Element contents
Nonlocal Continuum Limits of p-Laplacian Problems on Graphs
Published online by Cambridge University Press: 13 April 2023
Summary
In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.
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- Online ISBN: 9781009327862Publisher: Cambridge University PressPrint publication: 11 May 2023
References
Aksoylu, Burak, and Mengesha, Tadele. 2010. Results on nonlocal boundary value problems. Numerical Functional Analysis and Optimization, 31(12), 1301–1317.CrossRefGoogle Scholar
Alali, Bacim, Liu, Kuo, and Gunzburger, Max. 2015. A generalized nonlocal vector calculus. Zeitschrift für angewandte Mathematik und Physik, 66(5), 2807–2828.Google Scholar
Andreu, Fuensanta, Mazón, José, Rossi, Julio, and Toledo-Melero, Julián. 2008a. The Neumann problem for nonlocal nonlinear diffusion equations. Journal of Evolution Equations, 8(1), 189–215.CrossRefGoogle Scholar
Andreu, Fuensanta, Mazón, José, Rossi, Julio, and Toledo-Melero, Julián. 2008b. A nonlocal -Laplacian evolution equation with Neumann boundary conditions. Journal de Mathématiques Pures et Appliquées, 90(2), 201–227.Google Scholar
Andreu-Vaillo, Fuensanta, Mazón, José, Rossi, Julio, and Toledo-Melero, Julián. 2010. Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. American Mathematical Society.CrossRefGoogle Scholar
Attouch, Hédy. 1984. Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman Advanced Publishing Program.Google Scholar
Attouch, Hédy, and Peypouquet, Juan. 2016. The rate of convergence of Nesterov’s accelerated forward-backward method is actually faster than 1/k 2. SIAM Journal on Optimization, 26(3), 1824–1834.Google Scholar
Aubert, Gilles, and Kornprobst, Pierre. 2002. Mathematical Problems in Image Processing. Applied Mathematical Sciences, vol. 147. Springer.Google Scholar
Ayi, Nathalie, and Pouradier Duteil, Nastassia. 2021. Mean-field and graph limits for collective dynamics models with time-varying weights. Journal of Differential Equations, 299(2), 65–110.Google Scholar
Azé, Dominique, Attouch, Hédy, and Wets, Roger. 1988. Convergence of convex-concave saddle functions: Applications to convex programming and mechanics. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 5(6), 537–572.Google Scholar
Bates, Peter, and Chmaj, Adam. 1999. An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions. Journal of Statistical Physics, 95(5), 1119–1139.Google Scholar
Bates, Peter, Fife, Paul, Ren, Xiaofeng, and Wang, Xuefeng. 1997. Traveling waves in a convolution model for phase transitions. Archive for Rational Mechanics and Analysis, 138(2), 105–136.Google Scholar
Bauschke, Heinz, and Combettes, Patrick. 2011. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer.Google Scholar
Bažant, Zdeněk, and Jirásek, Milan. 2002. Nonlocal integral formulations of plasticity and damage: Survey of progress. Journal of Engineering Mechanics, 128(11), 1119–1149.CrossRefGoogle Scholar
Beck, Amir, and Teboulle, Marc. 2009. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183–202.Google Scholar
Beer, Gerald, and Lucchetti, Roberto. 1992. The epi-distance topology: Continuity and stability results with applications to convex optimization problems. Mathematics of Operations Research, 17(3), 715–726.Google Scholar
Bénilan, Philippe. 1972. Solutions intégrales d’équations d’évolution dans un espace de Banach. Comptes rendus de l’Académie des Sciences Series A–B, 274, A47–A50.Google Scholar
Bénilan, Philippe, and Crandall, Michael. 1991. Completely accretive operators. Pages 41–75 of Clément, Philippe, de Pagter, Ben, and Mitidieri, Enzo (eds.), Semigroup Theory and Evolution Equations. Lecture Notes in Pure and Applied Mathematics, vol. 135. Dekker.Google Scholar
Biccari, Umberto, Ko, Dongnam, and Zuazua, Enrique. 2019. Dynamics and control for multi-agent networked systems: A finite-difference approach. Mathematical Models and Methods in Applied Sciences, 29(4), 755–790.Google Scholar
Bognar, Gabriella. 2008. Numerical and analytic investigation of some nonlinear problems in fluid mechanics. Computer and Simulation in Modern Science, 2, 172–179.Google Scholar
