Research Areas

[A1] Nonlocal Vector Calculus and Calculus of Variations

[A2] Numerical Analysis for PDEs and Nonlocal Models

[A3] Meshfree Methods and their Applications

[A4] Multiscale Techniques and Nonlocal Modeling in materials

[A5] Stochastic Modeling and Uncertainty Quantification

Publication list


  1. Z. Han, T. Mengesha and X. Tian, Compactness results for a Dirichlet energy of nonlocal gradient with applications, preprint.

  2. Q. Du, J. Scott and X. Tian, Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems, preprint.

  3. X. Tian, Y. Wu and Y. Zhang, Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis, preprint.

  4. Y. Zhang, A. Cloninger, B. Li, and X. Tian, A neural network kernel decomposition for learning multiple steady states in parameterized dynamical systems, preprint.

Refereed Journal Papers

  1. Z. Han and X. Tian, Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications, Math. Models Methods Appl. Sci., 33(12),2507–2556, 2023. [A1] [A4]

  2. Q. Du, T. Mengesha and X. Tian, L^p compactness criteria with an application to variational convergence of some nonlocal energy functionals, Math. Eng., 5(6): 1-31, 2023. [A1]

  3. Q. Ye and X. Tian, Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation , J. Sci. Comput., 96(3), 85, 2023. [A2] [A3]

  4. Q. Du, X. Tian and Z. Zhou, Nonlocal diffusion models with consistent local and fractional limits, A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models: Proceedings of the 50th John H. Barrett Memorial Lectures,175-213, 2023. [A1] [A4]

  5. Y. Fan, H. You, X. Tian, X. Yang, X.H. Li, N. Prakash and Y. Yu, A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials, Comput. Methods Appl. Mech. Eng., 399, 115340, 2022. [A3] [A5]

  6. Y. Fan, X. Tian, X. Yang, X.H. Li, C. Webster and Y. Yu, An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems, J. Comput. Phys., 111376, 2022. [A3] [A5]

  7. M. Pasetto, Z. Shen, M. D’Elia, X. Tian, N.Trask and D. Kamensky, Efficient optimization-based quadrature for variational discretization of nonlocal problems, Comput. Methods Appl. Mech. Eng.,396, 115104, 2022. [A2]

  8. Q. Du, X. Tian, C. Wright and Y. Yu, Nonlocal trace spaces and extension results for nonlocal calculus, J. Funct. Anal., 282(12), 109453, 2022. [A1]

  9. Q. Du, T. Mengesha and X. Tian, Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localizations, Anal. Appl., 20(3), 579-614, 2022. [A1]

  10. M. D’Elia, X. Li, P. Seleson, X. Tian, and Y. Yu, A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics, J. Peridyn. Nonlocal Model., 4, 1–50, 2022. [A4]

  11. Y. Leng, X. Tian, L. Demkowicz, H. Gomez and J. Foster, A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems, J. Comput. Phys., 425, 110919, 2021. [A2]

  12. Y. Leng, X. Tian, N. Trask and J. Foster, Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion, SIAM J. Numer. Anal., 59(1), 88-118, 2021. [A2] [A3]

  13. L. Demkowicz, T. Führer, N. Heuer, and X. Tian, The Double Adaptivity Paradigm (How to circumvent the discrete inf-sup conditions of Babuška and Brezzi), Comput. Math. with Appl., 95, 41-66, 2021. [A2]

  14. M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian, and Z. Zhou, Numerical methods for nonlocal and fractional models, Acta Numerica, 29, 1-124, 2020. [A2]

  15. Y. Leng, X. Tian, N. Trask and J. Foster, Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation, Comput. Methods Appl. Mech. Eng., 370, 113264, 2020. [A2] [A3]

  16. M. D’Elia, X. Tian, and Y. Yu, A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints, SIAM J. Sci. Comput., 42(4), A1935-A1949, 2020. [A4]

