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Adaptive stepsize based on control theory for stochastic differential equations

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Title: Adaptive stepsize based on control theory for stochastic differential equations
Authors: Burrage, P.M. pmb@maths.uq.edu.au, Herdiana, R.1 ratna@maths.uq.edu.au, Burrage, K.1 kb@maths.uq.edu.au
Source: Journal of Computational & Applied Mathematics. Sep2004, Vol. 170 Issue 2, p317-336. 20p.
Subjects: Stochastic differential equations, Control theory (Engineering), Industrial efficiency, Stochastic processes
Abstract: The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case.In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. [Copyright &y& Elsevier]
Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 13624176
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PubTypeId: academicJournal
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  Data: Adaptive stepsize based on control theory for stochastic differential equations
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  Data: <searchLink fieldCode="AR" term="%22Burrage%2C+P%2EM%2E%22">Burrage, P.M.</searchLink><i> pmb@maths.uq.edu.au</i><br /><searchLink fieldCode="AR" term="%22Herdiana%2C+R%2E%22">Herdiana, R.</searchLink><relatesTo>1</relatesTo><i> ratna@maths.uq.edu.au</i><br /><searchLink fieldCode="AR" term="%22Burrage%2C+K%2E%22">Burrage, K.</searchLink><relatesTo>1</relatesTo><i> kb@maths.uq.edu.au</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Computational+%26+Applied+Mathematics%22">Journal of Computational & Applied Mathematics</searchLink>. Sep2004, Vol. 170 Issue 2, p317-336. 20p.
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  Data: <searchLink fieldCode="DE" term="%22Stochastic+differential+equations%22">Stochastic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Control+theory+%28Engineering%29%22">Control theory (Engineering)</searchLink><br /><searchLink fieldCode="DE" term="%22Industrial+efficiency%22">Industrial efficiency</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+processes%22">Stochastic processes</searchLink>
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  Data: The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case.In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. [Copyright &y& Elsevier]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.cam.2004.01.027
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      – Code: eng
        Text: English
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        PageCount: 20
        StartPage: 317
    Subjects:
      – SubjectFull: Stochastic differential equations
        Type: general
      – SubjectFull: Control theory (Engineering)
        Type: general
      – SubjectFull: Industrial efficiency
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      – SubjectFull: Stochastic processes
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      – TitleFull: Adaptive stepsize based on control theory for stochastic differential equations
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              Text: Sep2004
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