Topics in nonlinear approximation

非线性方法是近年来逼近理论和数值分析中的一个热点问题。这主要是由于非线性数值方法比线性方法表现得更好，并且非线性流形的函数近似阶比线性空间的元素近似阶好得多。提议者将继续他的调查各种非线性逼近流形（包括在单变量和多变量设置）的近似空间。特别是，新的自适应算法将进行研究和修改，以产生数值上更令人满意的结果。此外，提议人将继续他的研究在各种领域的约束非线性逼近进一步发展这一领域有许多应用在曲线插值和逼近，计算机辅助几何设计（CAGD），计算机辅助制造（CAM）等。

Nonlinear methods in Approximation Theory and Numerical Analysis received much attention in recent years. This is mostly due to the fact that nonlinear numerical methods perform much better than linear ones, and that the order of approximation of functions from nonlinear manifolds is much better than the order of approximation by the elements of linear spaces. The proposer will continue his investigation of approximation spaces for various nonlinear approximation manifolds (both in univariate and multivariate settings). In particular, new adaptive algorithms will be investigated and modified to yield numerically more satisfactory results. Also, the proposer will continue his research in various areas of constrained nonlinear approximation further developing this field having numerous applications in curve interpolation and approximation, computer-aided geometric design (CAGD), computer-aided manufacturing (CAM), etc.