More Like this

1. Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

   https://doi.org/10.1142/S0218202523500549  

   Han, Zhaolong; Tian, Xiaochuan (November 2023, Mathematical Models and Methods in Applied Sciences)

   This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. 

   more » « less
2. $$ L^{p} $$ compactness criteria with an application to variational convergence of some nonlocal energy functionals

   https://doi.org/10.3934/mine.2023097  

   Du, Qiang; Mengesha, Tadele; Tian, Xiaochuan (January 2023, Mathematics in Engineering)

   Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $$ L^{p} $$ vector fields defined on a domain $$ \Omega $$ that is either a bounded domain in $$ \mathbb{R}^{d} $$ or $$ \mathbb{R}^{d} $$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $$ L^{p} $$ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. 

   more » « less
3. Experience replay is associated with efficient nonlocal learning

   https://doi.org/10.1126/science.abf1357  

   Liu, Yunzhe; Mattar, Marcelo G.; Behrens, Timothy E. J.; Daw, Nathaniel D.; Dolan, Raymond J. (May 2021, Science)

   To make effective decisions, people need to consider the relationship between actions and outcomes. These are often separated by time and space. The neural mechanisms by which disjoint actions and outcomes are linked remain unknown. One promising hypothesis involves neural replay of nonlocal experience. Using a task that segregates direct from indirect value learning, combined with magnetoencephalography, we examined the role of neural replay in human nonlocal learning. After receipt of a reward, we found significant backward replay of nonlocal experience, with a 160-millisecond state-to-state time lag, which was linked to efficient learning of action values. Backward replay and behavioral evidence of nonlocal learning were more pronounced for experiences of greater benefit for future behavior. These findings support nonlocal replay as a neural mechanism for solving complex credit assignment problems during learning. 

   more » « less
4. Nonlocal Wasserstein distance: metric and asymptotic properties

   https://doi.org/10.1007/s00526-023-02576-6  

   Slepčev, Dejan; Warren, Andrew (September 2023, Calculus of Variations and Partial Differential Equations)

   Abstract The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. Here we consider a fundamental modification of the framework where the paths are solutions of nonlocal (jump) continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We characterize the basic properties of the distance and obtain sharp conditions on the (jump) kernel specifying the nonlocal transport that determine whether the topology metrized is the weak or the strong topology. A key result of the paper are the quantitative comparisons between the nonlocal and local Wasserstein distance. 

   more » « less
5. Compactness results for a Dirichlet energy of nonlocal gradient with applications

   https://doi.org/10.1002/num.23149  

   Han, Zhaolong; Mengesha, Tadele; Tian, Xiaochuan (November 2024, Numerical Methods for Partial Differential Equations)

   Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

   more » « less