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.

翻译：本文关注从可能受到噪音影响的有限许多测量中恢复一个未知函数U这一普遍存在的反常问题。近年来,在[MPPY15、BCDDPW17]中引入了基于线性近似空间的反向方法,并附有经认证的回收界限。然而,众所周知,线性空间对于类似简单和相关功能组合,例如通常在双曲PDEs(冲击)或图像(屏障)中出现的片状平滑函数,变得无效。对于这些家庭,已知非线性空间[DEvore98]可大大改善近似性能。本文的第一项贡献是为非线性近似空间的反向程序提供经认证的回收界限。第二个贡献是将这一框架应用于从单元格平均数据中回收一般的二维形状。我们还讨论了我们的结果对正弦值的应用如何与压缩感测的经典结果相关。