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Journal articles on the topic 'Nonlocal operators'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

DU, QIANG, MAX GUNZBURGER, R. B. LEHOUCQ, and KUN ZHOU. "A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS." Mathematical Models and Methods in Applied Sciences 23, no. 03 (January 14, 2013): 493–540. http://dx.doi.org/10.1142/s0218202512500546.

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A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is the posing of abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics.

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2

Chen, Yufu, and Hongqing Zhang. "NONLOCAL SYMMETRIES AND NONLOCAL RECURSION OPERATORS." Acta Mathematica Scientia 21, no. 1 (January 2001): 103–8. http://dx.doi.org/10.1016/s0252-9602(17)30582-9.

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3

Lee, Hwi, and Qiang Du. "Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 1 (January 2020): 105–28. http://dx.doi.org/10.1051/m2an/2019053.

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Nonlocal gradient operators are prototypical nonlocal differential operators that are very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on nonlocal gradient operators with a non-spherical interaction neighborhood. We show that the truncation of the spherical interaction neighborhood to a half sphere helps making nonlocal gradient operators well-defined and the associated nonlocal Dirichlet energies coercive. These become possible, unlike the case with full spherical neighborhoods, without any extra assumption on the strengths of the kernels near the origin. We then present some applications of the nonlocal gradient operators with non-spherical interaction neighborhoods. These include nonlocal linear models in mechanics such as nonlocal isotropic linear elasticity and nonlocal Stokes equations, and a nonlocal extension of the Helmholtz decomposition.

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4

Lee, Duckhwan, and Herschel Rabitz. "Scaling of nonlocal operators." Physical Review A 32, no. 2 (August 1, 1985): 877–82. http://dx.doi.org/10.1103/physreva.32.877.

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5

Lizama, Carlos, Marina Murillo-Arcila, and Alfred Peris. "Nonlocal operators are chaotic." Chaos: An Interdisciplinary Journal of Nonlinear Science 30, no. 10 (October 2020): 103126. http://dx.doi.org/10.1063/5.0018408.

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6

Lou, Yifei, Xiaoqun Zhang, Stanley Osher, and Andrea Bertozzi. "Image Recovery via Nonlocal Operators." Journal of Scientific Computing 42, no. 2 (August 27, 2009): 185–97. http://dx.doi.org/10.1007/s10915-009-9320-2.

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7

DI CECIO, G., and G. PAFFUTI. "SOME PROPERTIES OF RENORMALONS IN GAUGE THEORIES." International Journal of Modern Physics A 10, no. 10 (April 20, 1995): 1449–63. http://dx.doi.org/10.1142/s0217751x95000693.

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We find the explicit operatorial form of renormalon type singularities in Abelian gauge theory. Local operators of dimension six take care of the first UV renormalon; nonlocal operators are needed for IR singularities. In the effective Lagrangian constructed with these operators nonlocal imaginary parts appearing in the usual perturbative expansion at large orders are canceled.

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8

Xu, Xin-Jian, and Chuan-Fu Yang. "Inverse nodal problem for nonlocal differential operators." Tamkang Journal of Mathematics 50, no. 3 (September 2, 2019): 337–47. http://dx.doi.org/10.5556/j.tkjm.50.2019.3361.

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Inverse nodal problem consists in constructing operators from the given zeros of their eigenfunctions. The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary condition, and show that the potential function can be determined by nodal data.

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9

Ishikawa, Tomomi. "Perturbative matching of continuum and lattice quasi-distributions." EPJ Web of Conferences 175 (2018): 06028. http://dx.doi.org/10.1051/epjconf/201817506028.

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Matching of the quasi parton distribution functions between continuum and lattice is addressed using lattice perturbation theory specifically withWilson-type fermions. The matching is done for nonlocal quark bilinear operators with a straightWilson line in a spatial direction. We also investigate operator mixing in the renormalization and possible O(a) operators for the nonlocal operators based on a symmetry argument on lattice.

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10

Shakhmurov, Veli B. "Degenerate Differential Operators with Parameters." Abstract and Applied Analysis 2007 (2007): 1–27. http://dx.doi.org/10.1155/2007/51410.

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The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

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11

Yin, Xiang Feng, Jin Ming Duan, Zhen Kuan Pan, Wei Bo Wei, and Guo Dong Wang. "Nonlocal TV-L1 Inpainting Model and its Augmented Lagrangian Algorithm." Applied Mechanics and Materials 644-650 (September 2014): 4630–36. http://dx.doi.org/10.4028/www.scientific.net/amm.644-650.4630.

