Papers
Topics
Authors
Recent
Search
2000 character limit reached

Computational unique continuation with finite dimensional Neumann trace

Published 21 Feb 2024 in math.NA and cs.NA | (2402.13695v2)

Abstract: We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (37)
  1. The stability for the Cauchy problem for elliptic equations. Inverse Probl., 25(12):47, 2009. Id/No 123004.
  2. Lipschitz stability for the inverse conductivity problem. Adv. in Appl. Math., 35(2):207–241, 2005.
  3. L. Bourgeois. A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Problems, 21(3):1087–1104, 2005.
  4. Laurent Bourgeois. A remark on Lipschitz stability for inverse problems. C. R. Math. Acad. Sci. Paris, 351(5-6):187–190, 2013.
  5. On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates. ESAIM Math. Model. Numer. Anal., 54(2):493–529, 2020.
  6. A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM Math. Model. Numer. Anal., 52(1):123–145, 2018.
  7. A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comp., 66(219):935–955, 1997.
  8. Least-squares for second-order elliptic problems. volume 152, pages 195–210. 1998. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997).
  9. The mathematical theory of finite element methods, volume 15 of Texts Appl. Math. New York, NY: Springer, 3rd ed. edition, 2008.
  10. Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York, NY: Springer, 2011.
  11. Erik Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6):A2752–A2780, 2013.
  12. Erik Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655–659, 2014.
  13. Erik Burman. Stabilised finite element methods for ill-posed problems with conditional stability. In Building bridges: connections and challenges in modern approaches to numerical partial differential equations, volume 114 of Lect. Notes Comput. Sci. Eng., pages 93–127. Springer, [Cham], 2016.
  14. A hybridized high-order method for unique continuation subject to the Helmholtz equation. SIAM J. Numer. Anal., 59(5):2368–2392, 2021.
  15. Primal-dual mixed finite element methods for the elliptic Cauchy problem. SIAM Journal on Numerical Analysis, 56(6):3480–3509, 2018.
  16. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl. (9), 129:1–22, 2019.
  17. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime. Numer. Math., 144(3):451–477, 2020.
  18. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime. Numer. Math., 150(3):769–801, 2022.
  19. Optimal finite element approximation of unique continuation, 2023.
  20. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505–528, 2018.
  21. Finite element approximation of unique continuation of functions with finite dimensional trace, 2023.
  22. O. Chervova and L. Oksanen. Time reversal method with stabilizing boundary conditions for photoacoustic tomography. Inverse Problems, 32(12):125004, nov 2016.
  23. Least squares formulation for ill-posed inverse problems and applications. Appl. Anal., 101(15):5247–5261, 2022.
  24. P. Colli Franzone and E. Magenes. On the inverse potential problem of electrocardiology. Calcolo, 16(4):459–538 (1980), 1979.
  25. Least squares solvers for ill-posed PDEs that are conditionally stable. ESAIM Math. Model. Numer. Anal., 57(4):2227–2255, 2023.
  26. An Hd⁢i⁢vsubscript𝐻𝑑𝑖𝑣H_{{div}}italic_H start_POSTSUBSCRIPT italic_d italic_i italic_v end_POSTSUBSCRIPT-based mixed quasi-reversibility method for solving elliptic Cauchy problems. SIAM J. Numer. Anal., 51(4):2123–2148, 2013.
  27. Theory and practice of finite elements., volume 159 of Appl. Math. Sci. New York, NY: Springer, 2004.
  28. Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation. Math. Comp., 47(175):135–149, 1986.
  29. P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monogr. Stud. Math. Pitman, Boston, MA, 1985.
  30. The boundary-element method for an improperly posed problem. IMA J. Appl. Math., 47(1):61–79, 1991.
  31. Inverse problems, volume 22 of Series on Applied Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. Tikhonov theory and algorithms.
  32. Tomas Johansson. An iterative procedure for solving a Cauchy problem for second order elliptic equations. Math. Nachr., 272:46–54, 2004.
  33. Optimizational method for solving the Cauchy problem for an elliptic equation. J. Inverse Ill-Posed Probl., 3(1):21–46, 1995.
  34. R. Lattès and J.-L. Lions. The method of quasi-reversibility. Applications to partial differential equations. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co., Inc., New York, 1969. Translated from the French edition and edited by Richard Bellman.
  35. Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation. SIAM J. Numer. Anal., 36(3):890–905, 1999.
  36. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Mathematics of Computation, 54(190):483–493, 1990.
  37. Solutions of ill-posed problems. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto-London, 1977. Translated from the Russian, Preface by translation editor Fritz John.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up