  17. X. Tian and Q. Du, Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models, SIAM Rev., 62(1), 199–227, 2020 (SIAM Review SIGEST Award). [A2]

  18. Q. Du, B. Engquist and X. Tian, Multiscale Modeling, homogenization and nonlocal effects: mathematical and computational issues , Contemporary Mathematics, 754, 115-140, 75 Years of Mathematics of Computation, AMS, 2020. [A4]

  19. Q. Du, L. Ju, J. Lu and X. Tian A discontinuous Galerkin method with penalty for one-dimensional nonlocal diffusion problems, Comm. Appl. Math. Comput., 2(1):31-55, 2020. [A2]

  20. Q. Du and X. Tian, Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations, Foundations of Computational Mathematics, 20, 801-826, 2020. [A1] [A3] [A4]

  21. X. Tian and B. Engquist, Fast algorithm for computing nonlocal operators with finite interaction distance, Comm. Math. Sci., 17(6), 1653 – 1670, 2019. [A2]

  22. Y. Tao, X. Tian and Q. Du, Nonlocal models with heterogeneous localization and their application to seamless local-nonlocal coupling, Multiscale Modeling and Simulation, 17(3), 1052-1075, 2019. [A4]

  23. Y. Leng, X. Tian and J. Foster, Super-convergence of reproducing kernel approximation , Comput. Methods Appl. Mech. Eng., 352:488-507, 2019. [A3]

  24. Q. Du and X. Tian, Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling , SIAM J. Appl. Math., 78(3), 1536-1552 , 2018. [A1] [A4]

  25. Q. Du, Y. Tao, X. Tian and J. Yang, Asymptotically compatible numerical approximations of multidimensional nonlocal diffusion models and nonlocal Green's functions, IMA J. Numer. Anal., 39(2),607–625, 2018. [A2]

  26. Q. Du, X. H. Li, J. Lu, and X. Tian, A quasinonlocal coupling method for nonlocal and local diffusion models , SIAM J. Numer. Anal., 56 (3), 1386-1404, 2018. [A4]

  27. Q. Du, Y. Tao and X. Tian, A peridynamic model of fracture mechanics with bond-breaking , J. Elast.,132 (2), 197-218, 2018. [A4]

  28. X. Tian and Q. Du, Trace theorems for some nonlocal function spaces with heterogeneous localization , SIAM J. Math. Anal., 49(2):1621-1644, 2017. [A1]

  29. Y. Tao, X. Tian and Q. Du, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Appl. Math. Comput., 305:282-298, 2017. [A4]

  30. Q. Du, Y. Tao, X. Tian , and J. Yang, Robust a posteriori stress analysis for approximations of nonlocal models via nonlocal gradients, Comput. Methods Appl. Mech. Eng., 310:605-627, 2016. [A2]

  31. X. Tian and Q. Du, A class of high order nonlocal operators , Arch Rat. Mech. Anal., 222(3)1521-1553, 2016. [A1]

  32. X. Tian , Q. Du and M. Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains, Advances in Comp. Math., 42(6):1363-1380, 2016. [A2]

  33. X. Tian and Q. Du, Nonconforming discontinuous Galerkin methods for nonlocal varational problems, SIAM J. Numer. Anal., 53 (2015), 762-781. [A1] [A2]

  34. X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal. 52 (2014), 1641-1665. [A2]

  35. X. Tian and Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal. 51 (2013), 3458-3482. [A2]

Conference proceedings

  1. Q. Du and X. Tian, Heterogeneously localized nonlocal operators, boundary traces and variational problems, Proceedings of the International Congress of Chinese Mathematicians, International Press, 2018.

  2. Q. Du and X. Tian, Asymptotically compatible schemes for peridynamics based on numerical quadrature , Proc. ASME 2014.

  3. Q. Du and X. Tian, Robust discretization of nonlocal models related to peridynamics, Meshfree Methods for Partial Differential Equations VII, 97-113, 2014.