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Nonlocal differential operators have been extensively applied to variational models for image restoration due to its texture-preserving capability. In this paper, we propose a nonlocal TV (total variation)-L1 model for texture image inpainting, which, technically, combines nonlocal operators for regularization term and L1 norm for data term. The former is used to regularize texture and the latter to preserve contrast of images. In addition, we develop augmented Lagrangian algorithm for proposed model by introducing nonlocal auxiliary variable and Lagrangian multiplier. Finally, extensive experiments on synthetic and real texture images are presented to validate the effectiveness and efficiency of our proposed model and algorithm.

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12

Kassmann, Moritz, and Ante Mimica. "Intrinsic scaling properties for nonlocal operators." Journal of the European Mathematical Society 19, no. 4 (2017): 983–1011. http://dx.doi.org/10.4171/jems/686.

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13

Dyda, Bartłomiej, and Moritz Kassmann. "Regularity estimates for elliptic nonlocal operators." Analysis & PDE 13, no. 2 (March 19, 2020): 317–70. http://dx.doi.org/10.2140/apde.2020.13.317.

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14

Felsinger, Matthieu, and Moritz Kassmann. "Local Regularity for Parabolic Nonlocal Operators." Communications in Partial Differential Equations 38, no. 9 (September 2, 2013): 1539–73. http://dx.doi.org/10.1080/03605302.2013.808211.

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15

Chaker, Jamil, and Moritz Kassmann. "Nonlocal operators with singular anisotropic kernels." Communications in Partial Differential Equations 45, no. 1 (November 12, 2019): 1–31. http://dx.doi.org/10.1080/03605302.2019.1651335.

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16

Bogdan, Krzysztof, Tomasz Grzywny, Katarzyna Pietruska-Pałuba, and Artur Rutkowski. "Extension and trace for nonlocal operators." Journal de Mathématiques Pures et Appliquées 137 (May 2020): 33–69. http://dx.doi.org/10.1016/j.matpur.2019.09.005.

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17

Geyer, Bodo, and Markus Lazar. "Nonlocal LC-operators of definite twist." Nuclear Physics B - Proceedings Supplements 90 (December 2000): 28–30. http://dx.doi.org/10.1016/s0920-5632(00)00866-5.

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18

Kravchenko, K. V. "Differential operators with nonlocal boundary conditions." Differential Equations 36, no. 4 (April 2000): 517–23. http://dx.doi.org/10.1007/bf02754246.

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19

Mustafa, M. M., and M. W. Kermode. "The nonlocal tensor operatorS 12 N (?, ??)." Few-Body Systems 11, no. 2-3 (June 1991): 83–88. http://dx.doi.org/10.1007/bf01318553.

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20

Correa, Ernesto, and Arturo de Pablo. "Nonlocal operators of order near zero." Journal of Mathematical Analysis and Applications 461, no. 1 (May 2018): 837–67. http://dx.doi.org/10.1016/j.jmaa.2017.12.011.

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21

Xu, Xin-Jian, and Chuan-Fu Yang. "Trace formula for nonlocal differential operators." Indian Journal of Pure and Applied Mathematics 50, no. 4 (November 19, 2019): 1107–14. http://dx.doi.org/10.1007/s13226-019-0378-8.

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22

Albeverio, Sergio, and Leonid Nizhnik. "Schrödinger operators with nonlocal point interactions." Journal of Mathematical Analysis and Applications 332, no. 2 (August 2007): 884–95. http://dx.doi.org/10.1016/j.jmaa.2006.10.070.

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23

Zhang, Xicheng. "Fundamental Solutions of Nonlocal Hörmander’s Operators." Communications in Mathematics and Statistics 4, no. 3 (September 2016): 359–402. http://dx.doi.org/10.1007/s40304-016-0090-5.

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24

Felsinger, Matthieu, Moritz Kassmann, and Paul Voigt. "The Dirichlet problem for nonlocal operators." Mathematische Zeitschrift 279, no. 3-4 (November 2, 2014): 779–809. http://dx.doi.org/10.1007/s00209-014-1394-3.

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25

Nyeo, Su-Long. "Anomalous dimensions of nonlocal baryon operators." Zeitschrift für Physik C Particles and Fields 54, no. 4 (December 1992): 615–19. http://dx.doi.org/10.1007/bf01559489.

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26

Bogdan, Krzysztof, Paweł Sztonyk, and Victoria Knopova. "Heat Kernel of Anisotropic Nonlocal Operators." Documenta Mathematica 25 (2020): 1–54. http://dx.doi.org/10.4171/dm/736.

