Abstract
We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.
Authors
Erik Burman
Department of Mathematics, University College London, London, WC1E 6BT, UK
Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Department of Mathematics, Babeṣ-Bolyai University, Cluj-Napoca, Romania
Lauri Oksanen
Department of Mathematics and Statistics, University of Helsinki, P.O. 68, 00014, Helsinki, Finland
Keywords
Unique continuation; Ill-posed problems; Conditional stability; Approximation methods; Finite element methods; Stabilised methods; Regularisation; Error estimates; Optimality; Optimal convergence
Paper coordinates
E. Burman, M. Nechita, L. Oksanen, Optimal approximation of unique continuation. Found. Comput. Math. (2024). https://doi.org/10.1007/s10208-024-09655-w
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Foundations of Computational Mathematics
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[1] Alessandrini, G., Rondi, L., Rosset, E., Vessella, S., The stability for the Cauchy problem for elliptic equations. Inverse Problems 25(12), 123004, 47 (2009). https://doi.org/10.1088/0266-5611/25/12/123004.
[2] Armstrong, S., Kuusi, T., Smart, C., Optimal unique continuation for periodic elliptic equations on large scales (2021). Preprint arXiv:2107.14248
[3] Babuška, I., Error-bounds for finite element method. Numer. Math. 16, 322–333 (1970/71). https://doi.org/10.1007/BF02165003.
[4] Bernardi, C., Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26(5), 1212–1240 (1989). https://doi.org/10.1137/0726068. Article MathSciNet Google Scholar
[5] Boulakia, M., Burman, E., Fernández, M.A., Voisembert, C., Data assimilation finite element method for the linearized Navie-Stokes equations in the low Reynolds regime. Inverse Problems 36(8), 085003–85024 (2020). https://doi.org/10.1088/1361-6420/ab9161.Article MathSciNet Google Scholar
[6] Bourgeois, L., A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Problems 21(3), 1087–1104 (2005). https://doi.org/10.1088/0266-5611/21/3/018.Article MathSciNet Google Scholar
[7] Bourgeois, L., Recoquillay, A., A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM Math. Model. Numer. Anal. 52(1), 123–145 (2018). https://doi.org/10.1051/m2an/2018008.Article MathSciNet Google Scholar
[8] Brummelhuis, R., Three-spheres theorem for second order elliptic equations. Journal d’Analyse Mathematique 65(1), 179–206 (1995)Article MathSciNet Google Scholar
[9] Burman, E., Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. 35(6), A2752–A2780 (2013). https://doi.org/10.1137/130916862.Article MathSciNet Google Scholar
[10] Burman, E., Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris 352(7-8), 655–659 (2014). https://doi.org/10.1016/j.crma.2014.06.008.Article MathSciNet Google Scholar
[11] Burman, E., 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, Lect. Notes Comput. Sci. Eng., vol. 114, pp. 93–127. Springer, [Cham] (2016)
[12] Burman, E., Feizmohammadi, A., Oksanen, L., A finite element data assimilation method for the wave equation. Math. Comp. 89(324), 1681–1709 (2020). https://doi.org/10.1090/mcom/3508.Article MathSciNet Google Scholar
[13] Burman, E., Hansbo, P., Stabilized nonconforming finite element methods for data assimilation in incompressible flows. Math. Comp. 87(311), 1029–1050 (2018). https://doi.org/10.1090/mcom/3255.Article MathSciNet Google Scholar
[14] Burman, E., Hansbo, P., Larson, M.G., Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems 34(3), 035004, 36 (2018). https://doi.org/10.1088/1361-6420/aaa32b.
[15] Burman, E., Larson, M.G., Oksanen, L., Primal-dual mixed finite element methods for the elliptic Cauchy problem. SIAM J. Numer. Anal. 56(6), 3480–3509 (2018). https://doi.org/10.1137/17M1163335.Article MathSciNet Google Scholar
[16] Burman, E., Nechita, M., Oksanen, L., Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl. (9) 129, 1–22 (2019). https://doi.org/10.1016/j.matpur.2018.10.003.Article MathSciNet Google Scholar
[17] Burman, E., Nechita, M., Oksanen, L., 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). https://doi.org/10.1007/s00211-019-01087-x.Article MathSciNet Google Scholar
[18] Burman, E., Oksanen, L., Data assimilation for the heat equation using stabilized finite element methods. Numer. Math. 139(3), 505–528 (2018). https://doi.org/10.1007/s00211-018-0949-3.Article MathSciNet Google Scholar
[19] Burman, E., Oksanen, L., Finite element approximation of unique continuation of functions with finite dimensional trace (2023). Preprint arXiv:2305.06800
[20] Céa, J., Approximation variationnelle des problèmes aux limites. Ann. Inst. Fourier (Grenoble) 14(fasc. 2), 345–444 (1964)
[21] Dahmen, W., Monsuur, H., Stevenson, R., Least squares solvers for ill-posed PDEs that are conditionally stable. ESAIM Math. Model. Numer. Anal. 57(4), 2227–2255 (2023). https://doi.org/10.1051/m2an/2023050.Article MathSciNet Google Scholar
[22] Engl, H.W., Regularization by least-squares collocation. In: Numerical treatment of inverse problems in differential and integral equations (Heidelberg, 1982), Progr. Sci. Comput., vol. 2, pp. 345–354. Birkhäuser Boston, Boston, MA (1983)
