Related papers: Anisotropic smoothness classes : from finite eleme…

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given…

This article is devoted to nonlinear approximation and estimation via piecewise polynomials built on partitions into dyadic rectangles. The approximation rate is studied over possibly inhomogeneous and anisotropic smoothness classes that…

This article presents a multiscale, non-linear and directional statistical characterization of images based on the estimation of the skewness, flatness, entropy and distance from Gaussianity of the spatial increments. These increments are…

Mesh adaption procedures for finite element approximation allows one to adapt the resolution, by local refinement in the regions of strong variation of the function of interest. This procedure plays a key role in numerous applications of…

We refine a recently-proposed class of local entropic loss functions by restricting the smoothening regularization to only a subset of weights. The new loss functions are referred to as partial local entropies. They can adapt to the…

Mesh adaptation for finite element approximation is a procedure used in numerous applications. The use of thin and long anisotropic triangles improves the efficiency of the procedure. When piecewise linear finite elements are used, the…

The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients where the rate of decay depends on the smoothness of the function and vice-versa. The optimal number of degrees of freedom in the…

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several…

In previous works, we proposed a method to characterize jointly self-similarity and anisotropy properties of a large class of self--similar Gaussian random fields. We provide here a mathematical analysis of our approach, proving that the…

It has recently been proved that the popular nonlocal means (NLM) denoising algorithm does not optimally denoise images with sharp edges. Its weakness lies in the isotropic nature of the neighborhoods it uses to set its smoothing weights.…

A method for adaptive model order reduction for nonsmooth discrete element simulation is developed and analysed in numerical experiments. Regions of the granular media that collectively move as rigid bodies are substituted with rigid bodies…

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics,…

Generalized ellipsometry, a non-destructive optical characterization technique, is employed to determine geometrical structure parameters and anisotropic dielectric properties of highly spatially coherent three-dimensionally nanostructured…

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily…

This paper is concerned with the study of a class of nonsmooth cost functions subject to a quasi-linear PDE in Lipschitz domains in dimension two. We derive the Eulerian semi-derivative of the cost function by employing the averaged adjoint…

In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We prove the convergence and complexity of…

Anisotropic mesh quality measures and anisotropic mesh adaptation are studied for polygonal meshes. Three sets of alignment and equidistribution measures are developed, one based on least squares fitting, one based on generalized…

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spacial moduli of smoothness and their properties. In particular, we…

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on…

We consider a class of nonlocal conservation laws with an interaction kernel supported on the negative real half-line and featuring a decreasing jump at the origin. We provide, for the first time, an existence and uniqueness theory for said…