How Brent's Method is used in the OSD tool
• 28 Jul 2022
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# How Brent's Method is used in the OSD tool

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In order to provide some background to the optimisation method used, a brief description of the algorithm is provided here. First of all, to establish that a maximum of a function exists within a specified interval, we need to establish a bracketing triplet, i.e. the function at three storm durations, such that the function at the interior point is greater than the function at the extrema of the interval. The function [with respect to storm duration] in question here is the water level or flow at the selected node. (Note that for a minimum, a similar argument holds, although henceforth we shall assume that a maximum is being sought.) The function is then approximated by fitting a parabola through the three points, which will then necessarily contain a genuine maximum. The true function is then evaluated at the x-value (storm duration) corresponding to the parabola's maximum, at which step the bracket is narrowed by rejecting the appropriate end point. This process is continued until a maximum is found.

A backup method of a Golden Section interval search (reducing the bracketed interval in the proportion of the golden ratio) is also used in the case of the parabolic interpolation method failing to converge.

Two other considerations in the specific case of the storm duration are noted here:

Since the storm duration is constrained to be an odd integer multiple of the data interval, the calculated storm duration is always rounded to the nearest available.

Due to this, a local maximum is verified by checking that the function is greater than that at the nearest available greater and lesser storm durations.

Since the method will find a local minimum there is no guarantee that, if a function has two or more local maxima within the specified interval, the local maximum will be the global maximum. However, such instances are rare in this kind of analysis.