# zbMATH — the first resource for mathematics

Spectral approximation of fractional PDEs in image processing and phase field modeling. (English) Zbl 1431.65222
Summary: Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

##### MSC:
 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35R11 Fractional partial differential equations 35Q94 PDEs in connection with information and communication 35S05 Pseudodifferential operators as generalizations of partial differential operators 49K20 Optimality conditions for problems involving partial differential equations 49M05 Numerical methods based on necessary conditions 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
##### Software:
na28; scikit-image
Full Text:
##### References:
 [1] M. Ainsworth and Z. Mao, Analysis and Approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal. 55 (2017), no. 4, 1689-1718.; Ainsworth, M.; Mao, Z., Analysis and Approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55, 4, 1689-1718 (2017) [2] H. Antil, J. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization, preprint (2017), .; Antil, H.; Pfefferer, J.; Rogovs, S., Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization (2017) [3] H. Antil, J. Pfefferer and M. Warma, A note on semilinear fractional elliptic equation: Analysis and discretization, ESAIM Math. Model. Numer. Anal. (2017), 10.1051/m2an/2017023.; Antil, H.; Pfefferer, J.; Warma, M., A note on semilinear fractional elliptic equation: Analysis and discretization, ESAIM Math. Model. Numer. Anal. (2017) [4] U. Aßmann and A. Rösch, Regularization in Sobolev spaces with fractional order, Numer. Funct. Anal. Optim. 36 (2015), no. 3, 271-286.; Aßmann, U.; Rösch, A., Regularization in Sobolev spaces with fractional order, Numer. Funct. Anal. Optim., 36, 3, 271-286 (2015) [5] S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal. 50 (2012), no. 3, 1162-1180.; Bartels, S., Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal., 50, 3, 1162-1180 (2012) [6] S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer Ser. Comput. Math. 47, Springer, Cham, 2015.; Bartels, S., Numerical Methods for Nonlinear Partial Differential Equations (2015) [7] S. Bartels, Broken Sobolev space iteration for total variation regularized minimization problems, IMA J. Numer. Anal. 36 (2016), no. 2, 493-502.; Bartels, S., Broken Sobolev space iteration for total variation regularized minimization problems, IMA J. Numer. Anal., 36, 2, 493-502 (2016) [8] S. Bartels, R. Müller and C. Ortner, Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes, SIAM J. Numer. Anal. 49 (2011), no. 1, 110-134.; Bartels, S.; Müller, R.; Ortner, C., Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes, SIAM J. Numer. Anal., 49, 1, 110-134 (2011) [9] B. Benešová, C. Melcher and E. Süli, An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations, SIAM J. Numer. Anal. 52 (2014), no. 3, 1466-1496.; Benešová, B.; Melcher, C.; Süli, E., An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations, SIAM J. Numer. Anal., 52, 3, 1466-1496 (2014) [10] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, J. Royal Soc. Interface 11 (2014), no. 97, Article ID 20140352.; Bueno-Orovio, A.; Kay, D.; Grau, V.; Rodriguez, B.; Burrage, K., Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, J. Royal Soc. Interface, 11, 97 (2014) [11] L. Caffarelli and N. Muler, An $$L^{\infty}$$ bound for solutions of the Cahn-Hilliard equation, Arch. Ration. Mech. Anal. 133 (1995), no. 2, 129-144.; Caffarelli, L.; Muler, N., An $$L^{\infty}$$ bound for solutions of the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 133, 2, 129-144 (1995) [12] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260.; Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 7-9, 1245-1260 (2007) [13] A. Carasso and A. Vladár, Fractional diffusion, low exponent Lévy stable laws, and ‘slow motion’ denoising of helium ion microscope nanoscale imagery, J. Res. Natl. Inst. Stand. Technol. 117 (2012), 119-142.; Carasso, A.; Vladár, A., Fractional diffusion, low exponent Lévy stable laws, and ‘slow motion’ denoising of helium ion microscope nanoscale imagery, J. Res. Natl. Inst. Stand. Technol., 117, 119-142 (2012) [14] A. Carasso and A. Vladár, Recovery of background structures in nanoscale helium ion microscope imaging, J. Res. Natl. Inst. Stand. Technol. 119 (2014), 683-701.; Carasso, A.; Vladár, A., Recovery of background structures in nanoscale helium ion microscope imaging, J. Res. Natl. Inst. Stand. Technol., 119, 683-701 (2014) [15] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision 40 (2011), no. 1, 120-145.; Chambolle, A.; Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40, 1, 120-145 (2011) [16] T. Chan and P. Mulet, On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM J. Numer. Anal. 36 (1999), no. 2, 354-367.; Chan, T.; Mulet, P., On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM J. Numer. Anal., 36, 2, 354-367 (1999) [17] W. Chen, A speculative study of $$2/3$$-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos 16 (2006), no. 2, Article ID 023126.; Chen, W., A speculative study of $$2/3$$-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, 16, 2 (2006) [18] N. Condette, C. Melcher and E. Süli, Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth, Math. Comp. 80 (2011), no. 273, 205-223.; Condette, N.; Melcher, C.; Süli, E., Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth, Math. Comp., 80, 273, 205-223 (2011) [19] M. Dabkowski, Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal. 21 (2011), no. 1, 1-13.; Dabkowski, M., Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal., 21, 1, 1-13 (2011) [20] J. Dahl, P. C. Hansen, S. R. H. Jensen and T. L. M. Jensen, Algorithms and software for total variation image reconstruction via first-order methods, Numer. Algorithms 53 (2010), no. 1, 67-92.; Dahl, J.; Hansen, P. C.; Jensen, S. R. H.; Jensen, T. L. M., Algorithms and software for total variation image reconstruction via first-order methods, Numer. Algorithms, 53, 1, 67-92 (2010) [21] R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoam. 21 (2005), no. 3, 863-888.; Danchin, R., Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoam., 21, 3, 863-888 (2005) [22] D. del Castillo-Negrete, B. A. Carreras and V. E. