Partial differential equations are a versatile tool to model and eventually simulate physical phenomena. An important aspect in view of the reliability and relevance of such simulations are uncertainties arising from unknown parameters and measurement errors. In particular, the modelling and discretization of uncertainties of the computational domain requires special care. Such uncertainties emerge in a natural fashion when considering products fabricated by line production which are subject to manufacturing tolerances or shapes which are obtained by remote sensing techniques, like e.g. ultrasound or magnetic resonance imaging. This minisymposium is dedicated to recent developments in the numerical treatment of shape uncertainties in partial differential equations and welcomes contributions addressing analytical aspects, forward modelling, assimilation of measurement data, optimization, and applications.
14:00
Domain Uncertainty Quantification in Computational Electromagnetics
Carlos Jerez-Hanckes | Universidad Adolfo Ibáñez | Chile
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Carlos Jerez-Hanckes | Universidad Adolfo Ibáñez | Chile
We study the numerical approximation of time-harmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly high-dimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably-parametric, Maxwell-like cavity problems that are posed in a single domain, with inhomogeneous coefficients that possess finite, possibly low spatial regularity, but exhibit holomorphic parametric dependence in the differential operator. Our computational scheme is composed of a sparse-grid interpolation in the high-dimensional parameter domain and an H(curl)-conforming edge element discretization of the parametric problem in the nominal domain. As a stepping-stone in the analysis, we derive a novel Strang-type lemma for Maxwell-like problems in the nominal domain which is of independent interest. Moreover, we accommodate arbitrary small Sobolev regularity of the electric field and also cover uncertain isotropic constitutive or material laws. The shape holomorphy and edge-element consistency error analysis for the nominal problem are shown to imply convergence rates for Multi-level Monte-Carlo and for Quasi-Monte Carlo integration, as well as sparse grid approximations, in uncertainty quantification for computational electromagnetics. Finally, our computational experiments confirm the presented theoretical results.
14:30
Shape Uncertainty Quantification in Computational Nano-Optics
Ulrich Römer | TU Braunschweig | Germany
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Ulrich Römer | TU Braunschweig | Germany
Niklas Georg | TU Braunschweig | Germany
Quantifying uncertainties due to shape variability plays an important role in computational nano-plasmonics or nano-optics. We consider gratings as an important geometry for such applications, where the uncertainty in the shape may originate from resolution limitations in the measurement equipment or reflect manufacturing imperfections. We introduce Maxwell’s source problem with random input data as a mathematical model for these applications. Our final aim is to propagate these uncertainties onto system outputs such as scattering parameters and to carry out a sensitivity or yield analysis.
The main focus of the presentation is a framework for shape uncertainty which can be integrated with spline representations of CAD systems and which allows for the integration of measurement data. In particular, we compare different mechanisms for shape deformations, based either on PDE solutions or parametrization techniques. We also establish sparse polynomial approximation methods and briefly describe several techniques for convergence acceleration, such as adjoint correction. Numerical examples for benchmark gratings used in the physics literature are given for illustration.
15:00
Computation of Electromagnetic Fields Scattered From Objects of Uncertain Shapes Using Multilevel Monte Carlo
Alexander Litvinenko | RWTH Aachen | Germany
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Alexander Litvinenko | RWTH Aachen | Germany
Abdulkadir C. Yucel | KAUST | Saudi Arabia
Hakan Bagci | KAUST | Saudi Arabia
Jesper Oppelstrup | KTH Royal Institute of Technology | Sweden
Eric Michielssen | University of Michigan | United States
Raul F. Tempone | RWTH Aachen | Germany
Computational tools for characterizing scattering from objects of uncertain shapes are highly useful in the fields of electromagnetics, optics, and photonics, where device performance oftentimes is subject to manufacturing tolerances. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties.
For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational work. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.
15:30
A higher order perturbation approach for electromagnetic scattering problems on random domains
Jürgen Dölz | TU Darmstadt | Germany
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Jürgen Dölz | TU Darmstadt | Germany
We are interested in time-harmonic electromagnetic scattering problems on scatterers with uncertain shape. Thus, the scattered field will also be uncertain. Based on the knowledge of the two-point correlation of the domain boundary variations around a reference domain, we derive a perturbation analysis of the mean of the scattered field. The approach is based on the second shape derivative of the scattering problem and will be at least third order accurate in the perturbation amplitude of the domain variations. To compute the required second order correction term, a tensor product equation on the domain boundary has to be solved. We discuss its discretization and efficient solution using boundary integral equations. Numerical experiments in three dimensions will be presented.