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Applications such as unbalanced and fully shuffled regression can be approached by optimizing regularized optimal transport (OT) distances, including the entropic OT and Sinkhorn distances. A common approach for this optimization is to use a first-order optimizer, which requires the gradient of the OT distance. For faster convergence, one might also resort to a second-order optimizer, which additionally requires the Hessian. The computations of these derivatives are crucial for efficient and accurate optimization. However, they present significant challenges in terms of memory consumption and numerical instability, especially for large datasets and small regularization strengths. We circumvent these issues by analytically computing the gradients for OT distances and the Hessian for the entropic OT distance, which was not previously used due to intricate tensorwise calculations and the complex dependency on parameters within the bi-level loss function. Through analytical derivation and spectral analysis, we identify and resolve the numerical instability caused by the singularity and ill-posedness of a key linear system. Consequently, we achieve scalable and stable computation of the Hessian, enabling the implementation of the stochastic gradient descent (SGD)-Newton methods. Tests on shuffled regression examples demonstrate that the second stage of the SGD-Newton method converges orders of magnitude faster than the gradient descent-only method while achieving significantly more accurate parameter estimations. 

Inspired by the blending method developed by [P. Seleson, S. Beneddine, and S. Prudhome, A Force-Based Coupling Scheme for Peridynamics and Classical Elasticity, (2013)] for the nonlocal-to-local coupling, we create a symmetric and consistent blended force-based atomistic-to-continuum (a/c) scheme for the atomistic chain in one-dimensional space. The conditions for the well-posedness of the underlying model are established by analyzing an optimal blending size and blending type to ensure the H1{\$$}{\$$}H^1{\$$}{\$$}semi-norm stability for the blended force-based operator. We present several numerical experiments to test and confirm the theoretical findings.