Unscented Transform and Stochastic Collocation Methods for Stochastic Electromagnetic Compatibility Sebastien Lallechere 1,2 , Pierre Bonnet 1,2 , Ibrahim El Baba 1,2 , and Fran9oise Paladian 12 1 Clermont University, Blaise Pascal University, LASMEA, Clermont-Ferrand, 63000, France 2 CNRS, UMR 6602, LASMEA, Aubiere, 63177, France sebastien .lallechere @ univ-bpclermont .fr Abstract— This paper deals with the current growing interest concerning the use of stochastic techniques for electromagnetic compatability (EMC) issues. Various methods allow to face this problem: obviously, we may focus on the Monte Carlo (MC) formalism but other techniques have been implemented more recently (the unscented transform, UT, or stochastic collocation, SC, for instance). The aim of this work is to solve a stochastic electromagnetic compatibility problem (transmission line) with the UT and SC techniques and to face them with the reference MC results. I. AIMS AND MOTIVATION Nowadays, although one may note the growing interest of the electromagnetic community for measurement techniques and numerical codes with increasing accuracy, the trend is to improve their efficiency by optimizing them. Most of computational works in electromagnetics remain deterministic (i.e., one single result per set of exact input data). Although the uncertainties are intrinsic in electromagnetic analysis (slight variations from large production, environmental factors, repro- ducibility drifts), very few studies are achieved to take them into account efficiently. One of the most spread technique relies on Monte Carlo (MC) simulations. The aim of this work is to focus on two additional methods which present the simplicity of MC with better convergence rates. The two pre- vious formalisms, respectively the unscented transform (UT) method [1] and the stochastic collocation (SC) technique [2], will be detailed from their foundations to the electromagnetic compatability (EMC) application. II. THEORETICAL FOUNDATIONS Many methods are used to take uncertainties into account in electromagnetic simulations; one of the simplest remains as MC. Some probabilistic techniques are also available: without exhaustiveness we may think about the polynomial chaos [3] or the kriging technique [4]. In this article, UT [1] and SC [2] are evaluated and computed. They are two non-intrusive methods whose main advantages are simplicity and efficiency. Contrary to a classical MC simulation, well known for its slow convergence rate, the collocation methods are computationally very interesting since they need a limited number of well-chosen points related to the distribution of the random variables (RVs). From a straightforward point of view, the UT and SC techniques share many similarities, they only differ regarding two or more random parameters. In our model, a stochastic parameter Z will be defined in terms of a RV u as Z = Z°(l + au) (1) with a the magnitude of the uncertainty (percentage), Z° the initial value (central) and without any loss of generality u follows a given distribution law. The random value Z may stand for different parameters: material characteristics (dielec- tric), geometry (for instance size or location of a target) or straightforward the source parameters (magnitude, frequency, ...). Assuming Ii stands for the output given by random value Si (integration point), / may be given by a polynomial approximation using respectively a Taylor expansion [5] and a Lagrangian basis [6] (n degree, n + 1 integration points). For the sake of simplicity, in this paper, we will only consider independent RV even if the UT/SC methods allow considering dependencies. A. Principles of the UT Technique As explained in [1], the use of the UT method is similar to the MC technique. The main difference relies on the number of realizations needed to obtain the statistical moments of a given output. Thus, instead of several thousands of repetitions, only a few selected ones are necessary. 1) Single RV SC Case: Some conditions are required to compute UT for a single RV: we may know both the moments of the RV u and the nonlinear mapping of the random output (I(u)). Its nth order moment may be expressed as E{I(u) n }= I I(u) n pdf(u)du, (2) where pdf(u) is the probability density function of the RV u. A discrete equivalent of the relation (2) is used for the integration f I(u) n pdf(u)du « Yl UiHSi)", ( 3 ) ^ i where Si are the so-called sigma points (for the integration). If the nonlinear mapping I(u) is well behaved, it could be expressed from Taylor polynomial series (gj coefficients) as 978-1-4577-1686-4/11/$26.00 ©2011 IEEE 24