DELAY SYSTEMS IN BIOLOGICAL MODELS: APPROXIMATION TECHNIQUES**Invited lecture, International Conference on Nonlinear Systems and Applications, July 19-23, Arlington, Texas.

Banks, Harvey Thomas

doi:10.1016/b978-0-12-434150-0.50008-4

Cited by 19 publications

(10 citation statements)

References 17 publications

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“…The first of these is delay differential (also called functional differential) equations where sensitivity with respect to the delays or hysteresis kernels do not in general satisfy the analyticity requirements for use of the Cauchy-Riemann equations. Delay equations have been used in a wide variety of biological applications as well as in many engineering problems -see for examples the references [3,4,7,8,27,28,29,30,33,36,38,39,40,41,43,45,49,53,54,61]. Very early interest in the 1940's focused around the studies of mechanical systems by Minorsky [57,58,59] and slightly later those of Hutchinson [46,47] in biology.…”

Section: Numerical Examplesmentioning

confidence: 99%

The Complex-Step Method for Sensitivity Analysis of Non-Smooth Problems Arising in Biology

Banks¹,

Bekele-Maxwell²,

Bociu³

et al. 2015

EJMCA

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We investigate the complex-step method as applied to compute sensitivities with respect to model "parameters" for several types of examples. We first consider time delayed differential equations (DDEs) whose sensitivities are known to have lack of smoothness or even discontinuities with respect to parameters such as the delays. The second type of "parameter sensitivity" we consider is that of solutions to partial differential equations (PDEs) with respect to boundary conditions which again may not possess smoothness. These sensitivities are fundamental in any type of boundary control formulation such as those we motivate in Section 4 below.Our main focus here is to evaluate the so-called "complex-step methods" for computing such sensitivities. This is of interest since the complex-step method was derived based on the Cauchy-Riemann equations for analytic complex functions. Our computational findings are compared to those using the standard chain rule-based sensitivity differential equations which can be rigorously developed even for derivatives possessing much less regularity than analyticity. Our findings suggest that the complex-step methods are in very good agreement with the usual sensitivity equation results up to some critical step size we call h crit . They can offer significant savings in computational costs for problems driven by complicated dynamical systems with reasonable parameter size.Given an analytic function f , the following is an outline of the general steps for implementing the complex-step method for computing the first derivative, df /dx.1. Define all functions and operators that are not defined for complex arguments such as for example max, min and abs.

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Section: Numerical Examplesmentioning

confidence: 99%

The Complex-Step Method for Sensitivity Analysis of Non-Smooth Problems Arising in Biology

Banks¹,

Bekele-Maxwell²,

Bociu³

et al. 2015

EJMCA

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“…We note that the internal strain variable formulation (2), (5) is equivalent to the constitutive relationship proposed by Fung if one considers an approximation of the relaxation function G by a sum of exponential terms. Various internal strain variable models are investigated in [3] and a good agreement is demonstrated between a two internal strain variable model (e.g., of the form σ = Eε − E 1 ε 1 − E 2 ε 2 ) and undamped simulated data based on the Fung kernel G.…”

Section: Fung's Quasi-linear Modelsmentioning

confidence: 99%

“…For the latter there is a growing body of literature [11,12,13,45,54,68] on the Preisach and related theories for hysteretic control input such as arises in smart material systems [27,39,64]. Applications in which delays and hysteresis play a basic role in the underlying dynamics include sustained efforts in biology with early efforts employing delay systems [1,2,16,17,28,29,30,31,32,42,51,53,55,62] and more recent investigations involving hysteretic probabilistic structures [5,7] as well as classical materials and electromagnetics research (see [8] and the references therein). These applications drove a substantial amount of mathematical and computational research on hysteretic systems in the last half of the 20th century, e.g., see [26,40,41,46,47] among the many books, research monographs, and research articles written.…”

Section: Introductionmentioning

confidence: 99%

A Brief Review of Some Approaches to Hysteresis in Viscoelastic Polymers

Banks¹

2008

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“…Delay equations have been used in a wide variety of biological applications as well as in many engineering problems -see for examples the references [1,2,3,6,7,16,17,18,19,22]. Early applications of delay differential equations date back to 1940's for studies of mechanical systems mechanical systems by Minorsky [27,28,29] and slightly later for studies of population dynamics in biology by Hutchinson [20,21].…”

Section: Introductionmentioning

confidence: 99%

Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

et al. 2016

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In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.

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DELAY SYSTEMS IN BIOLOGICAL MODELS: APPROXIMATION TECHNIQUES**Invited lecture, International Conference on Nonlinear Systems and Applications, July 19-23, Arlington, Texas.

Cited by 19 publications

References 17 publications

The Complex-Step Method for Sensitivity Analysis of Non-Smooth Problems Arising in Biology

The Complex-Step Method for Sensitivity Analysis of Non-Smooth Problems Arising in Biology

A Brief Review of Some Approaches to Hysteresis in Viscoelastic Polymers

Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

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