Math. comput. , 46 (1986), pp. 1--26. C. Johnson, Discontinous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Enggrg. , 107 (1993), pp. 117--129. M. Wadati, T. Izuka, M. Hisakado, A coupled nonlinear Schrodirnger equation and optical solitons, J. Phys. Soc. 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Determination of join regions between carbon nanostructures using variational calculus

Abstract: We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two- dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other. doi:10.1017/S1446181113000217

An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

Abstract: We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele--Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman--Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour. References M. C. Dallaston and S. W. McCue. New exact solutions for Hele--Shaw flow in doubly connected regions. Phys. Fluids , 24:052101, 2012. doi:10.1063/1.4711274 . M. C. Dallaston and S. W. McCue. Bubble extinction in Hele--Shaw flow with surface tension and kinetic undercooling regularisation. Nonlinearity , 26:1639--1665, 2013. doi:10.1088/0951-7715/26/6/1639 . E. O. Dias and J. A. Miranda. Control of radial fingering patterns: A weakly nonlinear approach. Phys. Rev. E , 81(1):016312 (1--7), 2010. doi:10.1103/PhysRevE.81.016312 . E. O. Dias, E. Alvarez--Lacalle, M. S. Carvalho, and J. S. Miranda. Minimization of viscous fluid fingering: a variational scheme for optimal flow rates. Phys. Rev. Lett. , 109:144502, 2012. doi:10.1103/PhysRevLett.109.144502 . V. Entov and P. Etingof. On the breakup of air bubbles in a Hele--Shaw cell. Eur. J. Appl. Math. , 22:125--149, 2011. doi:10.1017/S095679251000032X . V. M. Entov and P. I. Etingof. Bubble contraction in Hele--Shaw cells. Q. J. Mech. Appl. Math. , 44:507--535, 1991. doi:10.1093/qjmam/44.4.507 . L. K. Forbes. A cylindrical Rayleigh--Taylor instability: radial outflow from pipes or stars. J. Eng. Math. , 70:205--224, 2011. doi:10.1007/s10665-010-9374-z . T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. Journal of Computational Physics , 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689 . S. D. Howison. Complex variable methods in Hele--Shaw moving boundary problems. Eur. J. Appl. Math. , 3:209--224, 1992. doi:10.1017/S0956792500000802 . J. R. King and S. W. McCue. Quadrature domains and p-Laplacian growth. Complex Anal. Oper. Th. , 3:453--469, 2009. doi:10.1007/s11785-008-0103-9 . S.-Y. Lee, E. Bettelheim, and P. Weigmann. Bubble break-off in Hele--Shaw flows---singularities and integrable structures. Physica D , 219:22--34, 2006. doi:10.1016/j.physd.2006.05.010 . S. Li, J. S. Lowengrub, and P. H. Leo. A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele--Shaw cell. J. Comp. Phys. , 225(1):554--567, 2007. doi:10.1016/j.jcp.2006.12.023 . S. Li, J. S. Lowengrub, J. Fontana, and P. Palffy--Muhoray. Control of viscous fingering patterns in a radial Hele--Shaw cell. Phys. Rev. Lett. , 102(17):174501, 2009. doi:10.1103/PhysRevLett.102.174501 . S. W. McCue and J. R. King. Contracting bubbles in Hele--Shaw cells with a power-law fluid. Nonlinearity , 24:613--641, 2011. doi:10.1088/0951-7715/24/2/009 . S. W. McCue, J. R. King, and D. S. Riley. Extinction behaviour of contracting bubbles in porous media. Q. J. Mech. Appl. Math. , 56:455--482, 2003. doi:10.1093/qjmam/56.3.455 . S. W. McCue, J. R. King, and D. S. Riley. Extinction behavior for two-dimensional inward-solidification problems. Proc. R. Soc. Lond. A , 459:977--999, 2003. doi:10.1098/rspa.2002.1059 . S. W. McCue, J. W. King, and D. S. Riley. The extinction problem for three-dimensional inward solidification. J. Eng. Math. , 52:389--409, 2005. doi:10.1007/s10665-005-3501-2 . M. Reissig, D. V. Rogosin, and F. Hubner. Analytical and numerical treatment of a complex model for Hele--Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math. , 10:561--579, 1999. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=44021&fulltextType=RA&fileId=S0956792599003939 P. G. Saffman and G. I. Taylor. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A , 245:312--329, 1958. doi:10.1098/rspa.1958.0085 .

