标题： 自适应双变量小波树逼近通过各向异性张量小波树 显示英文标题

标题： Adaptive Bivariate Quarklet Tree Approximation via Anisotropic Tensor Quarklets

摘要： 本文研究了利用准小波张量框架的元素来实现给定二元函数的近似最佳逼近。 为此，我们应用了Dahlke、Keding和Raasch于2017年左右引入的一元B样条准小波的各向异性张量积。 我们介绍了二元准小波树的概念，并开发了一种自适应算法，该算法允许通过选择的框架元素对给定的二元函数进行广义hp-逼近。 证明了此算法是近似最佳的，即只要满足一些关于局部误差的标准条件，它所提供的逼近误差就接近于最佳可能的准小波树逼近误差。 对这种算法的复杂性进行了研究。 此外，我们使用我们的技术以逆指数收敛率逼近一个二元测试函数。 可以预期，本文所呈现的结果将成为设计用于解决具有非常良好收敛特性的二元PDEs的自适应小波hp方法的重要组成部分。 显示英文摘要

摘要： This paper deals with near-best approximation of a given bivariate function using elements of quarkonial tensor frames. For that purpose we apply anisotropic tensor products of the univariate B-spline quarklets introduced around 2017 by Dahlke, Keding and Raasch. We introduce the concept of bivariate quarklet trees and develop an adaptive algorithm which allows for generalized hp-approximation of a given bivariate function by selected frame elements. It is proved that this algorithm is near-best, which means that as long as some standard conditions concerning local errors are fulfilled it provides an approximation with an error close to that one of the best possible quarklet tree approximation. For this algorithm the complexity is investigated. Moreover, we use our techniques to approximate a bivariate test function with inverse-exponential rates of convergence. It can be expected that the results presented in this paper serve as important building block for the design of adaptive wavelet-hp-methods for solving PDEs in the bivariate setting with very good convergence properties.