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27

Huang, Qiao, Jinqiao Duan, and Jiang-Lun Wu. "Maximum principles for nonlocal parabolic Waldenfels operators." Bulletin of Mathematical Sciences 09, no. 03 (December 2019): 1950015. http://dx.doi.org/10.1142/s1664360719500152.

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As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with [Formula: see text]-stable Lévy processes are presented to illustrate the maximum principles.

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28

Chen, Wenxiong, Congming Li, and Yan Li. "A direct blowing-up and rescaling argument on nonlocal elliptic equations." International Journal of Mathematics 27, no. 08 (July 2016): 1650064. http://dx.doi.org/10.1142/s0129167x16500646.

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In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray–Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.

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29

D’Elia, Marta, Mamikon Gulian, Hayley Olson, and George Em Karniadakis. "Towards a unified theory of fractional and nonlocal vector calculus." Fractional Calculus and Applied Analysis 24, no. 5 (October 1, 2021): 1301–55. http://dx.doi.org/10.1515/fca-2021-0057.

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Abstract Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

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30

Anguelov, Roumen, and Stephanus Marnus Stoltz. "Modelling of activator-inhibitor dynamics via nonlocal integral operators." Texts in Biomathematics 1 (December 28, 2017): 57. http://dx.doi.org/10.11145/texts.2017.12.233.

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This paper proposes application of nonlocal operators to represent the biological pattern formation mechanism of self-activation and lateral inhibition. The blue-green algae Anabaena is discussed as a model example. The patterns are determined by the kernels of the integrals representing the nonlocal operators. The emergence of patters when varying the size of the support of the kernels is numerically investigated.

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31

Izvarina, N. R. "ON THE SYMBOL OF NONLOCAL OPERATORS ASSOCIATED WITH A PARABOLIC DIFFEOMORPHISM." Eurasian Mathematical Journal 9, no. 2 (2018): 34–43. http://dx.doi.org/10.32523/2077-9879-2018-9-2-34-43.

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32

Piatnitski, A., V. Sloushch, T. Suslina, and E. Zhizhina. "On operator estimates in homogenization of nonlocal operators of convolution type." Journal of Differential Equations 352 (April 2023): 153–88. http://dx.doi.org/10.1016/j.jde.2022.12.036.

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33

Ma, Zaizhong, Ying-Hwa Kuo, Bin Wang, Wan-Shu Wu, and Sergey Sokolovskiy. "Comparison of Local and Nonlocal Observation Operators for the Assimilation of GPS RO Data with the NCEP GSI System: An OSSE Study." Monthly Weather Review 137, no. 10 (October 1, 2009): 3575–87. http://dx.doi.org/10.1175/2009mwr2809.1.

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Abstract In this study, an Observing System Simulation Experiment (OSSE) is performed to evaluate the performance of a nonlocal excess phase operator and a local refractivity operator for a GPS radio occultation (RO) sounding that passes through the eye of Hurricane Katrina as simulated by a high-resolution model, with significant horizontal refractivity gradients. Both observation operators are tested on the NCEP gridpoint statistical interpolation (GSI) data assimilation system at 12- and 36-km horizontal resolution. It is shown that the shape and magnitude of the analysis increments for sea level pressure, temperature, and water vapor mixing ratio exhibit significant differences between the use of local and nonlocal operators. The nonlocal operator produces more accurate analyses when verified against the “truth” derived from the ground truth run. It is found that the improvements of the analysis with the use of the nonlocal operator over that of the local operator are essentially the same at 12- and 36-km horizontal resolution. An additional experiment is performed over a region with small horizontal gradients. As expected, the use of both nonlocal and local operators produces similar results over such a region.

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34

Kim, Yong-Cheol. "Nonlocal Harnack inequalities for nonlocal Schrödinger operators with A1-Muckenhoupt potentials." Journal of Mathematical Analysis and Applications 507, no. 1 (March 2022): 125746. http://dx.doi.org/10.1016/j.jmaa.2021.125746.

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35

Shapovalov, Alexander V., Anton E. Kulagin, and Andrey Yu Trifonov. "The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve." Symmetry 12, no. 2 (February 1, 2020): 201. http://dx.doi.org/10.3390/sym12020201.

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We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.

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36

Buoninfante, Luca, Gaetano Lambiase, and Masahide Yamaguchi. "Enlarging local symmetries: A nonlocal Galilean model." International Journal of Geometric Methods in Modern Physics 17, supp01 (May 26, 2020): 2040009. http://dx.doi.org/10.1142/s0219887820400095.

