Applications of Magnus expansions and pseudospectra to Markov processes

Iserles, Arieh; MacNamara, Shev

doi:10.1017/s0956792518000177

Cited by 13 publications

(12 citation statements)

References 38 publications

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“…Note that the eigvals function that we use to compute the eigenvalues does not always return entirely real eigenvalues in every case (as one would physically expect) and often eigenvalues come in the form of a complex conjugate pairs. This is a common computational error, since we know theoretically that the eigenvalues should be real, and the resultant set of eigenvalues we obtain from eigvals is known as a pseudospectrum [44], which arises since the eigenvalues of these matrices are very sensitive to small perturbations. Importantly however, the usage of the pseudospectrum in Eqs.…”

Section: Master Equation and Analytical Time-dependent Solutionmentioning

confidence: 99%

“…In i(b) and i(c) we explore the various calibration times and plot the error on the calibrated parameters compared to the true values for Etot and f respectively defined in Eqs. ( 43)- (44). Although larger calibration times result in better parameter inference there are still large errors 1 even for large calibration times.…”

Section: Calibration From Multiple Trajectoriesmentioning

confidence: 99%

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Non-equilibrium time-dependent solution to discrete choice with social interactions

Holehouse,

Pollitt

2021

Preprint

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We solve the binary decision model of Brock and Durlauf [1] in time using methods often employed in studies of stochastic complex systems. This solution is valid when not at equilibrium and can be used to exemplify path-dependent behaviours of their model. The solution is computationally fast and is indistinguishable from Monte Carlo simulation. Lock-in effects are observed in some regions of the model's parameter space, and we calculate the time scale of the lock-ins. Curiously, we find that although altruistic agents coalesce more strongly on a particular decision, increasing their utility in the short-term, they are also more prone to being stuck in a non-optimal decision lock-in as compared to selfish agents. Finally, we construct a likelihood function that can be used on non-equilibrium data for model calibration. Even with a well-defined likelihood function, model calibration is difficult unless one has access to data representative of the underlying model.

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Section: Master Equation and Analytical Time-dependent Solutionmentioning

confidence: 99%

Section: Calibration From Multiple Trajectoriesmentioning

confidence: 99%

Non-equilibrium time-dependent solution to discrete choice with social interactions

Holehouse,

Pollitt

2021

Preprint

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“…This issue of stability is related to 'the hump' in the classical literature on the numerical analysis of the matrix exponential, and to the lognorm, and also to the subject of pseudospectra. Nonsymmetric graph Laplacians exhibit significant pseudospectra, manifesting in various ways, such as a more subtle stability analysis, and the failure of standard eigenvalue algorithms (Iserles & MacNamara 2019, MacNamara 2015, Macnamara, Blanes & Iserles 2020. A sufficient condition for stability of operator splitting methods is that each part separately be strongly stable, although this may be too pessimistic in practice.…”

Section: Graph Laplacians and Odesmentioning

confidence: 99%

“…Nonautonomous Laplacian systems, y ′ = A(t)y, have many important applications, including cardiac ion channel kinetics (Earnshaw & Keener 2010a, Earnshaw & Keener 2010b. In special cases, there are also exact solutions for the dynamical solutions, such as the explicit Magnus formulae in (Iserles & MacNamara 2019), and closely related invariant manifolds of binomial-like solutions.…”

Section: Illustrative Examplesmentioning

confidence: 99%

Positivity-preserving methods for population models

Blanes¹,

Iserles²,

MacNamara³

2021

Preprint

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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: our methods are of second order, and they are guaranteed to preserve positivity. 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.

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“…Here we have assumed that the rate constants are equal to unity, c 1 = c 2 = 1, and we have assumed that m = N − 1 where m is the maximum number of molecules. More details, including exact solutions, can be found in [8]. An instance of this matrix when m = 5 is…”

Section: Monomolecular Bimolecular and Trimolecular Modelsmentioning

confidence: 99%

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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Applications of Magnus expansions and pseudospectra to Markov processes

Cited by 13 publications

References 38 publications

Non-equilibrium time-dependent solution to discrete choice with social interactions

Non-equilibrium time-dependent solution to discrete choice with social interactions

Positivity-preserving methods for population models

Wider Contours and Adaptive Contours

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