A practical comparison between the spectral techniques in solving the SDEs

Abstract

Purpose The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes. Design/methodology/approach The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques. Findings The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems. Research limitations/implications Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions. Originality/value This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.

Uncertainty Quantification Spectral Technique for the Stochastic Point Reactor with Random Parameters

“…WIE kernels have exponential convergence in case of Gaussian and near-Gaussian processes [9]. This property of WIE guarantees the convergence of the truncated expansion.…”

“…Kernels of higher order can be deduced similarly. Using gPC in systems (9) and (10) to account for the randomness due to parameters. For example, the group decay constants λ j ; j = 1 • • • J are assumed to be independent random variables, for example:…”

“…Systems (13) and (14) can be solved using Runge-Kutta or simply using fixed-point (Picard's) technique. We can note that systems (13) and (14) will be reduced to systems (9) and 10…”

“…The random variations are parameters with given probability distributions and hence the statistical properties, such as mean and variance, are known. In case of randomness due to noise, many techniques (e.g., EM and WIE) can be used [9]. WIE has the advantage of having a high, sometimes exponential, rate of convergence.…”

Analysis of the Stochastic Population Model with Random Parameters

“…On the other hand, the decomposition Fourier-like techniques such as the Wiener–Hermite expansion (WHE) [ 9 ] and the Wiener chaos expansion (WCE) [ 10 , 11 ] have higher orders of convergence. The convergence order of the spectral techniques can be exponential for linear and near-Gaussian processes [ 12 ].…”