arXiv subjects

Shev MacNamara

Publications and source records attributed to Shev MacNamara.

8 recordsLinked to original sources

arXiv2020

The reflectionless properties of Toeplitz waves and Hankel waves: an analysis via Bessel functions

We study reflectionless properties at the boundary for the wave equation in one space dimension and time, in terms of a well-known matrix that arises from a simple discretisation of space. It is known that all matrix functions of the familiar second difference matrix representing the Laplacian in this setting are the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave equation is one such matrix function. Here, we study the behaviour of the corresponding waves that we call Toeplitz waves and Hankel waves. We show that these waves can be written as certain linear combinations of even Bessel functions of the first kind. We find exact and explicit formulae for these waves. We also show that the Toeplitz and Hankel waves are reflectionless on even, respectively odd, traversals of the domain. Our analysis naturally suggests a new method of computer simulation that allows control, so that it is possible to choose -- in advance -- the number of reflections. An attractive result that comes out of our analysis is the appearance of the well-known shift matrix, and also other matrices that might be thought of as Hankel versions of the shift matrix. By revealing the algebraic structure of the solution in terms of shift matrices, we make it clear how the Toeplitz and Hankel waves are indeed reflectionless at the boundary on even or odd traversals. Although the subject of the reflectionless boundary condition has a long history, we believe the point of view that we adopt here in terms of matrix functions is new.

math.NA↗

arXiv2018

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.

math.ST↗

arXiv2017

Wider contours and adaptive contours

Contour integrals in the complex plane are the basis of effective numerical methods for computing matrix functions, such as the matrix exponential and the Mittag-Leffler function. These methods provide successful ways to solve partial differential equations, such as convection--diffusion models. Part of the success of these methods comes from exploiting the freedom to choose the contour, by appealing to Cauchy's theorem. However, the pseudospectra of non-normal matrices or operators present a challenge for these methods: if the contour is too close to regions where the norm of the resolvent matrix is large, then the accuracy suffers. Important applications that involve non-normal matrices or operators include the Black--Scholes equation of finance, and Fokker--Planck equations for stochastic models arising in biology. Consequently, it is crucial to choose the contour carefully. As a remedy, we discuss choosing a contour that is wider than it might otherwise have been for a normal matrix or operator. We also suggest a semi-analytic approach to adapting the contour, in the form of a parabolic bound that is derived by estimating the field of values. To demonstrate the utility of the approaches that we advocate, we study three models in biology: a monomolecular reaction, a bimolecular reaction and a trimolecular reaction. Modelling and simulation of these reactions is done within the framework of Markov processes. We also consider non-Markov generalisations that have Mittag-Leffler waiting times instead of the usual exponential waiting times of a Markov process.

math.NA↗

arXiv2017

Magnus expansions and pseudospectra of Master Equations

New directions in research on master equations are showcased by example. Magnus expansions, time-varying rates, and pseudospectra are highlighted. Exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods in some cases. Isomerisation provides a running example and an illustrative application to chemical kinetics. We also give a brief example of the totally asymmetric exclusion process.

math.NA↗

arXiv2016

Error Control for Exponential Integration of the Master Equation

Error estimates for the numerical solution of the master equation are presented. Estimates are based on adjoint methods. We find that a good estimate can often be computed without spending computational effort on a dual problem. Estimates are applicable to both settings with time-independent, and time-dependent propensity functions. The Finite State Projection algorithm reduces the dimensionality of the problem and time propagation is based on an Arnoldi exponential integrator, which in the time-dependent setting is combined with a Magnus method. Local error estimates are devised for the truncation of both the Magnus expansion and the Krylov subspace in the Arnoldi algorithm. An issue with existing methods is that error estimates for truncation of the state space depend on measuring a loss of probability mass in a way that is not usually compatible with the approximation of the exponential. We suggest an alternative error estimate that is compatible with a Krylov approximation to the matrix exponential. Finally, we apply the new error estimates to develop an adaptive simulation algorithm. Numerical examples demonstrate the benefits of the approach.

math.NA↗

arXiv2016

Robustness analysis of spatiotemporal models in the presence of extrinsic fluctuations

We analyze the governing partial differential equations of a model of pole-to-pole oscillations of the MinD protein in a bacterial cell. The sensitivity to extrinsic noise in the parameters of the model is explored. Our analysis shows that overall, the oscillations are robust to extrinsic perturbations in the sense that small perturbations in reaction coefficients result in small differences in the frequency and in the amplitude. However, a combination of analysis and simulation also reveals that the oscillations are more sensitive to some extrinsic time-scales than to others.

math.NA↗

arXiv2016

A Local Inverse Formula and a Factorization

When a matrix has a banded inverse there is a remarkable formula that quickly computes that inverse, using only local information in the original matrix. This local inverse formula holds more generally, for matrices with sparsity patterns that are examples of chordal graphs or perfect eliminators. The formula has a long history going back at least as far as the completion problem for covariance matrices with missing data. Maximum entropy estimates, log-determinants, rank conditions, the Nullity Theorem and wavelets are all closely related, and the formula has found wide applications in machine learning and graphical models. We describe that local inverse and explain how it can be understood as a matrix factorization.

math.NA↗

arXiv2016

Fractional Euler Limits and Their Applications

Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An application to the Schlogl reactions with Mittag-Leffler waiting times is described.

math.CA↗