Benson, D., Wheatcraft, S., Meerschaert, M., Application of a fractional advection-dispersion equation. Water Resour. Res. 36:6 (2000), 1403–1412.
Benson, D., Schumer, R., Meerschaert, M., Wheatcraft, S., Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp. Porous Media 42 (2001), 211–240.
D'Elia, M., Gulian, M., Analysis of anisotropic nonlocal diffusion models: Well-posedness of fractional problems for anomalous transport. Numer. Math.: Theory Methods Appl., 2021 Accepted.
Deng, Z.-Q., Singh, V., Bengtsson, L., Numerical solution of fractional advection-dispersion equation. J. Hydraul. Eng., 130(5), 2004.
Schumer, R., Benson, D., Meerschaert, M., Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions. Water Resour. Res. 39:1 (2003), 1022–1032.
Schumer, R., Benson, D., Meerschaert, M., Wheatcraft, S., Eulerian derivation of the fractional advection-dispersion equation. J. Contam. Hydrol. 48 (2001), 69–88.
Ha, Y.D., Bobaru, F., Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78:6 (2011), 1156–1168.
D. Littlewood, Simulation of Dynamic Fracture using Peridynamics, Finite Element Modeling, and Contact, in: Proceedings of the ASME 2010 International Mechanical Engineering Congress and Exposition, Vancouver, British Columbia, Canada, Vol. 9, 2010, pp. 209–217.
Silling, S., Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000), 175–209.
Akhavan-Safaei, A., Samiee, M., Zayernouri, M., Data-driven fractional subgrid-scale modeling for scalar turbulence: A nonlocal LES approach. J. Comput. Phys., 2021, 110571.
Leoni, P.C.D., Zaki, T.A., Karniadakis, G., Meneveau, C., Two-point stress-strain rate correlation structure and non-local eddy viscosity in turbulent flows. J. Fluid Mech., 914, 2021, A6.
Pang, G., D'Elia, M., Parks, M., Karniadakis, G.E., nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications. J. Comput. Phys., 422, 2020, 109760.
Buades, A., Coll, B., Morel, J., Image denoising methods. A new nonlocal principle. SIAM Rev. 52 (2010), 113–147.
D'Elia, M., De Los Reyes, J.C., Miniguano-Trujillo, A., Bilevel parameter learning for nonlocal image denoising models. J. Math. Imaging Vision 63:6 (2021), 753–775.
Gilboa, G., Osher, S., Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6 (2007), 595–630.
Burch, N., D'Elia, M., Lehoucq, R., The exit-time problem for a Markov jump process. Eur. Phys. J. Spec. Top. 223 (2014), 3257–3271.
D'Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes. Comput. Methods Appl. Math. 29 (2017), 71–103.
Meerschaert, M., Sikorskii, A., Stochastic Models for Fractional Calculus. 2012, Studies in Mathematics, Gruyter.
Metzler, R., Klafter, J., The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.
Metzler, R., Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004), 161–208.
D'Elia, M., Gulian, M., Olson, H., Karniadakis, G.E., Towards a unified theory of fractional and nonlocal vector calculus. Fract. Calc. Appl. Anal. 24:5 (2021), 1301–1355.
Capodaglio, G., D'Elia, M., Bochev, P., Gunzburger, M., An energy-based coupling approach to nonlocal interface problems. Comput. & Fluids, 207, 2020, 104593.
Xu, X., Glusa, C., D'Elia, M., Foster, J.T., A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems. Comput. Methods Appl. Mech. Engrg., 2021.
Xu, X., D'Elia, M., Foster, J.T., A machine-learning framework for peridynamic material models with physical constraints. Comput. Methods Appl. Mech. Engrg., 2021.
Burkovska, O., Glusa, C., D'Elia, M., An optimization-based approach to parameter learning for fractional type nonlocal models. Comput. Math. Appl., 2021.
D'Elia, M., Gunzburger, M., Optimal distributed control of nonlocal steady diffusion problems. SIAM J. Control Optim. 55 (2014), 667–696.
D'Elia, M., Gunzburger, M., Identification of the diffusion parameter in nonlocal steady diffusion problems. Appl. Math. Optim. 73 (2016), 227–249.
Gulian, M., Raissi, M., Perdikaris, P., Karniadakis, G.E., Machine learning of space-fractional differential equations. SIAM J. Sci. Comput. 41:4 (2019), A2485–A2509.
Pang, G., Perdikaris, P., Cai, W., Karniadakis, G.E., Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization. J. Comput. Phys. 348 (2017), 694–714.
Xu, X., D'Elia, M., Foster, J.T., A machine-learning framework for peridynamic material models with physical constraints. Comput. Methods Appl. Mech. Engrg., 386, 2021, 114062.
You, H., Yu, Y., Trask, N., Gulian, M., D'Elia, M., Data-driven learning of nonlocal physics from high-fidelity synthetic data. Comput. Methods Appl. Mech. Engrg., 374, 2021, 113553.