A new formula for adomian polynomials and the analysis of its truncated series solution for fractional non-differentiable initial value problems

Abstract: In this article, a new formula for Adomian's polynomials is introduced. It is applied to obtain the truncated series solutions for the fractional initial value problems with non-differentiable functions. This kind of equations contains a fractional single-term which is examined using Jumarie fractional derivatives and fractional Taylor series for non-differentiable functions. The property of non-locality of these equations is examined. The discuss of the existence and the uniqueness of the solutions for these equations is clarified. Also, the convergence and the error analysis of Adomian series solution of the presented formula are studied. Numerical examples are proposed to show the accuracy and the eciency of this formula for solving high-order fractional differential equations with initial values. doi:10.1017/S1446181113000321

A dual-reciprocity boundary element method for steady infiltration problems

Abstract: Steady water infiltration in homogeneous soils is governed by the Richards equation. This equation can be studied more conveniently by transforming to a type of Helmholtz equation. In this study, a dual-reciprocity boundary element method (DRBEM) is employed to solve the Helmholtz equation numerically. Using the solutions obtained, numerical values of the suction potential are then computed. The proposed method is tested on problems involving infiltration from different types of periodic channels in a homogeneous soil. Moreover, the method is also examined using infiltration from periodic trapezoidal channels in three different types of homogeneous soil. doi:10.1017/S1446181113000187

Maximum a posteriori density estimation and the sparse grid combination technique

Abstract: We study a novel method for maximum a posteriori (MAP) estimation of the probability density function of an arbitrary, independent and identically distributed \(d\)-dimensional data set. We give an interpretation of the MAP algorithm in terms of regularised maximum likelihood. We also present numerical experiments using a sparse grid combination technique and the `opticom' method. The numerical results demonstrate the viability of parallelisation for the combination technique. References H. J. Bungartz, M. Griebel, D. Roschke and C. Zenger. Pointwise convergence of the combination technique for the Laplace equation. East-West J. Numer. Math , 2:21--45 (1994). http://zbmath.org/?q=an:00653220 J. Garcke. Regression with the optimised combination technique. In Proceedings of the 23rd international conference on Machine learning , ICML '06, pages 321--328 (2006). doi:10.1145/1143844.1143885 J. Garcke. Sparse grid tutorial. Technical report (2011). http://page.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf M. Griebel and M. Hegland. A finite element method for density estimation with Gaussian process priors. SIAM J. Numer. Anal. , 47:4759--4792 (2010). doi:10.1137/080736478 M. Griebel, M. Schneider and C. Zenger. A combination technique for the solution of sparse grid problems. In Iterative methods in linear algebra (Brussels, 1991) , pages 263--281. North-Holland, Amsterdam (1992). M. Hegland. Adaptive sparse grids. ANZIAM J. , 44:C335--C353 (2003). http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 M. Hegland. Approximate maximum a posteriori with Gaussian process priors. Constr. Approx. , 26:205--224 (2007). doi:10.1007/s00365-006-0661-4 M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra Appl. , 420:249--275 (2007). doi:10.1016/j.laa.2006.07.014 C. T. Kelley. Solving nonlinear equations with Newton's method . Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003). H. Kobayashi, B.L. Mark, and W. Turin. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance . Cambridge University Press (2012). C. Pflaum and A. Zhou. Error analysis of the combination technique. Numerische Mathematik , 84:327--350 (1999). doi:10.1007/s002110050474 D. W. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization . John Wiley and Sons (2004). C. Zenger. Sparse grids. Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar , 31 (1990).

