Uncertainty Quantification for Low-Frequency, Time-Harmonic Maxwell Equations with Stochastic Conductivity Models

We consider an Uncertainty Quantification (UQ) problem for the low-frequency, time-harmonic Maxwell equations with conductivity that is modelled by a fixed layer and a lognormal random field layer. We formulate and prove the well-posedness of the stochastic and the parametric problem; the latter obtained using a Karhunen-Loève expansion for the random field with covariance function belonging to the anisotropic Whittle-Matérn class. For the approximation of the infinite-dimensional integrals in the forward UQ problem, we employ the Sparse Quadrature (SQ) method and we prove dimension-independent convergence rates for this model. These rates depend on the sparsity of the parametric representation for the random field and can exceed the convergence rate of the Monte-Carlo method, thus enabling a computationally tractable calculation for Quantities of Interest. To further reduce the computational cost involved in large-scale models, such as those occurring in the Controlled-Source Electromagnetic Method, this work proposes a combined SQ and model reduction approach using the Reduced Basis (RB) and Empirical Interpolation (EIM) methods. We develop goal-oriented, primal-dual based, a posteriori error estimators that enable an adaptive, greedy construction of the reduced problem using training sets that are selected from a sparse grid algorithm. The performance of the SQ algorithm is tested numerically and shown to agree with the estimates. We also give numerical evidence for the combined SQ-EIM-RB method that suggests a similar convergence rate. Finally, we report numerical results that exhibit the behaviour of quantities in the algorithm.