Bollobás, Béla, and Riordan, Oliver. 2009. Metrics for sparse graphs. Pages 211–288 of Huczynska, Sophie, Mitchell, James, and Roney-Dougal, Colva (eds.), Surveys in Combinatorics 2009. London Mathematical Society Lecture Note Series. Cambridge University Press.Google Scholar
Bollobás, Béla, Janson, Svante, and Riordan, Oliver. 2007. The phase transition in inhomogeneous random graphs. Random Structures & Algorithms, 31(1), 3–122.Google Scholar
Borgs, Christian, Chayes, Jennifer, Cohn, Henry, and Zhao, Yufei. 2018. An theory of sparse graph convergence II: LD convergence, quotients and right convergence. Annals of Probability, 46(1), 337–396.Google Scholar
Borgs, Christian, Chayes, Jennifer, Cohn, Henry, and Zhao, Yufei. 2019. An theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. Transactions of the American Mathematical Society, 372(5), 3019–3062.Google Scholar
Borgs, Christian, Chayes, Jennifer, Lovász, László, Sós, Vera, and Vesztergombi, Katalin. 2008. Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 219(6), 1801–1851.Google Scholar
Borgs, Christian, Chayes, Jennifer, Lovász, László, Sós, Vera, and Vesztergombi, Katalin. 2011. Limits of randomly grown graph sequences. European Journal of Combinatorics, 32(7), 985–999.Google Scholar
Borwein, Jonathan, and Fitzpatrick, Simon. 1989. Mosco convergence and the Kadec property. Proceedings of the American Mathematical Society, 106(3), 843–851.Google Scholar
Braides, Andrea. 2002. Gamma-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Clarendon Press.Google Scholar
Brézis, Haїm. 1973. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland.Google Scholar
Buades, Antoni, Coll, Bartomeu, and Morel, Jean-Michel. 2005. A review of image denoising algorithms, with a new one. Multiscale Modeling & Simulation, 4(2), 490–530.Google Scholar
Bühler, Thomas, and Hein, Matthias. 2009. Spectral clustering based on the graph -Laplacian. Pages 81–88 of Proceedings of the 26th Annual International Conference on Machine Learning. ICML ’09. Association for Computing Machinery.CrossRefGoogle Scholar
Bungert, Leon, Calder, Jeff, and Roith, Tim. 2021. Uniform convergence rates for Lipschitz learning on graphs. arXiv:2111.12370.CrossRefGoogle Scholar
Byström, Johan. 2005. Sharp constants for some inequalities connected to the -Laplace operator. Journal of Inequalities in Pure and Applied Mathematics, 6(2).Google Scholar
Calder, Jeff. 2018. The game theoretic -Laplacian and semi-supervised learning with few labels. Nonlinearity, 32(1), 301.Google Scholar
Calder, Jeff. 2019. Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data. SIAM Journal on Mathematics of Data Science, 1(4), 780–812.CrossRefGoogle Scholar
Calder, Jeff, Slepcev, Dejan, and Thorpe, Matthew. 2020. Rates of convergence for Laplacian semi-supervised learning with low labeling rates. arXiv:2006.02765.Google Scholar
Carrillo, C., and Fife, P. 2005. Spatial effects in discrete generation population models. Journal of Mathematical Biology, 50(2), 161–188.Google Scholar
Carrillo, José Antonio, Manuel, del Pino, Figalli, Alessio, Mingione, Giuseppe, and Vázquez, Juan Luis. 2017. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics, vol. 2186. Springer.Google Scholar
Chambolle, Antonin, and Dossal, Charles. 2015. On the convergence of the iterates of the fast iterative shrinkage/thresholding algorithm. Journal of Optimization Theory and Applications, 166(3), 968–982.Google Scholar
Chambolle, Antonin, and Pock, Thomas. 2011. A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1), 120–145.Google Scholar
Cortazar, Carmen, Elgueta, Manuel, Rossi, Julio, and Wolanski, Noemi. 2008. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Archive for Rational Mechanics and Analysis, 187(1), 137–156.Google Scholar
Crandall, Michael, and Liggett, Thomas. 1971. Generation of semigroups of nonlinear transformations on general Banach spaces. American Journal of Mathematics, 93(2), 265–298.Google Scholar
DeVore, Ronald, and Lorentz, George. 1993. Constructive Approximation. Grundlehren der Mathematischen, vol. 303. Springer.Google Scholar
Drábek, Pavel. 2007. The -Laplacian – mascot of nonlinear analysis. Acta Mathematica Universitatis Comenianae, 76(1), 85–98.Google Scholar
Du, Qiang. 2019. Nonlocal Modeling, Analysis, and Computation: Nonlocal Modeling, Analysis, and Computation. SIAM.Google Scholar
