Cauchy integrals for computational solutions of master equations

MacNamara, Shev

doi:10.21914/anziamj.v56i0.9345

Cited by 8 publications

(15 citation statements)

References 43 publications

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“…Such an unfavourable situation can certainly arise. It is demonstrated to happen on the parabolic contour displayed here in Figure 6 for bimolecular reactions [12]. The issue is also addressed by In'T Hout and Weideman [7] for Black-Scholes models.…”

Section: Estimating the Field Of Valuesmentioning

confidence: 68%

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Wider Contours and Adaptive Contours

2019

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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

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Section: Estimating the Field Of Valuesmentioning

confidence: 68%

“…The process is a Markov process if and only if the waiting times are exponential random variables, in which case the K(j, t) appearing in the convolutions in (2) are Dirac delta distributions and (2) reduces to the usual equation in (1). In the special case of Mittag-Leffler waiting times, (2) can be re-written as [12,13]:…”

Section: Introductionmentioning

confidence: 99%

Wider Contours and Adaptive Contours

2019

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“…A simple example of a graph Laplacian on a line of nodes appears in [40], and, like the matrix exponential, it has been shown that a Mittag-Leffler function [16] of a graph Laplacian matrix is also a stochastic matrix [33]. All of this suggests research into non-Markovian generalisations of Gillespie-like stochastic simulation algorithms allowing waiting times not exclusively drawn from an exponential distribution [31]. It is known that if we generalise (2.1) to a Caputo fractional derivative of order 0 < α < 1, d α /dt α , then the matrix exponential is generalised to the Mittag-Leffler function E α , so that (2.1) becomes d α p/dt α = Ap with solution p(t) = E α (t α A)p(0).…”

Section: Matrix Functions Of Graph Laplaciansmentioning

confidence: 99%

“…Of course, similar reasoning applies also to a product y = Zu. The only difference vis-á-vis (30) is that now y…”

Section: Evaluating the Exponential Via (29)mentioning

confidence: 99%

Applications of Magnus expansions and pseudospectra to Markov processes

Iserles

MacNamara

2018

Eur. J. Appl. Math

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New directions in Markov processes and research on master equations are showcased by example. The utility of Magnus expansions for handling time-varying rates is demonstrated. The useful notion in applied mathematics often turns out to be the pseudospectra and not simply the eigenvalues. We highlight that general principle with our own examples of Markov processes where exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods. As a motivating application, isomerisation provides a running example and an illustration of our approaches to chemical kinetics. We also present a brief example of a totally asymmetric exclusion process.

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“…Putting α ← 1 in these formulas recovers the survival time distribution, cdf , and waiting time density, corresponding to an exponential distribution with parameter λ and mean value 1/λ. Unlike the exponential case, equation (37) shows that the Mittag-Leffler function and hence also the solution of (7) is not differentiable at t = 0 + . In general care must be taken when differentiating near zero, as happens later in (59).…”

Section: A Probabilistic Interpretationmentioning

confidence: 99%