Subgrid parameterisation of the eddy-meanfield interactions in a baroclinic quasi-geostrophic ocean

Abstract: We present parameterisations of the subgrid eddy-eddy and eddy-meanfield interactions in a baroclinic ocean representative of the Antarctic Circumpolar Current. Benchmark direct numerical simulations were undertaken using a quasi-geostrophic spectral spherical harmonic code of maximum zonal and total truncation wavenumber of \(T=252\,\). The eddy-eddy interactions are represented by both stochastic and deterministic parameterisations, with model coefficients determined from the direct numerical simulations truncated back to the large eddy simulation truncation wavenumber \(T_R\) less than \(T\). Coefficients of the deterministic eddy-meanfield model are determined by a new least squares regression method. Truncations were repeated for various \(T_R\), with the dependence of the coefficients on \(T_R\) identified. Kinetic energy spectra from the large eddy simulations using these coefficients agree with the direct numerical simulations. References K. Bryan and J. L. Lewis. A water mass model of the global ocean. J. Geophys. Res. , 84:2503--2517 (1979). doi:10.1029/JC084iC05p02503 J. S. Frederiksen. Precursors to blocking anomalies: the tangent linear and inverse problems. J. Atmos. Sci. , 55:2419--2436 (1998). 2.0.CO;2">doi:10.1175/1520-0469(1998)055 2.0.CO;2 J. S. Frederiksen. Self-energy closure for inhomogeneous turbulence and subgrid modeling. Entropy , 14:769--799 (2012). doi:10.3390/e14040769 J. S. Frederiksen and S. M. Kepert. Dynamical subgrid-scale parameterizations from direct numerical simulations. J. Atmos. Sci. , 63:3006--3019 (2006). doi:10.1175/JAS3795.1 P. R. Gent and J. C. McWilliams. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. , 20:150--155 (1990). 2.0.CO;2">doi:10.1175/1520-0485(1990)020 2.0.CO;2 S. M. Griffies, A. Gnanadesikan, K. W. Dixon, J. P. Dunne, R. Gerdes, M. J. Harrison, A. Rosati, J. L. Russell, B. L. Samuels, M. J. Spelman, M. Winton, and R. Zhang. Formulation of an ocean model for global climate simulations. Ocean Science , 1:45--79 (2005). doi:10.5194/os-1-45-2005 V. Kitsios, J. S. Frederiksen, and M. J. Zidikheri. Scaling laws for parameterisations of subgrid eddy-eddy interactions in simulations of oceanic circulations. Ocean Modelling , in print (2013). J. N. Koshyk and K. Hamilton. The horizontal kinetic energy spectrum and spectral budget simulated by a high-resolution troposphere-stratosphere-mesosphere GCM. J. Atmos. Sci. , 58:329--348 (2001). 2.0.CO;2">doi:10.1175/1520-0469(2001)058 2.0.CO;2 M. H. Redi. Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr. , 12:1154--1158 (1982). 2.0.CO;2">doi:10.1175/1520-0485(1982)012 2.0.CO;2 M. J. Zidikheri and J. S. Frederiksen. Stochastic modelling of unresolved eddy fluxes. Geophysical and Astrophysical Fluid Dynamics , 104:323--348 (2010). doi:10.1080/03091921003694701 M. J. Zidikheri and J. S. Frederiksen. Stochastic subgrid-scale modelling for non-equilibrium geophysical flows. Phil. Trans. Royal Soc. A , 368:145--160 (2010). doi:10.1098/rsta.2009.0192