Du, Qiang, Gunzburger, Max, Lehoucq, Richard, and Zhou, Kun. 2012. Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Review, 54(4), 667–696.Google Scholar
Du, Qiang, Gunzburger, Max, Lehoucq, Richard, and Zhou, Kun. 2013a. A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Mathematical Models and Methods in Applied Sciences, 23(3), 493–540.Google Scholar
Du, Qiang, Gunzburger, Max, Lehoucq, Richard, and Zhou, Kun. 2013b. A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Mathematical Models and Methods in Applied Sciences, 23(3), 493–540.Google Scholar
Dunlop, Matthew, Slepcev, Dejan, and Stuart, Andrew. 2020. Large data and zero noise limits of graph-based semi-supervised learning algorithms. Applied and Computational Harmonic Analysis, 49(2), 655–697.Google Scholar
Ahmed, El Alaoui, Xiang, Cheng, Aaditya, Ramdas, Martin, Wainwright, and Jordan, Michael. 2016. Asymptotic behavior of -based Laplacian regularization in semi-supervised learning. Pages 879–906 of Conference on Learning Theory.Google Scholar
Imad, El Bouchairi, Jalal, Fadili, and Abderrahim, Elmoataz. 2020. Continuum limit of -Laplacian evolution problems on graphs: graphons and sparse graphs. arXiv:2010.08697.Google Scholar
Abdallah, El Chakik, Abdderahim, Elmoataz, and Desquesnes, Xavier. 2014. Mean curvature flow on graphs for image and manifold restoration and enhancement. Signal Processing, 105, 449–463.Google Scholar
Elmoataz, Abderrahim, Desquesnes, Xavier, and Lézoray, Olivier. 2012. Non-local morphological PDEs and p-Laplacian equation on graphs with applications in image processing and machine learning. IEEE Journal of Selected Topics in Signal Processing, 6(7), 764–779.Google Scholar
Elmoataz, Abderrahim, Desquesnes, Xavier, Lakhdari, Zakaria, and Lézoray, Olivier. 2014. Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning. Mathematics and Computers in Simulation, 102, 153–163.Google Scholar
Elmoataz, Abderrahim, Lezoray, Olivier, Bougleux, Sébastien, and Ta, Vinh Thong. 2008. Unifying local and nonlocal processing with partial difference operators on weighted graphs. Pages 11–26 of International Workshop on Local and Non-Local Approximation in Image Processing.Google Scholar
Elmoataz, Abderrahim, Toutain, Matthieu, and Tenbrinck, Daniel. 2015. On the -Laplacian and -Laplacian on graphs with applications in image and data processing. SIAM Journal on Imaging Sciences, 8(4), 2412–2451.Google Scholar
Erdös, Paul, and Rényi, Alfréd. 1960. On the evolution of random graphs. Publication of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.Google Scholar
Eringen, Ahmet Cemal, and Wegner, J. L. 2003. Nonlocal continuum field theories. Applied Mechanics Reviews, 56(2), B20–B22.Google Scholar
Fadili, Jalal, Forcadel, Nicolas, Nguyen, Thi Tuyen, and Zantout, Rita. 2021. Limits and consistency of non-local and graph approximations to the Eikonal equation. arXiv:2105.01977.Google Scholar
Fadili, Jalal, and Peyré, Gabriel. 2010. Total variation projection with first order schemes. IEEE Transactions on Image Processing, 20(3), 657–669.Google Scholar
Fife, Paul, and Wang, Xuefeng. 1998. A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions. Advances in Difference Equations, 3(1), 85–110.Google Scholar
Fife, Paul. 2002. Some nonclassical trends in parabolic and parabolic-like evolutions. In Fiedler, B. (ed.), Trends in Nonlinear Analysis. Springer.Google Scholar
Figalli, Alessio, Peral, Ireneo, and Valdinoci, Enrico. 2018. Partial Differential Equations and Geometric Measure Theory. Springer.Google Scholar
Flores, Mauricio, Calder, Jeff, and Lerman, Gilad. 2022. Analysis and algorithms for -based semi-supervised learning on graphs. Applied and Computational Harmonic Analysis, 60, 77–122.Google Scholar
Frieze, Alan, and Kannan, Ravi. 1999. Quick approximation to matrices and applications. Combinatorica, 19(2), 175–220.Google Scholar
Trillos, Garcia, Nicolas. 2019. Variational limits of -NN graph-based functionals on data clouds. SIAM Journal on Mathematics of Data Science, 1(1), 93–120.Google Scholar
Trillos, Garcia, Nicolas, and Murray, Ryan. 2020. A maximum principle argument for the uniform convergence of graph Laplacian regressors. SIAM Journal on Mathematics of Data Science, 2(3), 705–739.Google Scholar
