Multilevel domain uncertainty quantification in computational electromagnetics

Rubén Aylwin, Carlos Jerez-Hanckes, Christoph Schwab, Jakob Zech

Research output: Contribution to journalArticlepeer-review


We continue our study [R. Aylwin, C. Jerez-Hanckes, C. Schwab and J. Zech, Domain uncertainty quantification in computational electromagnetics, SIAM/ASA J. Uncertain. Quant. 8 (2020) 301–341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback solutions on the nominal domain is characterized in piecewise Sobolev spaces. We prove error convergence rates and optimize the algorithmic steering of parameters for edge-element discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution mappings. All calculations are performed on the polyhedral nominal domain, which enables the use of standard simplicial finite element meshes. We provide a rigorous fully discrete error analysis and show, in all cases, that dimension-independent algebraic convergence is achieved. For the multilevel sparse-grid quadrature methods, we prove higher order convergence rates free from the so-called curse of dimensionality. Numerical experiments confirm our theoretical results and verify the superiority of the sparse-grid methods.

Original languageEnglish
Pages (from-to)877-921
Number of pages45
JournalMathematical Models and Methods in Applied Sciences
Issue number4
StatePublished - 1 Apr 2023
Externally publishedYes


  • Computational electromagnetics
  • Smolyak quadrature
  • finite elements
  • shape holomorphy
  • uncertainty quantification


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