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas 11 (2004), no. 8, 3854-3864.; del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E., Fractional diffusion in plasma turbulence, Phys. Plasmas, 11, 8, 3854-3864 (2004) [23] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math. 94 (2003), no. 1, 33-65.; Feng, X.; Prohl, A., Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94, 1, 33-65 (2003) [24] P. Gatto and J. Hesthaven, Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising, J. Sci. Comput. 65 (2015), no. 1, 249-270.; Gatto, P.; Hesthaven, J., Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising, J. Sci. Comput., 65, 1, 249-270 (2015) [25] J. Ge, M. Everett and C. Weiss, Fractional diffusion analysis of the electromagnetic field in fractured media. Part 2: 3D approach, Geophys. 80 (2015), no. 3, E175-E185.; Ge, J.; Everett, M.; Weiss, C., Fractional diffusion analysis of the electromagnetic field in fractured media. Part 2: 3D approach, Geophys., 80, 3, E175-E185 (2015) [26] T. Goldstein and S. Osher, The split Bregman method for $$L1$$-regularized problems, SIAM J. Imaging Sci. 2 (2009), no. 2, 323-343.; Goldstein, T.; Osher, S., The split Bregman method for $$L1$$-regularized problems, SIAM J. Imaging Sci., 2, 2, 323-343 (2009) [27] Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?, SIAM J. Math. Anal. 33 (2001), no. 3, 634-648.; Gousseau, Y.; Morel, J.-M., Are natural images of bounded variation?, SIAM J. Math. Anal., 33, 3, 634-648 (2001) [28] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Math. 64 (2004), no. 4, 1311-1333.; Hintermüller, M.; Kunisch, K., Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Math., 64, 4, 1311-1333 (2004) [29] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445-453.; Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 3, 445-453 (2007) [30] T. Kühn and T. Schonbek, Compact embeddings of Besov spaces into Orlicz and Lorentz-Zygmund spaces, Houston J. Math. 31 (2005), no. 4, 1221-1243.; Kühn, T.; Schonbek, T., Compact embeddings of Besov spaces into Orlicz and Lorentz-Zygmund spaces, Houston J. Math., 31, 4, 1221-1243 (2005) [31] A. Marquina and S. Osher, Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput. 22 (2000), no. 2, 387-405.; Marquina, A.; Osher, S., Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput., 22, 2, 387-405 (2000) [32] R. Nochetto, E. Otárola and A. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math. 15 (2015), no. 3, 733-791.; Nochetto, R.; Otárola, E.; Salgado, A., A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15, 3, 733-791 (2015) [33] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. 18 (2016), no. 3, Article ID 1550033.; Roncal, L.; Stinga, P. R., Fractional Laplacian on the torus, Commun. Contemp. Math., 18, 3 (2016) [34] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), no. 1-4, 259-268.; Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60, 1-4, 259-268 (1992) [35] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monogr. Math., Springer, Berlin, 2002.; Saranen, J.; Vainikko, G., Periodic Integral and Pseudodifferential Equations with Numerical Approximation (2002) [36] O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 4, 479-500.; Savin, O.; Valdinoci, E., Γ-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29, 4, 479-500 (2012) [37] F. Song, C. Xu and G. E. Karniadakis, A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations, Comput. Methods Appl. Mech. Engrg. 305 (2016), 376-404.; Song, F.; Xu, C.; Karniadakis, G. E., A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations, Comput. Methods Appl. Mech. Engrg., 305, 376-404 (2016) [38] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092-2122.; Stinga, P. R.; Torrea, J. L., Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations, 35, 11, 2092-2122 (2010) [39] P. R. Stinga and B. Volzone, Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 1009-1042.; Stinga, P. R.; Volzone, B., Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations, 54, 1, 1009-1042 (2015) [40] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007.; Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces (2007) [41] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math. 25, Springer, Berlin, 1997.; Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (1997) [42] J. Toft, Embeddings for modulation spaces and Besov spaces, Report, Blekinge Institute of Technology, 2001.; Toft, J., Embeddings for modulation spaces and Besov spaces (2001) [43] S. van der Walt, J. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. Warner, N. Yager, E. Gouillart, T. Yu and the SciKit-image contributors, SciKit-image: Image processing in Python, PeerJ 2 (2014), Article ID e453.; van der Walt, S.; Schönberger, J.; Nunez-Iglesias, J.; Boulogne, F.; Warner, J.; Yager, N.; Gouillart, E.; Yu, T., SciKit-image: Image processing in Python, PeerJ, 2 (2014)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.