Yves van, Gennip, and Andrea, Bertozzi. 2012. -convergence of graph Ginzburg–Landau functionals. Advances in Differential Equations, 17(4), 1115–1180.Google Scholar
Gilboa, Guy, and Osher, Stanley. 2007. Nonlocal linear image regularization and supervised segmentation. Multiscale Modeling & Simulation, 6(2), 595–630.Google Scholar
Gilboa, Guy, and Osher, Stanley. 2008. Nonlocal operators with applications to image processing. SIAM Journal on Multiscale Modeling and Simulation, 7(3), 1005–1028.Google Scholar
Glowinski, Roland, and Marrocco, A. 1975. Sur l’approximation par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non-linéaires. RAIRO: Analyse Numérique, 9(R2), 41–76.Google Scholar
Gunzburger, Max, and Lehoucq, Richard. 2010. A nonlocal vector calculus with applications to nonlocal boundary value problems. SIAM Journal on Multiscale Modeling and Simulation, 8(5), 1581–1620.Google Scholar
Hafiene, Yosra, Fadili, Jalal, and Elmoataz, Abderrahim. 2018. Nonlocal -Laplacian evolution problems on graphs. SIAM Journal on Numerical Analysis, 56(2), 1064–1090.Google Scholar
Hafiene, Yosra, Fadili, Jalal, and Elmoataz, Abderrahim. 2019. Nonlocal -Laplacian Variational problems on graphs. SIAM Journal on Imaging Sciences, 12(4), 1772–1807.Google Scholar
Hafiene, Yosra, Fadili, Jalal, Chesneau, Christophe, and Elmoataz, Abderrahim. 2020. Continuum limit of the nonlocal -Laplacian evolution problem on random inhomogeneous graphs. ESAIM: Mathematical Modelling and Numerical Analysis, 54(2), 565–589.Google Scholar
Hinds, Brittney, and Radu, Petronela. 2012. Dirichlet’s principle and wellposedness of solutions for a nonlocal -Laplacian system. Applied Mathematics and Computation, 219(4), 1411–1419.Google Scholar
Hoeffding, Wassily. 1961. The Strong Law of Large Numbers for U-Statistics. Institute of Statistics Mimeograph Series, vol. 302. North Carolina State University.Google Scholar
Ibragimov, Rustam, and Sharakhmetov, Shoturgun. 2002. The exact constant in the Rosenthal inequality for random variables with mean zero. Theory of Probability & Its Applications, 46(1), 127–132.Google Scholar
Janson, Svante. 2013. Graphons, Cut Norm and Distance, Couplings and Rearrangements. New York Journal of Mathematics, vol. 4. NYJM Monographs, State University of New York, University at Albany.Google Scholar
Kaliuzhnyi-Verbovetskyi, Dmitry, and Medvedev, Georgi. 2017. The semilinear heat equation on sparse random graphs. SIAM Journal on Mathematical Analysis, 49(2), 1333–1355.Google Scholar
Kawohl, Bernd. 2011. Variations on the -Laplacian. Nonlinear Elliptic Partial Differential Equations. Contemporary Mathematics, 540, 35–46.Google Scholar
Kindermann, Stefan, Osher, Stanley, and Jones, Peter. 2006. Deblurring and denoising of images by nonlocal functionals. SIAM Multiscale Modeling and Simulations, 4(4), 25.Google Scholar
Koabayashi, Yoshikazu. 1975. Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups. Journal of Mathematical Society of Japan, 27(4), 640–665.Google Scholar
Kusolitsch, Norbert. 2010. Why the theorem of Scheffé should be rather called a theorem of Riesz. Periodica Mathematica Hungarica, 61(1–2), 225–229.Google Scholar
Laux, Tim, and Lelmi, Jona. 2021. Large data limit of the MBO scheme for data clustering: -convergence of the thresholding energies. arXiv:2112.06737.Google Scholar
Lee, Jong-Sen. 1983. Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Image Processing, 24(2), 255–269.Google Scholar
Lovász, László. 2012. Large Networks and Graph Limits. Colloquium Publications, vol. 60. American Mathematical Society.Google Scholar
Lovász, László, and Szegedy, Balázs. 2006. Limits of dense graph sequences. Journal of Combinatorial Theory, Series B, 96(6), 933–957.Google Scholar
Madenci, Erdogan, and Oterkus, Erkan. 2014. Peridynamic theory. Pages 19–43 of Peridynamic Theory and Its Applications. Springer.Google Scholar
McLinden, Lynn, and Bergstrom, Roy. 1981. Preservation of convergence of convex sets and functions in finite dimensions. Transactions of the American Mathematical Society, 268(1), 127–142.Google Scholar
