## Preprints

- Q. Ye, X. Tian, and D. Wang,
*A fast and accurate solver for the fractional Fokker-Planck equation with Dirac-delta initial conditions*, preprint. - Q. Du, J. Scott and X. Tian,
*Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems*, preprint. - X. Tian, Y. Wu and Y. Zhang,
*Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis*, preprint. - 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

- Z. Han, T. Mengesha and X. Tian,
*Compactness results for a Dirichlet energy of nonlocal gradient with applications*, Numer. Methods Partial Differ. Equ., 40, e23149, 2024.*Dedicated to Professor Qiang Du on the occasion of his 60th birthday.* - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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). - 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. - 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. - Q. Du and X. Tian,
*Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations*, Found. Comput. Math., 20, 801-826, 2020. - X. Tian and B. Engquist,
*Fast algorithm for computing nonlocal operators with finite interaction distance*, Comm. Math. Sci., 17(6), 1653 – 1670, 2019. - 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. - Y. Leng, X. Tian and J. Foster,
*Super-convergence of reproducing kernel approximation*, Comput. Methods Appl. Mech. Eng., 352:488-507, 2019. - 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. - 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. - 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. - Q. Du, Y. Tao and X. Tian,
*A peridynamic model of fracture mechanics with bond-breaking*, J. Elast.,132 (2), 197-218, 2018. - X. Tian and Q. Du,
*Trace theorems for some nonlocal function spaces with heterogeneous localization*, SIAM J. Math. Anal., 49(2):1621-1644, 2017. - 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. - 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. - X. Tian and Q. Du,
*A class of high order nonlocal operators*, Arch Rat. Mech. Anal., 222(3)1521-1553, 2016. - 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. - X. Tian and Q. Du,
*Nonconforming discontinuous Galerkin methods for nonlocal varational problems*, SIAM J. Numer. Anal., 53 (2015), 762-781. - X. Tian and Q. Du,
*Asymptotically compatible schemes and applications to robust discretization of nonlocal models*, SIAM J. Numer. Anal. 52 (2014), 1641-1665. - 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.

## Conference proceedings

- 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. - Q. Du and X. Tian,
*Asymptotically compatible schemes for peridynamics based on numerical quadrature*, Proc. ASME 2014. - Q. Du and X. Tian,
*Robust discretization of nonlocal models related to peridynamics*, Meshfree Methods for Partial Differential Equations VII, 97-113, 2014.