Convergent numerical approximation of the stochastic total variation flow. (English) Zbl 1468.60075
Stoch. Partial Differ. Equ., Anal. Comput. 9, No. 2, 437-471 (2021); correction ibid. 11, No. 4, 1732-1739 (2023).
Summary: We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter \(\varepsilon\) we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully discrete finite element approximation for the regularized gradient flow equation and show that the numerical solution converges to the solution of the unregularized STVF equation. We perform numerical experiments to demonstrate the practicability of the proposed numerical approximation.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35A15 | Variational methods applied to PDEs |
| 35Q30 | Navier-Stokes equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
Keywords:
stochastic variational inequalities; convergent numerical approximation; finite element method; stochastic total variation flow; image processingReferences:
| [1] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001 |
| [2] | Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA. Applications to PDEs and optimization (2006) · Zbl 1095.49001 |
| [3] | Barbu, V.; Röckner, M., Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal., 209, 3, 797-834 (2013) · Zbl 1286.35202 · doi:10.1007/s00205-013-0632-x |
| [4] | Bartels, S.; Milicevic, M., Stability and experimental comparison of prototypical iterative schemes for total variation regularized problems, Comput. Methods Appl. Math., 16, 3, 361-388 (2016) · Zbl 1342.65152 · doi:10.1515/cmam-2016-0014 |
| [5] | Brenner, SC; Scott, LR, The Mathematical Theory of Finite Element Methods (2002), New York: Springer, New York · Zbl 1012.65115 · doi:10.1007/978-1-4757-3658-8 |
| [6] | Emmrich, E.; Šiška, D., Nonlinear stochastic evolution equations of second order with damping, Stoch. Partial Differ. Equ. Anal. Comput., 5, 1, 81-112 (2017) · Zbl 1362.60061 |
| [7] | Feng, X.; Prohl, A., Analysis of total variation flow and its finite element approximations, M2AN Math. Model. Numer. Anal., 37, 3, 533-556 (2003) · Zbl 1050.35004 · doi:10.1051/m2an:2003041 |
| [8] | Gess, B.; Tölle, J., Stability of solutions to stochastic partial differential equations, J. Differ. Equ., 260, 6, 4973-5025 (2016) · Zbl 1338.60157 · doi:10.1016/j.jde.2015.11.039 |
| [9] | Gyöngy, I.; Millet, A., On discretization schemes for stochastic evolution equations, Potential Anal., 23, 2, 99-134 (2005) · Zbl 1067.60049 · doi:10.1007/s11118-004-5393-6 |
| [10] | Juan, O.; Keriven, R.; Postelnicu, G., Stochastic motion and the level set method in computer vision: stochastic active contours, Int. J. Comput. Vis., 69, 7-25 (2006) · Zbl 1477.68380 · doi:10.1007/s11263-006-6849-5 |
| [11] | Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. In: Stochastic Differential Equations: Theory and Applications, Volume 2 of Interdiscip. Math. Sci., pp. 1-69. World Sci. Publ., Hackensack, NJ (2007) · Zbl 1130.60069 |
| [12] | Liu, W.; Röckner, M., Stochastic Partial Differential Equations: an Introduction. Universitext (2015), Cham: Springer, Cham · Zbl 1361.60002 · doi:10.1007/978-3-319-22354-4 |
| [13] | Rudin, LI; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60, 1-4, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F |
| [14] | Sixou, B.; Wang, L.; Peyrin, F., Stochastic diffusion equation with singular diffusivity and gradient-dependent noise in binary tomography, J. Phys Conf. Ser., 69, 012001 (2010) |
| [15] | Temam, R.: Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Studies in Mathematics and its Applications, Vol. 2 · Zbl 0383.35057 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