You, H., Yu, Y., Silling, S., D'Elia, M., Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws. 2020 Accepted in AAAI MLPS Symposium.
You, H., Yu, Y., Silling, S., D'Elia, M., A data-driven peridynamic continuum model for upscaling molecular dynamics. Comput. Methods Appl. Mech. Engrg., 2021.
Ainsworth, M., Glusa, C., Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains. Contemporary Computational Mathematics-a Celebration of the 80th Birthday of Ian Sloan, 2018, Springer, 17–57.
Capodaglio, G., D'Elia, M., Gunzburger, M., Bochev, P., Klar, M., Vollmann, C., A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions. Numer. Methods Partial Differential Equations, 2020.
D'Elia, M., Du, Q., Glusa, C., Tian, X., Zhou, Z., Numerical methods for nonlocal and fractional models. ACTA Numer., 29, 2020.
D'Elia, M., Gunzburger, M., Vollmann, C., A cookbook for approximating euclidean balls and for quadrature rules in finite element methods for nonlocal problems. Math. Models Methods Appl. Sci. 31:08 (2021), 1505–1567.
Pasetto, M., Enhanced Meshfree Methods for Numerical Solution of Local and Nonlocal Theories of Solid Mechanics. (Ph.D. thesis), 2019, University of California, San Diego, CA.
Pasetto, M., Leng, Y., Chen, J., Foster, J., Seleson, P., A reproducing kernel enhanced approach for peridynamic solutions. Comput. Methods Appl. Mech. Engrg. 340 (2018), 1044–1078.
Silling, S.A., Askari, E., A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83:17–18 (2005), 1526–1535.
Wang, H., Wang, K., Sircar, T., A direct O(Nlog2N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229:21 (2010), 8095–8104.
Chen, Y., Lee, J., Eskandarian, A., Meshless Methods in Solid Mechanics. 2006, Springer Science & Business Media.
Parks, M., Littlewood, D., Mitchell, J., Silling, S., Peridigm Users Guide: Technical Report SAND2012-7800., 2012, Sandia National Laboratories, NM, USA.
Parks, M., Seleson, P., Plimpton, S., Lehoucq, R., Silling, S., Peridynamics with LAMMPS:A User Guide: Technical Report SAND2010-5549., 2010, Sandia National Laboratories, NM, USA.
Trask, N., You, H., Yu, Y., Parks, M.L., An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics. Comput. Methods Appl. Mech. Engrg. 343 (2019), 151–165.
Gunzburger, M., Lehoucq, R., A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model. Simul. 8 (2010), 1581–1598.
Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K., A nonlocal vector calculus, nonlocal volume constrained problems, and nonlocal balance laws. Math. Models Appl. Sci. 23:3 (2013), 493–540.
Aulisa, E., Capodaglio, G., Chierici, A., D'Elia, M., Efficient quadrature rules for finite element discretizations of nonlocal equations. Numer. Methods Partial Differential Equations, 2021.
Leng, Y., Tian, X., Trask, N., Foster, J., Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion. SIAM J. Numer. Anal. 59 (2021), 88–118.
Gross, B., Trask, N., Kuberry, P., Atzberger, P., Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A generalized moving least-squares (GMLS) approach. J. Comput. Phys. 409 (2020), 109–340.
D'Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes. Comput. Methods Appl. Math. 17:4 (2017), 707–722.
Felsinger, M., Kassmann, M., Voigt, P., The Dirichlet problem for nonlocal operators. Math. Z. 279:3–4 (2015), 779–809.
Mengesha, T., Du, Q., Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete Contin. Dyn. Syst. Ser. B, 18(5), 2013, 1415.
Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54:4 (2012), 667–696.
Gunzburger, M., Lehoucq, R., A nonlocal vector calculus with applications to nonlocal boundary value problems. Multiscale Model. Simul. 8:5 (2010), 1581–1598.
D'Elia, M., Tian, X., Yu, Y., A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints. SIAM J. Sci. Comput. 42:4 (2020), A1935–A1949.
D'Elia, M., Yu, Y., On the prescription of boundary conditions for nonlocal Poisson's and peridynamics models. 2021 arXiv preprint arXiv:2107.04450.
Mirzaei, D., Schaback, R., Dehghan, M., On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32 (2012), 983–1000.
Salehi, R., Dehghan, M., A generalized moving least square reproducing kernel method. J. Comput. Appl. Math. 249 (2013), 120–132.
Mirzaei, D., Schaback, R., Direct meshless local Petrov–Galerkin (DMLPG) method:A generalized MLS approximation. Appl. Numer. Math. 68 (2013), 73–82.
Chen, J., Hillman, M., Chi, S.-W., Meshfree methods: Progress made after 20 years. J. Eng. Mech., 143(4), 2017.
Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012), 667–696.
Tian, X., Du, Q., Asymptotically compatible schemes and applications to robust discretizations of nonlocal models. SIAM J. Numer. Anal. 52:4 (2014), 1641–1665.