[23] Engl, H.W., Neubauer, A., On projection methods for solving linear ill-posed problems. In: Model optimization in exploration geophysics (Berlin, 1986), Theory Practice Appl. Geophys., vol. 1, pp. 73–92. Friedr. Vieweg, Braunschweig (1987)
[24] Engl, H.W., Neubauer, A., Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales. Proc. Amer. Math. Soc. 102(3), 587–592 (1988). https://doi.org/10.2307/2047228.Article MathSciNet Google Scholar
[25] Escauriaza, L., Fernández, F.J., Vessella, S., Doubling properties of caloric functions. Appl. Anal. 85(1-3), 205–223 (2006). https://doi.org/10.1080/00036810500277082.Article MathSciNet Google Scholar
[26] Hämarik, U., Avi, E., Ganina, A., On the solution of ill-posed problems by projection methods with a posteriori choice of the discretization level. Math. Model. Anal. 7(2), 241–252 (2002)Article MathSciNet Google Scholar
[27] Helfrich, H.P., Optimale lineare Approximation beschränkter Mengen in normierten Räumen. J. Approximation Theory 4, 165–182 (1971). https://doi.org/10.1016/0021-9045(71)90027-x.Article MathSciNet Google Scholar
[28] Ito, K., Jin, B., Inverse problems, Series on Applied Mathematics, vol. 22. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015). Tikhonov theory and algorithms
[29[ John, F., Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math. 13, 551–585 (1960)Article MathSciNet Google Scholar
[30] Kaltenbacher, B., Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inverse Problems 16(5), 1523–1539 (2000). https://doi.org/10.1088/0266-5611/16/5/322.Article MathSciNet Google Scholar
[31] Lattès, R., Lions, J.L., The method of quasi-reversibility. Applications to partial differential equations. Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co., Inc., New York (1969)
[32] Lax, P.D., Milgram, A.N., Parabolic equations. In: Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, pp. 167–190. Princeton University Press, Princeton, N. J. (1954)
[33] Lin, C.L., Nakamura, G., Wang, J.N., Optimal three-ball inequalities and quantitative uniqueness for the Lamé system with Lipschitz coefficients. Duke Math. J. 155(1), 189–204 (2010). https://doi.org/10.1215/00127094-2010-054.Article MathSciNet Google Scholar
[34] Lin, C.L., Uhlmann, G., Wang, J.N., Optimal three-ball inequalities and quantitative uniqueness for the Stokes system. Discrete Contin. Dyn. Syst. 28(3), 1273–1290 (2010). https://doi.org/10.3934/dcds.2010.28.1273.Article MathSciNet Google Scholar
[35] Lukas, M.A., Convergence rates for regularized solutions. Math. Comp. 51(183), 107–131 (1988). https://doi.org/10.2307/2008582.Article MathSciNet Google Scholar
[36] Mathé, P., Pereverzev, S.V., Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods. SIAM J. Numer. Anal. 38(6), 1999–2021 (2001). https://doi.org/10.1137/S003614299936175X.Article MathSciNet Google Scholar
[37] Miller, K., Three circle theorems in partial differential equations and applications to improperly posed problems. Ph.D. thesis, Rice University (1962)
[38] Miller, K., Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems. In: Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), pp. 161–176. Lecture Notes in Math., Vol. 316 (1973)
[39] Mishra, S., Molinaro, R., Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA Journal of Numerical Analysis 42(2), 981–1022 (2022). https://doi.org/10.1093/imanum/drab032.Article MathSciNet Google Scholar
[40] Monk, P., Süli, E., The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal. 36(1), 251–274 (1999). https://doi.org/10.1137/S0036142997315172.Article MathSciNet Google Scholar
[41] Natterer, F., The finite element method for ill-posed problems. RAIRO Anal. Numér. 11(3), 271–278 (1977). https://doi.org/10.1051/m2an/1977110302711.Article MathSciNet Google Scholar
[42] Natterer, F., Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math. 28(3), 329–341 (1977). https://doi.org/10.1007/BF01389972.Article MathSciNet Google Scholar
[43] Natterer, F., Error bounds for Tikhonov regularization in Hilbert scales. Applicable Anal. 18(1-2), 29–37 (1984). https://doi.org/10.1080/00036818408839508. Article MathSciNet Google Scholar
[44] Nitsche, J., Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Randwertprobleme. Arch. Rational Mech. Anal. 36, 348–355 (1970). https://doi.org/10.1007/BF00282271.Article MathSciNet Google Scholar
[45] Tikhonov, A.N., Arsenin, V.Y., Solutions of ill-posed problems. V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London (1977). Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics
[46] Trefethen, L.N., Quantifying the ill-conditioning of analytic continuation. BIT Numerical Mathematics 60(4), 901–915 (2020). https://doi.org/10.1007/s10543-020-00802-7.Article MathSciNet Google Scholar
[47] Trefethen, L.N., Numerical analytic continuation. Japan Journal of Industrial and Applied Mathematics 40(3), 1587–1636 (2023). https://doi.org/10.1007/s13160-023-00599-2. Article MathSciNet Google Scholar
[48] Zlámal, M., On the finite element method. Numer. Math. 12, 394–409 (1968). https://doi.org/10.1007/BF02161362. Article MathSciNet Google Scholar
[49] Zlámal, M., Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10, 229–240 (1973). https://doi.org/10.1137/0710022.