Efficient computation of stochastic electromagnetic problems using unscented transforms L.R.A.X.de Menezes, A. Ajayi, C. Christopoulos, P. Sewell and G.A. Borges Abstract: The objective of this work is to present a new approach to the modelling of uncertainty in electromagnetic simulations. This contribution enhances the practical utilisation of regular time- and frequency-domain models. The approach is based on the unscented transform (UT) method. The procedure shows results with accuracy similar to the Monte Carlo approach, but uses a smaller number of simulations. This work uses standard transmission line matrix-based solvers. Furthermore, the combination of the UT approach with other time- or frequency-domain electro- magnetic simulators is straightforward. The validation of the technique used both time- and frequency-domain tests. The results show good agreement compared with Monte Carlo outcomes with much reduced simulator use. 1 Introduction Electromagnetic modelling usually assumes a complete knowledge of all necessary relevant factors of the problem. Therefore, it is implicit in this approach that par- ameter uncertainty is a negligible aspect. In most cases, such approximation provides good results. However, there are problems in which uncertainty plays a major part. In certain cases, important factors are known only up to certain accuracy. One example is the uncertainty of electric parameters of living tissue. In other problems, specific par- ameters may even change over time in a random fashion. The causes may be related to aging, moisture and/or temp- erature. All these effects introduce uncertainty to the problem and the calculated results. In these cases, the inclusion of uncertainty in simulations is usually performed with the Monte Carlo technique [1]. This method uses a large set of random variables as input parameters to the simulation. These variables are chosen according to a previously known distribution relevant to the model. Once the simulator produces the results for the set, the program or a post-processing procedure calculates the statistical parameters of the solution. The typical Monte Carlo approach utilises several hundred thousand simulations to obtain the statistics of the final result. Although this procedure is quite accurate, the number of simulations needed is unfeasible for most practical appli- cations of electromagnetic simulation. However, there are other possible approaches. One possi- bility is to condense the Monte Carlo minimal set with par- ticle filters [2]. The new set is usually constructed from selection of random samples with appropriate character- istics. Therefore the number of necessary simulations is reduced. Unfortunately, the overall set can still be quite large. In a different approach, it is possible to combine a Taylor series representation of the uncertainty with the simulation method. This is the basic idea of the first-order-second-moment method (FOSM). This has been performed for the finite element technique (FEM) for uncer- tainty modelling [3]. The same concept is applicable to the transmission line matrix (TLM) method with the direct sol- ution technique (DST) [4]. The DST uses a Taylor expan- sion similar to the FOSM method. The solution is calculated considering the separate matrix equations for the expected value and variance. The DST provides good results for practical variances. However, FOSM-based approaches usually require modifications to the algorithm and global representation of variables, which may be a drawback considering storage. Instead of the usual approaches described above, this work proposes a different solution. The idea is an efficient combination of the electromagnetic simulation with the unscented transform (UT). The UT was developed by Julier and Uhlman in 1997 [5]. It has been used in several domains of electrical engineering [6, 7]. The idea is to approximate a nonlinear mapping by a set of selected points (sigma points). Both expected value and variance of the mapping are available through a weighted average of the sigma points. The method is quite similar to the moment design technique (MDT) originally developed by Zhang [8], Taguchi [9] and John and Nicholas [10]. The technique uses the moments of the probability distribution function to develop a selected set of points. In [11], these points are called design values. In the UT approach, they are called sigma points. This work will use the sigma point nomenclature and the moment design approach to find the weights and points. This combination shows that the UT and the MDT are equivalent if the Taylor approxi- mation is truncated in the same form. The paper is organised as follows: Section 2 details the basic theory behind the combination of UT and TLM. Section 3 details some results from the combination of UT and TLM and Section 4 is the conclusion. The first part of the next section demonstrates that the UT and MDT approaches may be combined with different kinds of time and frequency domain electromagnetic simulators. # The Institution of Engineering and Technology 2008 doi:10.1049/iet-smt:20070050 Paper first received 23rd May and in revised form 17th August 2007 L.R.A.X. de Menezes and G.A. Borges are with the Departamento de Engenharia Ele ´trica, Faculdade de Tecnologia, Universidade de Brasilia, Brası ´lia-DF 70910-900, Brazil A. Ajayi, C. Christopoulos and P. Sewell are with George Green Institute for Electromagnetics Research, the University of Nottingham, Nottingham NG7 2RD, UK E-mail: leonardo@ene.unb.br IET Sci. Meas. Technol., 2008, 2, (2), pp. 88–95 88