Kernel Density Estimation with Linked Boundary Conditions

Kernel density estimation on a finite interval poses an outstanding challenge because of the well-recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we consider a boundary constraint linking the values of the unknown target density function at the boundaries. We provide a kernel density estimator (KDE) that successfully incorporates this linked boundary condition, leading to a non-self-adjoint diffusion process and expansions in non-separable generalized eigenfunctions. The solution is rigorously analyzed through an integral representation given by the unified transform (or Fokas method). The new KDE possesses many desirable properties, such as consistency, asymptotically negligible bias at the boundaries, and an increased rate of approximation, as measured by the AMISE. We apply our method to the motivating example in biology and provide numerical experiments with synthetic data, including comparisons with state-of-the-art KDEs (which currently cannot handle linked boundary constraints). Results suggest that the new method is fast and accurate. Furthermore, we demonstrate how to build statistical estimators of the boundary conditions satisfied by the target function without apriori knowledge. Our analysis can also be extended to more general boundary conditions that may be encountered in applications

Positivity-preserving methods for ordinary differential equations

[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056

Cauchy integrals for computational solutions of master equations

Cauchy contour integrals are demonstrated to be effective in computationally solving master equations. A fractional generalization of a bimolecular master equation is one interesting application. References A. Andreychenko, L. Mikeev, D. Spieler, and V. Wolf. Approximate maximum likelihood estimation for stochastic chemical kinetics. EURASIP J. Bioinf. Sys. Biol., 2012:9, 2012. doi:10.1186/1687-4153-2012-9 C. N. Angstmann, I. C. Donnelly, B. I. Henry, and J. A. Nichols. A discrete time random walk model for anomalous diffusion. J. Comput. Phys., 293:53–69, 2014. doi:10.1016/j.jcp.2014.08.003 Y. Berkowitz, Y. Edery, H. Scher, and B. Berkowitz. Fickian and non-Fickian diffusion with bimolecular reactions. Phys. Rev. E, 87:032812, 2013. doi:10.1103/PhysRevE.87.032812 J. C. Butcher. On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines, 117:1–8, 1957. D. Ding, D. A. Benson, A. Paster, and D. Bolster. Modeling bimolecular reactions and transport in porous media via particle tracking. Adv. Water Resour., 53:56–65, 2013. doi:10.1016/j.advwatres.2012.11.001 B. Drawert, S. Engblom, and A. Hellander. URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst. Biol., 6:76, 2012. doi:10.1186/1752-0509-6-76 T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. Pafnuty Publications, 2014. http://www.chebfun.org/docs/guide/ N. Dunford and J. Schwartz. Linear Operators I, II, III. Wiley New York, 1971. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608483.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608475.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608467.html D. Fulger, E. Scalas, and G. Germano. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E, 77:021122, 2008. doi:10.1103/PhysRevE.77.021122 D. Gillespie. Markov Processes: An Introduction for Physical Scientists. Academic Press, 1991. http://www.elsevier.com/books/markov-processes/gillespie/978-0-12-283955-9 M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708–724, 2006. doi:10.1016/j.cam.2006.02.053 B. I. Henry, T. A. M. Langlands, and S. L. Wearne. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, 74(3):031116, 2006. doi:10.1103/PhysRevE.74.031116 N. J. Higham. Functions of Matrices. SIAM, 2008. doi:10.1137/1.9780898717778 R. Hilfer and L. Anton. Fractional master equations and fractal time random walks. Phys. Rev. E, 51:R848, 1995. doi:10.1103/PhysRevE.51.R848 K. J. in 't Hout and J. A. C. Weideman. A contour integral method for the Black–Scholes and Heston equations. SIAM J. Sci. Comput., 33:763–785, 2011. doi:10.1137/090776081 T. Jahnke and D. Altintan. Efficient simulation of discrete stochastic reaction systems with a splitting method. BIT, 50:797–822, 2010. doi:doi:10.1007/s10543-010-0286-0 T. Kato. Perturbation theory for linear operators. Springer-Verlag, 1976. http://link.springer.com/book/10.1007%2F978-3-642-66282-9 V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger. Generalized master equations for continuous-time random walks. J. Stat. Phys., 9(1):45, 1973. doi:10.1007/BF01016796 M. Lopez-Fernandez and C. Palencia. On the numerical inversion of the Laplace transform in certain holomorphic mappings. Appl. Numer. Math., 51:289–303, 2004. doi:10.1016/j.apnum.2004.06.015 S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. Simul., 6(4):1146–1168, 2008. doi:10.1137/060678154 M. Magdziarz, A. Weron, and K. Weron. Fractional Fokker–Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E, 75:016708, 2007. doi:10.1103/PhysRevE.75.016708 F. Mainardi, R. Gorenflo, and A. Vivoli. Beyond the Poisson renewal process: A tutorial survey. J. Comput. Appl. Math., 2007. doi:10.1016/j.cam.2006.04.060 W. McLean. Regularity of solutions to a time-fractional diffusion equation. ANZIAM J., 52(2):123–138, 2010. doi:10.1017/S1446181111000617 W. McLean and V. Thomee. Time discretization of an evolution equation via Laplace transforms. IMA J. Numer. Anal., 24:439–463, 2004. doi:10.1093/imanum/24.3.439 R. Metzler and J. Klafter. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339:1–77, 2000. doi:10.1016/S0370-1573(00)00070-3 C. Moler and C. Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix, 25 Years Later. SIAM Rev., 45(1):3–49, 2003. doi:10.1137/S00361445024180 E. W. Montroll and G. H. Weiss. Random Walks on Lattices. II. J. Math. Phys., 6(2):167–181, 1965. doi:10.1063/1.1704269 I. Moret and P. Novati. On the Convergence of Krylov Subspace Methods for Matrix Mittag–Leffler Functions. SIAM J. Numer. Anal., 49(5):2144–2164, 2011. doi:10.1137/080738374 I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999. http://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9 M. Raberto, F. Rapallo, and E. Scalas. Semi-Markov Graph Dynamics. PLoS ONE, 6(8):e23370, 2011. doi:10.1371/journal.pone.0023370 S. C. Reddy and L. N. Trefethen. Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math., 54(6):1634–1649, 1994. doi:10.1137/S0036139993246982 E. B. Saff and A. D. Snider. Fundamentals of complex analysis with applications to engineering and science. Pearson Education, 2003. http://www.pearsonhighered.com/educator/product/Fundamentals-of-Complex-Analysis-with-Applications-to-Engineering-Science-and-Mathematics/9780139078743.page E. Scalas, R. Gorenflo, and F. Mainardi. Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E, 69:011107, 2004. doi:10.1103/PhysRevE.69.011107 D. Sheen, I. H. Sloan, and V. Thomee. A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal., 23:269–299, 2003. doi:10.1093/imanum/23.2.269 M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425 G. Strang and S. MacNamara. Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev., 56(3):525–546, 2014. doi:10.1137/120897572 A. Talbot. The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl., 23:97–120, 1979. doi:10.1093/imamat/23.1.97 L. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, Philadelphia, 2013. http://bookstore.siam.org/ot128/ L. N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005. http://press.princeton.edu/titles/8113.html L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule. SIAM Rev., 56(3):385–458, 2014. doi:10.1137/130932132 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier Science, 2001. http://www.elsevier.com/books/stochastic-processes-in-physics-and-chemistry/van-kampen/978-0-444-52965-7 J. A. C. Weideman. Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal., 30:334–350, 2010. doi:10.1093/imanum/drn074 J. A. C. Weideman and L. N. Trefethen. Parabolic and hyperbolic contours for computing the bromwich integral. Math. Comput., 76:1341–1356, 2007. doi:10.1090/S0025-5718-07-01945-X T. G. Wright. Eigtool, 2002. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/. Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu. Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions. ANZIAM J., 52:395–409, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3791/146

Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian

An important framework for modelling and simulation of chemical reactions is a Markov process sometimes known as a master equation. Explicit solutions of master equations are rare; in general the explicit solution of the governing master equation for a bimolecular reaction remains an open question. We show that a solution is possible in special cases. One method of solution is diagonalization. The crucial class of matrices that describe this family of models are non-symmetric graph Laplacians. We illustrate how standard numerical algorithms for finding eigenvalues fail for the non-symmetric graph Laplacians that arise in master equations for models of chemical kinetics. We propose a novel way to explore the pseudospectra of the non-symmetric graph Laplacians that arise in this class of applications, and illustrate our proposal by Monte Carlo. Finally, we apply the Magnus expansion, which provides a method of simulation when rates change in time. Again the graph Laplacian structure presents some unique issues: standard numerical methods of more than second-order fail to preserve positivity. We therefore propose a method that achieves fourth-order accuracy, and maintain positivity. References A. Basak, E. Paquette, and O. Zeitouni. Regularization of non-normal matrices by gaussian noise–-the banded toeplitz and twisted toeplitz cases. In Forum of Mathematics, Sigma, volume 7. Cambridge University Press, 2019. doi:10.1017/fms.2018.29. S. Blanes, F. Casas, J. A. Oteo, and J. Ros. The magnus expansion and some of its applications. Phys. Rep., 470(5-6):151–238, 2009. doi:10.1016/j.physrep.2008.11.001. B. A. Earnshaw and J. P. Keener. Invariant manifolds of binomial-like nonautonomous master equations. SIAM J. Appl. Dyn. Sys., 9(2):568–588, 2010. doi10.1137/090759689. J. Gunawardena. A linear framework for time-scale separation in nonlinear biochemical systems. PloS One, 7(5):e36321, 2012. doi:10.1371/journal.pone.0036321. A. Iserles and S. MacNamara. Applications of magnus expansions and pseudospectra to markov processes. Euro. J. Appl. Math., 30(2):400–425, 2019. doi:10.1017/S0956792518000177. S. MacNamara. Cauchy integrals for computational solutions of master equations. ANZIAM Journal, 56:32–51, 2015. doi:10.21914/anziamj.v56i0.9345. S. MacNamara, A. M. Bersani, K. Burrage, and R. B. Sidje. Stochastic chemical kinetics and the total quasi-steady-state assumption: Application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys., 129:095105, 2008. doi:10.1063/1.2971036. S. MacNamara and K. Burrage. Stochastic modeling of naive T cell homeostasis for competing clonotypes via the master equation. SIAM Multiscale Model. Sim., 8(4):1325–1347, 2010. S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. and Sim., 6(4):1146–1168, 2008. doi:10.1137/060678154. S. MacNamara, Wi. McLean, and K. Burrage. Wider contours and adaptive contours, pages 79–98. Springer International Publishing, 2019. doi:10.1007/978-3-030-04161-8_7. M. J. Shon. Trapping and manipulating single molecules of DNA. PhD thesis, Harvard University, 2014. http://nrs.harvard.edu/urn-3:HUL.InstRepos:11744428. M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425. C. Timm. Random transition-rate matrices for the master equation. Phys. Rev. E, 80(2):021140, 2009. doi:10.1103/PhysRevE.80.021140. L. N. Trefethen and M. Embree. Spectra and pseudospectra: The behavior of nonnormal matrices and operators. Princeton University Press, 2005. https://press.princeton.edu/books/hardcover/9780691119465/spectra-and-pseudospectra

Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation

Stochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have been proposed in the literature by Stirk, Molina-París and van den Berg (2008). A stochastic modelling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probability density function (PDF) approaches can be expensive but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by the Finite State Projection and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise in some parameter regimes. Time-dependent propensities naturally arise in immunological processes due to, for example, age-dependent effects. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology

Proceedings of the 2020 Computational Techniques and Applications Conference

CTAC-2020 Sydney, Australia 30 August -- 2 September 2020 The 20th biennial Computational Techniques and Applications Conference was originally planned to take place at the University of New South Wales from 30 August until 2 September, 2020. As the Covid-19 pandemic worsened, it became clear that travel restrictions and other measures meant that the conference could not be held in its usual form, and the organising committee decided to move the conference online. The ANZIAM Special Interest Group in Computational Mathematics is responsible for this series of conferences, the first of which was held in 1981. Participants of the conference are able to submit a short article based on their presentation for publication in this special section of the ANZIAM Journal (Electronic Supplement). The editors, William McLean and Shev MacNamara, thank the referees whose efforts have helped improve the quality of these conference proceedings. This Special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed conference papers. The eight keynote presentations were as follows. Konstantin Brenner, University of Nice Sophia-Antipolis Numerical modeling of two-phase flow in fractured porous media Michael Feischl, TU ViennaNumerical analysis and machine learning. David Harvey, UNSWFast Fourier transforms of prime length. Trevor McDougall, UNSWSome mathematical aspects of physical oceanography. Kate Smith-Miles, University of MelbourneIn search of algorithmic trust \dots show us the stress-testing! Catherine Powell, University of ManchesterAdaptive and multilevel stochastic Galerkin methods for PDEs with uncertain inputs. Aretha Teckentrup, University of EdinburghConvergence of Gaussian process emulators with estimated hyper-parameters. Alex Townsend, Cornell UniversityThe ultraspherical spectral method. The presentation by Trevor McDougall was a public lecture. The conference attracted 131 registered participants, and featured a total of 81 contributed talks. CTAC2020 Organising Committee Josef Dick, UNSW Bishnu Lamichhane, University of Newcastle Ngan Le, Monash University Quoc Thong Le Gia (Chair), UNSW Shev MacNamara, UTS William McLean, UNSW Vera Roschchina, UNSW Thanh Tran, UNSW CTAC2020 Scientific Committee Steve Armfield, University of Sydney Jerome Droniou, Monash University Frances Kuo, UNSW Markus Hegland, ANU Stephen Roberts, ANU Ian H. Sloan (Chair), UNSW Ian Turner, QUT Acknowledgements We gratefully acknowledge support from the following sponsors: School of Mathematics and Statistics, UNSW. The New South Wales Government. The Modelling and Simulation Society of Australia and New Zealand. The Mathematics of Computation and Optimization (MoCaO) special interest group of the Australian Mathematical Society

Stochastic Modeling of Naive T Cell Homeostasis for Competing Clonotypes via the Master Equation

Stochastic models for competing clonotypes of T cells by multivariate, continuoustime, discrete state, Markov processes have recently been proposed in the literature. A stochastic modeling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probabilitydensity function (PDF) approaches can be expensive, but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by projections and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise. Both experimental and theoretical investigations have emphasized the need to take into account effects due to aging. Thus time-dependent propensities naturally arise in immunological processes. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods, but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology. © 2010 Society for Industrial and Applied Mathematics