Medvedev, Georgi. 2014a. The nonlinear heat equation on dense graphs. SIAM Journal on Mathematical Analysis, 46(4), 2743–2766.Google Scholar
Medvedev, Georgi. 2014b. The nonlinear heat equation on W-random graphs. Archive for Rational Mechanics and Analysis, 212(3), 781–803.Google Scholar
Medvedev, Georgi. 2019. The continuum limit of the Kuramoto model on sparse random graphs. Communications in Mathematical Sciences, 17(4), 883–898.Google Scholar
Mosco, Umberto. 1969. Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics, 3(4), 510–585.Google Scholar
Nesterov, Yurii. 1983. A method for solving the convex programming problem with convergence rate . Proceedings of the USSR Academy of Sciences, 269(3), 543–547.Google Scholar
Nochetto, Ricardo, and Savaré, Giuseppe. 2006. Nonlinear evolution governed by accretive operators in Banach spaces: Error control and applications. Mathematical Models and Methods in Applied Sciences, 16(3), 439–477.Google Scholar
Osher, Stanley, and Paragios, Nikos. 2003. Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer.Google Scholar
Pardoux, Etienne. 2009. Cours intégration et probabilité. Lecture notes (Aix-Marseille Universtité), 783.Google Scholar
Rabczuk, Timon, Ren, Huilong, and Zhuang, Xiaoying. 2019. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Materials & Continua, 59(1), 31–55.Google Scholar
Radu, Petronela, Toundykov, Daniel, and Trageser, Jeremy. 2017. A nonlocal biharmonic operator and its connection with the classical analogue. Archive for Rational Mechanics and Analysis, 223(2), 845–880.Google Scholar
Reich, Simeon, and Shoiykhet, David. 2005. Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press.Google Scholar
Roith, Tim, and Bungert, Leon. 2022. Continuum limit of Lipschitz learning on graphs. Foundations of Computational Mathematics, 1–39. DOI: https://doi.org/10.1007/s10208-022-09557-9.Google Scholar
Salinetti, Gabriella, and Wets, Roger. 1977. On the relations between two types of convergence for convex functions. Journal of Mathematical Analysis and Applications, 60(1), 211–226.Google Scholar
Scherzer, Otmar, Grasmair, Markus, Grossauer, Harald, Haltmeier, Markus, and Lenzen, Frank. 2009. Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer.Google Scholar
Silling, Stewart. 2000. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 48(1), 175–209.Google Scholar
Slepcev, Dejan, and Thorpe, Matthew. 2019. Analysis of -Laplacian regularization in semisupervised learning. SIAM Journal on Mathematical Analysis, 51(3), 2085–2120.Google Scholar
Tao, Yunzhe, Tian, Xiaochuan, and Du, Qiang. 2017. Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. Applied Mathematics and Computation, 305, 282–298.Google Scholar
Tian, Xiaochuan, and Du, Qiang. 2014. Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM Journal on Numerical Analysis, 52(4), 1641–1665.Google Scholar
Tian, Xiaochuan, and Du, Qiang. 2020. Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models. SIAM Review, 62(1), 199–227.Google Scholar
Trillos, Nicolás García, and Slepčev, Dejan. 2016. Continuum limit of total variation on point clouds. Archive for Rational Mechanics and Analysis, 220(1), 193–241.Google Scholar
Vaiter, Samuel, Peyré, Gabriel, and Fadili, Jalal. 2015. Low complexity regularization of linear inverse problems. In Pfander, Gotz (ed.), Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis (ANHA). Birkhäuser/Springer. DOI: https://doi.org/10.1007/978-3-319-19749-4.Google Scholar
Yves, Van Gennip, Guillen, Nestor, Osting, Braxton, and Bertozzi, Andrea. 2014. Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan Journal of Mathematics, 82(1), 3–65.Google Scholar
Wang, Xuefeng. 2002. Metastability and stability of patterns in a convolution model for phase transitions. Journal of Differential Equations, 183(2), 434–461.Google Scholar
Wijsman, Robert. 1966. Convergence of sequences of convex sets, cones and functions. II. Transactions of the American Mathematical Society, 123(1), 32–45.Google Scholar
You, Huaiqian, Lu, XinYang, Task, Nathaniel, and Yu, Yue. 2020. An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems. ESAIM: Mathematical Modelling and Numerical Analysis, 54(4), 1373–1413.Google Scholar
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