Skip to main content
SpringerLink
Log in

Shape Holomorphy of the Calderón Projector for the Laplacian in \({\mathbb {R}}^2\)

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

We establish the holomorphic dependence of the boundary integral operators (BIOs) comprising the Calderón projector for the Laplacian in two dimensions on the boundary shape. More precisely, we show that the Calderón projector, as an element of the Banach space of bounded linear operators satisfying suitable mapping properties, depends holomorphically on a set of boundaries given by a collection of \({\mathscr {C}}^2\)–regular Jordan curves in \({\mathbb {R}}^2\). In turn, this result implies that the solution of a well-posed first or second kind boundary integral equation (BIE) arising from the boundary reduction of the Laplace problem set on a domain of class \({\mathscr {C}}^2\) in two spatial dimensions depends holomorphically on the shape of the boundary, provided that the corresponding right-hand side does so as well. This property of shape holomorphy is of crucial significance to mathematically justify the construction of sparse parametric shape surrogates of polynomial chaos type, and to prove dimension-independent convergence rates for the approximation of parametric solution families of BIEs in forward and inverse computational shape uncertainty quantification.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In this section we study the complex Fréchet differentiability of maps between complex Banach spaces. For the sake of readability, we drop the word “complex” as it is already implied that Fréchet differentiability only in this sense is established here, as we work only with complex Banach spaces.

References

  1. Alfonseca, M.A., Auscher, P., Axelsson, A., Hofmann, S., Kim, S.: Analyticity of layer potentials and \(L^2\) solvability of boundary value problems for divergence form elliptic equations with complex \(L^\infty \) coefficients. Adv. Math. 226(5), 4533–4606 (2011)

    Article  MathSciNet  Google Scholar 

  2. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, vol. 39, 3rd edn. Springer, New York (2009)

  3. Aylwin, R., Jerez-Hanckes, C., Schwab, Ch., Zech, J.: Domain uncertainty quantification in computational electromagnetics. SIAM/ASA J. Uncertain. Quantif. 8(1), 301–341 (2020)

    Article  MathSciNet  Google Scholar 

  4. Chapko, R., Gintides, D., Mindrinos, L.: The inverse scattering problem by an elastic inclusion. Adv. Comput. Math. 44(2), 453–476 (2018)

    Article  MathSciNet  Google Scholar 

  5. Charalambopoulos, A.: On the Fréchet differentiability of boundary integral operators in the inverse elastic scattering problem. Inverse Prob. 11(6), 1137 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chkifa, A., Cohen, A., DeVore, R., Schwab, Ch.: Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Math. Model. Numer. Anal. 47(1), 253–280 (2013)

  7. Chkifa, A., Cohen, A., Schwab, Ch.: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103(2), 400–428 (2015)

    Article  MathSciNet  Google Scholar 

  8. Choi, J.H., Kwak, B.M.: Shape design sensitivity analysis of elliptic problems in boundary integral equation formulation. Mech. Struct. Mach. 16(2), 147–165 (1988)

    Article  MathSciNet  Google Scholar 

  9. Cohen, A., Schwab, Ch., Zech, J.: Shape holomorphy of the stationary Navier-Stokes equations. SIAM J. Math. Anal. 50(2), 1720–1752 (2018)

    Article  MathSciNet  Google Scholar 

  10. Conway, J.: Functions of One Complex Variable I, vol. 159, 2nd edn. Springer (1978)

  11. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)

    Article  MathSciNet  Google Scholar 

  12. Costabel, M., Le. Louër, F, : Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: shape differentiability of pseudo-homogeneous boundary integral operators. Integral Equ. Oper. Theory 72(4), 509–535 (2012)

  13. Costabel, M., Le. Louër, F, : Shape derivatives of boundary integral operators in electromagnetic scattering. Part II: Application to scattering by a homogeneous dielectric obstacle. Integral Equ. Oper. Theory 73(1), 17–48 (2012)

  14. Delfour, M., Zolesio, J.P.: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, vol. 22. SIAM (2011)

  15. Dick, J., Gantner, R.N., Le. Gia, Q.T., Schwab, C. : Higher order Quasi-Monte Carlo integration for bayesian estimation. Comput. Math. Appl. 77(1), 144–172 (2019)

  16. Dick, J., Le. Gia, Q.T., Schwab, C. : Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations. SIAM/ASA J. Uncertain. Quantif. 4(1), 48–79 (2016)

  17. Dudley, R.M., Norvaiša, R.: Concrete Functional Calculus. Springer, New York (2011)

    Book  Google Scholar 

  18. Eckel, H., Kress, R.: Nonlinear integral equations for the inverse electrical impedance problem. Inverse Prob. 23(2), 475 (2007)

