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

Abstract

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 infinitedimensional 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.

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

“…There, also the parametric regularity analysis of the parametric electric and magnetic fields is discussed, albeit by real-variable methods. The setting in [71] is, however, so that the presently developed, complex variable methods can be brought to bear on it. We refrain from developing the details.…”

“…Similar models are available for time-harmonic, electromagnetic waves in dielectric media with uncertain conductivity. We refer to [71], where log-Gaussian models are employed. There, also the parametric regularity analysis of the parametric electric and magnetic fields is discussed, albeit by real-variable methods.…”

“…Numerical analysis of PDEs with GRF inputs refers on the one hand to questions of efficient, numerical simulation methods of GRF inputs (see, e.g., [83,45,53,11,13,92] and the references there), and on the other hand to the efficient numerical approximation of corresponding PDE solution ensembles, which arise for GRF inputs, in particular efficient representation of such solution ensembles (see [67,9,8]), and numerical quadrature of corresponding solution fields (see [67,77,46,54,63,62,90] and the references there). Applications include for instance subsurface flow models (see, e.g., [45,52]) but also other PDE models for media with uncertain properties (see, e.g., [71] for electromagnetics).…”

Analysis and Application of Single Level, Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs

“…In 2016, Römer et al [37] discussed a stochastic nonlinear magnetostatic problem solved by the stochastic collocation method. In 2018, Kamilis and Polydorides [25] considered an UQ problem for the low-frequency, time-harmonic Maxwell's equations with lognormal random conductivity. Also Jerez et al [23] and Hao et al [20] investigated the time-harmonic Maxwell's equations with random interfaces.…”