Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that linear convergence is possible on suitably constructed meshes.
Keywords: Nonsmooth minimization, graded meshes, adaptivity, total variation, error estimates
@article{M2AN_2022__56_6_1871_0,
author = {Bartels, S\"oren and Tovey, Robert and Wassmer, Friedrich},
title = {Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1871--1888},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {6},
doi = {10.1051/m2an/2022056},
mrnumber = {4467102},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022056/}
}
TY - JOUR AU - Bartels, Sören AU - Tovey, Robert AU - Wassmer, Friedrich TI - Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1871 EP - 1888 VL - 56 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022056/ DO - 10.1051/m2an/2022056 LA - en ID - M2AN_2022__56_6_1871_0 ER -
%0 Journal Article %A Bartels, Sören %A Tovey, Robert %A Wassmer, Friedrich %T Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1871-1888 %V 56 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022056/ %R 10.1051/m2an/2022056 %G en %F M2AN_2022__56_6_1871_0
Bartels, Sören; Tovey, Robert; Wassmer, Friedrich. Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 1871-1888. doi: 10.1051/m2an/2022056
[1] , and , A characterization of convex calibrable sets in . Math. Ann. 332 (2005) 329–366. | MR | Zbl | DOI
[2] , and , Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). | MR | Zbl | DOI
[3] , and , Variational Analysis in Sobolev and BV Spaces – Applications to PDEs and Optimization. Vol. 4 of MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2006). | MR | Zbl
[4] , Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. 50 (2012) 1162–1180. | MR | Zbl | DOI
[5] , Error control and adaptivity for a variational model problem defined on functions of bounded variation. Math. Comput. 84 (2015) 1217–1240. | MR | DOI
[6] , Numerical Methods for Nonlinear Partial Differential Equations. Vol. 47 of Springer Series in Computational Mathematics. Springer, Cham (2015). | MR | Zbl | DOI
[7] , Numerical Approximation of Partial Differential Equations. Vol. 64 of Texts in Applied Mathematics. Springer, Berlin, Heidelberg (2016). | MR | DOI
[8] , Error estimates for a class of discontinuous Galerkin methods for nonsmooth problems via convex duality relations. Math. Comput. 90 (2021) 2579–2602. | MR | DOI
[9] , Nonconforming discretizations of convex minimization problems and precise relations to mixed methods. Comput. Math. App. 93 (2021) 214–229. | MR
[10] and , Primal-dual gap estimators for a posteriori error analysis of nonsmooth minimization problems. ESAIM: Math. Model. Numer. Anal. 54 (2020) 1635–1660. | MR | Zbl | Numdam | DOI
[11] , and , A total variation diminishing interpolation operator and applications. Math. Comput. 84 (2015) 2569–2587. | MR | DOI
[12] , and , Unconditional stability of semi-implicit discretizations of singular flows. SIAM J. Numer. Anal. 56 (2018) 1896–1914. | MR | DOI
[13] , and , A posteriori error control for the binary Mumford-Shah model. Math. Comput. 86 (2017) 1769–1791. | MR | DOI
[14] , Forty years of the Crouzeix-Raviart element. Numer. Methods Part. Differ. Equ. 31 (2015) 367–396. | MR | DOI
[15] and , The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York (2008). | MR | Zbl
[16] , Bregman distances in inverse problems and partial differential equations. In: Advances in Mathematical Modeling, Optimization and Optimal Control. Vol. 109 of Springer Optim. Appl. Springer, Cham (2016) 3–33. | MR | DOI
[17] and , Error estimates for finite differences approximations of the total variation. HAL preprint nr. 02539136 (2020). | MR
[18] , and , The Discontinuity Set of Solutions of the TV Denoising Problem and Some Extensions. Multiscale Model. Simul. 6 (2007) 879–894. | MR | Zbl | DOI
[19] and , Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167–188. | MR | Zbl | DOI
[20] and , Crouzeix-Raviart approximation of the total variation on simplicial meshes. J. Math. Imaging Vision 62 (2020) 872–899. | MR | DOI
[21] and , Approximating the total variation with finite differences or finite elements. In: Handbook of Numerical Analysis: Geometric Partial Differential Equations II. Elsevier (2021). | MR
[22] , , , and , An introduction to total variation for image analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery. Vol. 9 of Radon Ser. Comput. Appl. Math. Walter de Gruyter, Berlin (2010) 263–340. | MR | Zbl | DOI
[23] , and , An upwind finite-difference method for total variation-based image smoothing. SIAM J. Imaging Sci. 4 (2011) 277–299. | MR | Zbl | DOI
[24] and , Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. R.A.I.R.O. 7 (1973) 33–75. | MR | Numdam | Zbl
[25] and , A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal. 41 (2003) 2032–2055. | MR | Zbl | DOI
[26] , , , and , Discrete total variation with finite elements and applications to imaging. J. Math. Imaging Vision 61 (2019) 411–431. | MR | DOI
[27] and , Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 64 (2004) 1311–1333. | MR | Zbl | DOI
[28] and , Piecewise linear approximation of the continuous Rudin–Osher–Fatemi model for image denoising. SIAM J. Numer. Anal. 50 (2012) 2446–2466. | MR | Zbl | DOI
[29] and , A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, edited by and . Springer, Berlin, Heidelberg (1977) 292–315. | MR | Zbl | DOI
[30] , and , Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60 (1992) 259–268. | MR | Zbl | DOI
[31] , Mathematical challenges in electron microscopy. Ph.D. thesis, University of Cambridge (2020).
[32] and , Error bounds for finite-difference methods for Rudin–Osher–Fatemi image smoothing. SIAM J. Numer. Anal. 49 (2011) 845–868. | MR | Zbl | DOI
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