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The stochastic computation of electromagnetic (EM) problems is a relatively new topic, yet very important in understanding the true physics due to uncertainties associated with them. To deal with these uncertainties, the traditional Monte Carlo (MC) method can be applied. However, it requires a very large number of simulations to reach convergence, which makes it very computationally expensive. This dissertation discusses alternative stochastic methods which are more efficient than the MC method as well as their applications in EM modeling. It consists of three major parts. The first part presents the use of the generalized polynomial chaos (gPC) method for stochastic computation. In the gPC method, the stochastic solutions we are interested in are approximated by polynomial expansion in terms of input random variables, truncated at a finite order. Based on the distribution of random inputs, there is an optimal choice for a polynomial basis to achieve the fastest convergence. By taking the inner product of the testing basis, we seek to solve the Maxwell equations in a weak form. The second part focuses on applying the Stochastic Collocation (SC) method for stochastic computation. In the SC method, the stochastic solution is constructed via polynomial interpolation. One only needs a small number of repeated simulations to get accurate statistics, which makes it computationally favorable. The selection of collocation points is of greatest importance in the SC method, especially in the multi-dimensional problems, since the total simulation cost is proportional to the number of collocation points. A sparse grid (SG) technique can be used for generating collocation points much easier than the tensor product rule in the multi-dimensional problems. The third part emphasizes analyzing uncertainty problems with correlations. Most of the stochastic methods are based on the assumption that the probability space can be characterized by a set of independent random variables (RV). However, this requirement may not be met in some cases. For example, the random process is a function of spatial coordinates or RVs that are correlated in the probability space. To deal with spatial correlation, the Karhunen-Loeve expansion technique can be applied. And for correlated Gaussian RVs, a linear mapping technique can transform them into uncorrelated RVs.



Collocation, Stochastic, Electromagnetics, FDTD, Monte Carlo, GPC, Sparse grid, Correlations, KLE, Linear mapping