论文标题
非线性近似空间的反问题
Nonlinear approximation spaces for inverse problems
论文作者
论文摘要
本文涉及从有限的许多可能受噪声影响的测量中恢复未知函数u的无处不在的逆问题。近年来，基于线性近似空间的反转方法在[MPPY15，BCDDPW17]中引入了经认证的恢复边界。但是，众所周知，线性空间无法近似于简单且相关的函数系列，例如通常发生在双曲PDE（冲击）或图像（边缘）中的分段平滑函数。对于此类家庭，已知非线性空间[Devore98]可显着提高近似性能。本文的第一个贡献是为基于非线性近似空间的反转程序提供认证的恢复界。第二个贡献是将该框架应用于从细胞平均数据中恢复一般的二维形状。我们还讨论了我们的结果在N期限近似中的应用如何与压缩传感的经典结果相关。
This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were introduced in [MPPY15, BCDDPW17] with certified recovery bounds. It is however known that linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions that typically occur in hyperbolic PDEs (shocks) or images (edges). For such families, nonlinear spaces [Devore98] are known to significantly improve the approximation performance. The first contribution of this paper is to provide with certified recovery bounds for inversion procedures based on nonlinear approximation spaces. The second contribution is the application of this framework to the recovery of general bidimensional shapes from cell-average data. We also discuss how the application of our results to n-term approximation relates to classical results in compressed sensing.
