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Multilevel domain uncertainty quantification in computational electromagnetics

Rubén Aylwin

Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Diagonal Las Torres 2700 Santiago, 7941169, Chile

E-mail Address: [email protected]

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Carlos Jerez-Hanckes

Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Diagonal Las Torres 2700 Santiago, 7941169, Chile

E-mail Address: [email protected]

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Christoph Schwab

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, and

Jakob Zech

Interdisciplinary Center for Scientific Computing (IWR), Universität Heidelberg, Heidelberg 69120, Germany

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218202523500264Cited by:1 (Source: Crossref)

Abstract

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.

Communicated by F. Brezzi

Keywords:

AMSC: 35A20, 35B30, 32D05, 35Q61

References

1. G. S. Alberti and Y. Capdeboscq, Elliptic regularity theory applied to time harmonic anisotropic Maxwell’s equations with less than Lipschitz complex coefficients, SIAM J. Math. Anal. 46 (2014) 998–1016. Crossref, Web of Science, Google Scholar
2. R. Aylwin and C. Jerez-Hanckes, The effect of quadrature rules on finite element solutions of Maxwell variational problems, Numer. Math. 147 (2021) 903–936. Crossref, Web of Science, Google Scholar
3. 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. Crossref, Google Scholar
4. A. Barth, C. Schwab and N. Zollinger, Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients, Numer. Math. 119 (2011) 123–161. Crossref, Web of Science, Google Scholar
5. V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000) 273–288. Crossref, Web of Science, Google Scholar
6. A. Buffa, M. Costabel and D. Sheen, On traces for $H (c u r l, Ω)$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002) 845–867. Crossref, Web of Science, Google Scholar
7. A. Buffa and R. Hiptmair, Galerkin boundary element methods for electromagnetic scattering, in Topics in Computational Wave Propagation, Lecture Notes in Computational Science and Engineering, Vol. 31 (Springer, 2003), pp. 83–124. Crossref, Google Scholar
8. A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math. 95 (2003) 459–485. Crossref, Web of Science, Google Scholar
9. M. A. Chkifa, On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection, J. Approx. Theory 166 (2013) 176–200. Crossref, Web of Science, Google Scholar
10. A. Chkifa, A. Cohen and C. Schwab, High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs, J. Found. Comp. Math. 14 (2013) 601–633. Crossref, Web of Science, Google Scholar
11. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (SIAM, 2002). Crossref, Google Scholar
12. P. G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, ed. A. Aziz (Academic Press, 1972), pp. 409–474. Crossref, Google Scholar
13. P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Appl. Mech. Engrg. 1 (1972) 217–249. Crossref, Google Scholar
14. K. A. Cliffe, M. B. Giles, R. Scheichl and A. L. Teckentrup, Multilevel monte carlo methods and applications to elliptic PDEs with random coefficients, Comput. Vis. Sci. 14 (2011) 3. Crossref, Google Scholar
15. A. Cohen, C. Schwab and J. Zech, Shape holomorphy of the stationary Navier–Stokes equations, SIAM J. Math. Anal. 50 (2018) 1720–1752. Crossref, Web of Science, Google Scholar
16. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 2nd edn. (Cambridge Univ. Press, 2014). Crossref, Google Scholar
17. M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Advances in Design and Control, Vol. 22, 2nd edn. (SIAM, 2011). Crossref, Google Scholar
18. J. Diestel and J. Uhl, Vector Measures (Amer. Math. Soc., 1977). Crossref, Google Scholar
19. P. Dular and C. Geuzaine, GetDP reference manual: The documentation for GetDP, a general environment for the treatment of discrete problems, http://getdp.info. Google Scholar
20. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Vol. 159 (Springer Science & Business Media, 2004). Crossref, Google Scholar
21. A. Ern and J.-L. Guermond, Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions, Comput. Math. Appl. 75 (2018) 918–932. Crossref, Web of Science, Google Scholar
22. R. N. Gantner, Computational higher-order quasi-monte carlo for random partial differential equations, Ph.D. thesis, ETH Zurich (2017). Google Scholar
23. C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Eng. 79 (2009) 1309–1331. Crossref, Web of Science, Google Scholar
24. R. Hiptmair, L. Scarabosio, C. Schillings and C. Schwab, Large deformation shape uncertainty quantification in acoustic scattering, Adv. Comput. Math. 44 (2018) 1475–1518. Crossref, Web of Science, Google Scholar
25. T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. Vol. I. Martingales and Littlewood–Paley Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics Vol. 63 (Springer, 2016). Google Scholar
26. C. Jerez-Hanckes, C. Schwab and J. Zech, Electromagnetic wave scattering by random surfaces: Shape holomorphy, Math. Models Methods Appl. Sci. 27 (2017) 2229–2259. Link, Web of Science, Google Scholar
27. A. Klimke, Uncertainty modeling using fuzzy arithmetic and sparse grids, Ph.D. thesis, Universität Stuttgart (2005). Google Scholar
28. R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien, Math. Z. 106 (1968) 213–224. Crossref, Web of Science, Google Scholar
29. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge Univ. Press, 2000). Google Scholar
30. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford Univ. Press, 2003). Crossref, Google Scholar
31. H. Royden and P. Fitzpatrick, Real Analysis, 4th edn. (Prentice-Hall Inc, 2010). Google Scholar
32. S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, Vol. 39 (Springer-Verlag, 2011), translated and expanded from the 2004 German original. Crossref, Google Scholar
33. C. Schwab and J. Zech, Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ, Anal. Appl. 17 (2019) 19–55. Link, Web of Science, Google Scholar
34. A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd edn. (John Wiley & Sons, 1980). Google Scholar
35. C. Weber, Regularity theorems for Maxwell’s equations, Math. Methods Appl. Sci. 3 (1981) 523–536. Crossref, Google Scholar
36. J. Zech, Sparse-Grid Approximation of High-Dimensional Parametric PDEs, Dissertation, ETH Zürich (2018). Google Scholar
37. J. Zech, D. Dũng and C. Schwab, Multilevel approximation of parametric and stochastic PDEs, Math. Models Methods Appl. Sci. 29 (2019) 1753–1817. Link, Web of Science, Google Scholar