3 research outputs found
Golden Ratio Phenomenon of Random Data Obeying von Karman Spectrum
von Karman originally deduced his spectrum of wind speed fluctuation based on the Stokes-Navier equation. Taking into account, the practical issues of measurement and/or computation errors, we suggest that the spectrum can be described from the point of view of the golden ratio. We call it the golden ratio phenomenon of the von Karman spectrum. To depict that phenomenon, we derive the von Karman spectrum based on fractional differential equations, which bridges the golden ratio to the von Karman spectrum and consequently provides a new outlook of random data following the von Karman spectrum in turbulence. In addition, we express the fractal dimension, which is a measure of local self-similarity, using the golden ratio, of random data governed by the von Karman spectrum
Golden Ratio Phenomenon of Random Data Obeying von Karman Spectrum
von Karman originally deduced his spectrum of wind speed fluctuation based on the Stokes-Navier equation. Taking into account, the practical issues of measurement and/or computation errors, we suggest that the spectrum can be described from the point of view of the golden ratio. We call it the golden ratio phenomenon of the von Karman spectrum. To depict that phenomenon, we derive the von Karman spectrum based on fractional differential equations, which bridges the golden ratio to the von Karman spectrum and consequently provides a new outlook of random data following the von Karman spectrum in turbulence. In addition, we express the fractal dimension, which is a measure of local self-similarity, using the golden ratio, of random data governed by the von Karman spectrum
Optimization with Discrete Simultaneous Perturbation Stochastic Approximation Using Noisy Loss Function Measurements
Discrete stochastic optimization considers the problem of minimizing (or
maximizing) loss functions defined on discrete sets, where only noisy
measurements of the loss functions are available. The discrete stochastic
optimization problem is widely applicable in practice, and many algorithms have
been considered to solve this kind of optimization problem. Motivated by the
efficient algorithm of simultaneous perturbation stochastic approximation
(SPSA) for continuous stochastic optimization problems, we introduce the middle
point discrete simultaneous perturbation stochastic approximation (DSPSA)
algorithm for the stochastic optimization of a loss function defined on a
p-dimensional grid of points in Euclidean space. We show that the sequence
generated by DSPSA converges to the optimal point under some conditions.
Consistent with other stochastic approximation methods, DSPSA formally
accommodates noisy measurements of the loss function. We also show the rate of
convergence analysis of DSPSA by solving an upper bound of the mean squared
error of the generated sequence. In order to compare the performance of DSPSA
with the other algorithms such as the stochastic ruler algorithm (SR) and the
stochastic comparison algorithm (SC), we set up a bridge between DSPSA and the
other two algorithms by comparing the probability in a big-O sense of not
achieving the optimal solution. We show the theoretical and numerical
comparison results of DSPSA, SR, and SC. In addition, we consider an
application of DSPSA towards developing optimal public health strategies for
containing the spread of influenza given limited societal resources