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1. Optimal Control, Numerics, and Applications of Fractional PDEs

   https://doi.org/10.1016/bs.hna.2021.12.003  

   Antil, H.; Brown T.; Khatri, R.; Onwunta, A.; Verma, D.; Warma, M. (February 2022, Handbook of numerical analysis)

   Trélat, E.; Zuazua, E. (Ed.)

   This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems. 

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2. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

   https://doi.org/10.3934/cpaa.2021174  

   Scott, James M.; Mengesha, Tadele (January 2021, Communications on Pure & Applied Analysis)

   We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

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3. Uniqueness in an inverse problem of fractional elasticity

   https://doi.org/10.1098/rspa.2023.0474  

   Covi, Giovanni; de_Hoop, Maarten; Salo, Mikko (October 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We study a nonlinear inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated with a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis, we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction to the case of fractional elasticity. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity. 

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4. Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

   https://doi.org/10.1515/cmam-2021-0118  

   Shi, Tianyi; Antil, Harbir; Kouri, Drew P. (February 2022, Computational Methods in Applied Mathematics)

   Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces.However, this nonlocality makes it challenging to design efficient solvers for such problems.In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains.We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs.In addition, we introduce both serial and parallel domain decomposition solvers.We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints. 

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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. 

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