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We consider the possibility to enlarge the class of symmetries realized in standard local field theories by introducing infinite order derivative operators in the actions, which become nonlocal. In particular, we focus on the Galilean shift symmetry and its generalization in nonlocal (infinite derivative) field theories. First, we construct a nonlocal Galilean model which may be UV finite, showing how the ultraviolet behavior of loop integrals can be ameliorated. We also discuss the pole structure of the propagator which has infinitely many complex conjugate poles, but satisfies tree level unitarity. Moreover, we will introduce the same kind of nonlocal operators in the context of linearized gravity. In such a scenario, the graviton propagator turns out to be ghost-free and the spacetime metric generated by a point-like source is non-singular.

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37

Ruzhansky, Michael, Niyaz Tokmagambetov, and Berikbol T. Torebek. "On a non–local problem for a multi–term fractional diffusion-wave equation." Fractional Calculus and Applied Analysis 23, no. 2 (April 28, 2020): 324–55. http://dx.doi.org/10.1515/fca-2020-0016.

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AbstractThis paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions.

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38

Fernández Bonder, Julián, Antonella Ritorto, and Ariel Martin Salort. "A class of shape optimization problems for some nonlocal operators." Advances in Calculus of Variations 11, no. 4 (October 1, 2018): 373–86. http://dx.doi.org/10.1515/acv-2016-0065.

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AbstractIn this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

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39

Shen, Wenxian, and Xiaoxia Xie. "Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations." Journal of Differential Equations 259, no. 12 (December 2015): 7375–405. http://dx.doi.org/10.1016/j.jde.2015.08.026.

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40

Yangari, Miguel. "Monotone systems involving variable-order nonlocal operators." Publicacions Matemàtiques 66 (January 1, 2022): 129–58. http://dx.doi.org/10.5565/publmat6612205.

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41

Gilboa, Guy, and Stanley Osher. "Nonlocal Operators with Applications to Image Processing." Multiscale Modeling & Simulation 7, no. 3 (January 2009): 1005–28. http://dx.doi.org/10.1137/070698592.

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42

Chiu, Ting-Wai. "Can nonlocal Dirac operators be topologically proper?" Physics Letters B 498, no. 1-2 (January 2001): 111–16. http://dx.doi.org/10.1016/s0370-2693(00)01365-4.

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43

Zhang, Xicheng. "Fundamental solutions of nonlocal Hörmander’s operators II." Annals of Probability 45, no. 3 (May 2017): 1799–841. http://dx.doi.org/10.1214/16-aop1102.

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44

Nazaikinskii, V. E., A. Yu Savin, and B. Yu Sternin. "On the index of nonlocal elliptic operators." Doklady Mathematics 77, no. 3 (June 2008): 441–45. http://dx.doi.org/10.1134/s1064562408030320.

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45

Savin, A. Yu, and B. Yu Sternin. "Nonlocal elliptic operators for compact Lie groups." Doklady Mathematics 81, no. 2 (April 2010): 258–61. http://dx.doi.org/10.1134/s1064562410020262.

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46

Karlovich, Yuri I., and Iván Loreto-Hernández. "C⁎-algebra of nonlocal convolution type operators." Journal of Mathematical Analysis and Applications 475, no. 2 (July 2019): 1130–61. http://dx.doi.org/10.1016/j.jmaa.2018.11.085.

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47

Yurko, Vjacheslav Anatoljevich, and Chuan-Fu Yang. "Recovering differential operators with nonlocal boundary conditions." Analysis and Mathematical Physics 6, no. 4 (November 14, 2015): 315–26. http://dx.doi.org/10.1007/s13324-015-0120-6.

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48

Tian, Xiaochuan, and Qiang Du. "A Class of High Order Nonlocal Operators." Archive for Rational Mechanics and Analysis 222, no. 3 (July 6, 2016): 1521–53. http://dx.doi.org/10.1007/s00205-016-1025-8.

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49

Capri, M. A. L., V. E. R. Lemes, R. F. Sobreiro, S. P. Sorella, and R. Thibes. "Local renormalizable gauge theories from nonlocal operators." Annals of Physics 323, no. 3 (March 2008): 752–67. http://dx.doi.org/10.1016/j.aop.2007.07.002.

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50

Fan, Zhenbin, and Gisèle Mophou. "Nonlocal Problems for Fractional Differential Equations via Resolvent Operators." International Journal of Differential Equations 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/490673.

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We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

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