Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in an earlier published version of this paper meet such objectives for a class of parametrized problems. An extended version of the abstract mathematical framework is established rigorously here, together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal models of diffusion and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical discretizations to problems involving nonlocal interactions on multiple scales. This work provides new insight in this regard through a careful analysis of related conforming finite element discretizations, and the finding is valid under minimal regularity assumptions on exact solutions. It reveals that for the nonlocal models under consideration and their local limit, as long as the finite element space contains continuous piecewise linear functions, the Galerkin finite element approximation is always asymptotically compatible. For piecewise constant finite elements, whenever applicable, it is shown that a correct local limit solution can also be obtained as long as the discretization (mesh) parameter decreases faster than the modeling (horizon) parameter does. These results can be used to guide future computational studies of nonlocal problems. Some other applications, such as the fractional PDE limit of nonlocal models, and open questions are also presented

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the l2 operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical Bakry–Émery curvature criterion implies subgaussian concentration for “matrix Lipschitz” functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron–Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.

The sparsity-based approaches have demonstrated promising performance in image processing. In this paper, for better preservation of the salient edge structures of images, we propose an l_0 + l_2-norm based analysis model, which requires solving a challenging non-separable l0-norm related minimiza- tion problem, and we also propose an inexact augmented Lagrangian method with proven convergence to a local minimum. Extensive experiments in image smoothing, including texture removal and context smoothing, show that our method achieves better visual results over various sparsity-based models and the CNN method. Also, experiments on sparse view CT reconstruction further validate the advantage of the proposed method.

This paper is devoted to the theoretical and numerical investigation of the local minimum problem in an inverse boundary value problem. We provide a mathematical analysis for the objective function of the optimal transportation type in the context of seismic full waveform inversion. In particular, we prove that the gradient obtained using the adjoint-state method does not depend on the specific choice of the Kantorovich potentials. Moreover, our frequency analysis results show a decreasing sensitivity of the reconstruction as the data misfit is concentrated in the high-frequency part. This confirms previous observations in many numerical experiments. We also propose a new method using the softplus encoding, which maps the seismic data into probability measures and, therefore, the Wasserstein metric can be applied. The softplus encoding retains the convexity of the data misfit with respect to translations and provides a parameter to tune the landscape of the objective function. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to bound- edness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of ε-Fano type. Furthermore, we show birational boundedness of 3-folds of ε-Fano type.