one-dimensional optimization

Abstract. The method of one-dimensional optimization is explained in the context of capillary-gravity waves.

1. Introduction.

R provides functions for both one-dimensional and multi-dimensional optimization. The second topic is much more complicated than the former (see e.g. Nocedal 1999) and will be left for another day.

A convenient function for 1D optimization is optimize(), also known as optimise(). Its first argument is a function whose minimum (or maximum) is sought, and the second is a two-element vector giving the range of values of the independent variable to be searched. (See ?optimize for more.)

2. Application.

As an example, consider the phase speed of deep gravity-capillary waves, which is given by \omega/k where \omega is the frequency and k is the wavenumber, and the two are bound together with the dispersion relationship \omega^2=gk+\sigma k^3/\rho, where g is the acceleration due to gravity, \sigma is the surface tension parameter (0.074 N/m for an air-water interface) and \rho is the water density (1000 kg/m^3 for fresh water). This yields wave speed given by the following R function.

phaseSpeed <- function(k) {
    g <- 9.8
    sigma <- 0.074  # water-air
    rho <- 1000  # fresh water
    omega2 <- g * k + sigma * k^3/rho
    sqrt(omega2)/k
}

Readers with a background in the topic of waves may know that there is a minimum phase speed at wavelengths \lambda of about 0.02m, or a wavenumber k=2\pi/\lambda of roughly 300. It always makes sense to plot a function to be optimized, if only to check that it has been coded correctly, so that's the next step. We’ll use a range of half an order of magnitude (factor of 3) smaller and larger k.

k <- seq(100, 1000, length.out = 100)
par(mar = c(3, 3, 1, 1), mgp = c(2, 0.7, 0))
plot(k, phaseSpeed(k), type = "l",
     xlab = "Wavenumber", ylab = "Phase speed")
Wave speed for capillary-gravity waves in infinitely deep water

Wave speed for capillary-gravity waves in infinitely deep water

The results suggest that the range of k illustrated in the diagram contains the minimum, so we provide that to optimize().

o <- optimize(phaseSpeed, range(k))
phaseSpeed(o$minimum)
## [1] 0.2321

This speed is not especially fast; it would take about a heartbeat to move past your hand. (Or, about as fast as the waves that pulse your heart … presumably with different physics, though!)

3. Exercises

  1. Use str(o) to learn about the contents of the optimized solution.

  2. Use abline() to indicate the wavenumber at the speed minimum.

  3. Try other functions that are of interest to you.

  4. Use the multi-dimensional optimizer named optim() on this problem.

References

Jorge Nocedal and Stephen J. Wright, 1999. Numerical optimization. Springer series in operations research. New York, NY, USA.

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