Model completion and validation using inversion of grey box models

Abstract: A grey box model uses some theoretical structure that is not complete and it is thus necessary to complete the model using data. Unfortunately, as the models are generally nonlinear, the efficient linear methods of selecting regression terms were not available for the completion of these models. However, the method of model inversion converts the unknown parts within a nonlinear model into an approximate linear model such that efficient linear term selection methods can be applied. In addition to completing grey box models, the inversion technique can be used to test if the model adequately describes the available data. References Bohlin, T. P., Practical grey-box process identification, Theory and applications (Advances in industrial control , Springer (London), 2006. ISBN: 978-1-846-28402-1. Draper, N. R., and Smith, H., Applied regression analysis , Wiley (New York), 1998. ISBN: 978-0-471-17082-2. Gustafsson, F., and Hjalmarsson, H., Twenty-one ML estimators for model selection, Automatica , 31 (10) 1377--1392, 1995. doi:10.1016/0005-1098(95)00058-5 Kojovic, T., The davelopment and application of Model--an automated model builder for mineral processing , PhD thesis, The University of Queensland, 1989. Kojovic, T., and Whiten W. J., Evaluation of the quality of simulation models , Innovations in mineral processing, (Lauretian University, Sudbury) p437--446, 1994. ISBN: 088667025X. Konishi, S., and Kitagawa, G., Information criteria and statistical Modeling , Springer (New York) 2010. ISBN 978-1-441-92456-8 Lawson, C. L., and Hanson, R. J., Solving least squares problems , SIAM (Philadelphia), 1995. ISBN: 978-0-898-71356-5. Napier--Munn, T. J., Morrell, S., Morrison, R. D. and Kojovic, T., Mineral comminution circuits--their operation and optimisation , Julius Kruttschnitt Mineral Research Centre (Brisbane), 1996. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical recipes , Cambridge (New York), 2007. ISBN: 978-0-521-88068-8. Tran, V., Improved calibration techniques for nuclear analysers , PhD thesis, The University of Queensland, 1998. Weisburg, S., Applied linear regression , Wiley-Interscience (Hoboken, N.J.), 2005. ISBN: 978-0-471-66379-9. Whiten, W. J., Model building techniques applied to mineral treatment processes, Symp. on Automatic Control Systems in Mineral Processing Plants , (Australas. Inst. Min. Metall., S. Queensland Branch, Brisbane), 129--148, 1971. Whiten, W.J., A matrix theory of comminution machines, Chem. Eng. Sci. , 29 , 589-599., 1974. doi:10.1016/0009-2509(74)80070-9 . Whiten, W. J., Determination of parameter relations within non-linear models, SIGNUM Newsletter , 29 (3--4), 2--5, 1994. doi:10.1145/192527.192535 . Xiao, J., Extensions of model building techniques and their applications in mineral processing , PhD thesis, The University of Queensland, 1998.

Critical timescales and time intervals for coupled linear processes

Abstract: In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion– reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes. doi:10.1017/S1446181113000059

An improved second-order numerical method for the generalized burgers–fisher equation

Abstract: A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.

A CARTopt method for bound constrained global optimization

Abstract: A stochastic method for bound constrained global optimization is described. The method can be applied to objective functions which are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines desirable subsets where the objective function is relatively low. Further samples are drawn directly from these low subsets before a new partition is formed. Alternating between sampling and partition phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to a global minimizer of the objective function with probability one under mild conditions. Non-probabilistic results are also give when random sampling is replaced with samples taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice. doi:10.1017/S1446181113000412

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Abstract: Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

Calculus from the past: multiple delay systems arising in cancer cell modelling

Abstract: Nonlocal calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of nonlocal problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including a model of the cell cycle developed recently by Simms, Bean and Koeber. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents. doi:10.1017/S1446181113000102

On Laguerre-Sobolev type orthogonal polynomials, zeros and electrostatic interpretation

Abstract: We study the sequence of monic polynomials orthogonal with respect to inner product $$ \langle p,q\rangle = \int_0^\infty p(x)q(x)e^{-x}x^{\alpha}\;dx +Mp(\zeta )q(\zeta )+Np^\prime(\zeta )q^\prime(\zeta), $$ where \(\alpha >-1\), \(M\geq 0\), \(N\geq 0\), \(\zeta doi:10.1017/S1446181113000308

Design of a multi-electrode array to measure cardiac conductivities

Abstract: Accurate determination of cardiac tissue parameters is essential in bidomain models that simulate the electrical activity of the heart and thereby contribute to understanding cardiovascular disease. Recent experimental work indicated the need for six parameters, which measure electrical conductivity in two domains (extracellular and intracellular), along and across the cardiac fibres within a sheet and also between sheets. This is in contrast to the available experimentally determined conductivities, which are sets of four values, where it is assumed that conductivities across the fibres within a sheet and between the fibre sheets are equal. This study presents a mathematical model that incorporates six bidomain conductivities. It also discusses the design of a multi-electrode array and inversion method to retrieve these