    Article  MathSciNet  Google Scholar 

  19. Eppler, K.: Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim. 20(1), 63–78 (2000)

    Article  MathSciNet  Google Scholar 

  20. Eppler, K.: Optimal shape design for elliptic equations via BIE-methods. Int. J. Appl. Math. Comput. Sci. 10(3), 487–516 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Eppler, K., Harbrecht, H.: A regularized Newton method in electrical impedance tomography using shape hessian information. Control Cybern. 34(1), 203–225 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Eppler, K., Harbrecht, H.: Second-order shape optimization using wavelet BEM. Optim. Methods Softw. 21(1), 135–153 (2006)

    Article  MathSciNet  Google Scholar 

  23. Gantner, R., Peters, M.: Higher order Quasi-Monte Carlo for Bayesian shape inversion. SIAM/ASA J. Uncertain. Quantif. 6(2), 707–736 (2016)

    Article  MathSciNet  Google Scholar 

  24. Henríquez, F.: Shape Uncertainty Quantification in Acoustic Scattering. Ph.D. thesis, Dissertation 27480, ETH Zürich (2021)

  25. Hiptmair, R., Li, J.: Shape derivatives for scattering problems. Inverse Probl. 34(10):105001, 25 (2018)

  26. Hiptmair, R., Scarabosio, L., Schillings, C., Schwab, Ch.: Large deformation shape uncertainty quantification in acoustic scattering. Adv. Comput. Math. 44(5), 1475–1518 (2018)

    Article  MathSciNet  Google Scholar 

  27. Hsiao, G., Wendland, W.: Boundary Integral Equations, vol. 164. Springer, Berlin (2008)

    Book  Google Scholar 

  28. Ivanyshyn, O., Kress, R.: Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. J. Integral Equ. Appl. 18(1), 13–38 (2006)

    Article  MathSciNet  Google Scholar 

  29. Jerez-Hanckes, C., Schwab, Ch.: Electromagnetic wave scattering by random surfaces: uncertainty quantification via sparse tensor boundary elements. IMA J. Numer. Anal. 37(3), 1175–1210 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Jerez-Hanckes, C., Schwab, Ch., Zech, J.: Electromagnetic wave scattering by random surfaces: shape holomorphy. Math. Models Methods Appl. Sci. 27(12), 2229–2259 (2016)

    Article  MathSciNet  Google Scholar 

  31. Kress, R.: Inverse elastic scattering from a crack. Inverse Prob. 12(5), 667 (1996)

    Article  MathSciNet  Google Scholar 

  32. Kress, R.: Linear integral equations. In: Applied Mathematical Sciences, vol. 82. Springer, New York (1999)

  33. Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Prob. 21(4), 1207 (2005)

    Article  MathSciNet  Google Scholar 

  34. Kress, R., Rundell, W.: A nonlinear integral equation and an iterative algorithm for an inverse source problem. J. Integral Equ. Appl. 27(2), 179–197 (2015)

    Article  MathSciNet  Google Scholar 

  35. Mujica, J.: Complex Analysis in Banach Spaces, vol. 120. North-Holland Publishing Co., Amsterdam (1986)

    MATH  Google Scholar 

  36. Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Prob. 10(2), 431–447 (1994)

    Article  MathSciNet  Google Scholar 

  37. Potthast, R.: Domain derivatives in electromagnetic scattering. Math. Methods Appl. Sci. 19(15), 1157–1175 (1996)

    Article  MathSciNet  Google Scholar 

  38. Potthast, R.: Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain. J. Inverse Ill-Posed Probl. 4(1), 67–84 (1996)

    Article  MathSciNet  Google Scholar 

  39. Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations, volume 92 of Unitext. Springer, Cham (2016). An introduction, La Matematica per il 3+2

  40. Reed, M., Simon, B.: Functional Analysis, vol. I. Academic press, New York (1980)

    MATH  Google Scholar 

  41. Saranen, J., Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer (2013)

  42. Sauter, S., Schwab, Ch.: Boundary Element Methods. Springer, Berlin (2010)

    Book  Google Scholar 

  43. Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Springer, Berlin (1992)

    Book  Google Scholar 

  44. Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer (2007)

  45. Yaman, O., Le. Louër, F.: Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems. Inverse Probl. 32(9), 095003 (2016)

Download references

Author information

Authors and Affiliations

  1. Seminar for Applied Mathematics ETH Zurich, Raemistrasse 101, 8092, Zurich, Switzerland

    Fernando Henríquez & Christoph Schwab

Authors
  1. Fernando Henríquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Schwab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fernando Henríquez.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially funded by ETH Zürich through Grant ETH-44 17-1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Henríquez, F., Schwab, C. Shape Holomorphy of the Calderón Projector for the Laplacian in \({\mathbb {R}}^2\). Integr. Equ. Oper. Theory 93, 43 (2021). https://doi.org/10.1007/s00020-021-02653-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-021-02653-5

Keywords

Mathematics Subject Classification

Search

Navigation