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 modeled 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--Loeve expansion for the random field with covariance function belonging to the anisotropic Whittle--Matern 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, thi...

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