Needlet algorithms for estimation in inverse problems
Needlet algorithms for estimation in inverse problems
复制标题
用于反问题估计的针算法
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
T. Willer
中科院分区：
文献类型：
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作者：
G. Kerkyacharian;P. Petrushev;D. Picard;T. Willer
我们提供了一种处理反演问题的新算法，该算法在精心选择的新基础上将传统的 SVD 反演与适当的阈值技术相结合。我们的目标是设计一种反演程序，它具有小波表示的局部化和多尺度分析的优点，同时又不损失 SVD 分解的稳定性和可计算性。为此，我们利用建立在 SVD 基础上的局部框架（称为“针”）的结构。我们考虑两种不同的情况：“小波”场景，其中假设针的行为与真实小波类似，以及“雅可比型”场景，其中我们假设框架的属性真正取决于手头的 SVD 基础（因此取决于运算符）。为了说明每种情况，我们分别将估计算法应用于反卷积问题和 Wicksell 问题。在后一种情况下，SVD 基础是雅可比多项式基础，我们表明我们的方案能够实现在 L2 情况下最优的收敛速度，我们获得了文献中新的（据我们所知）其他 Lp 范数的有趣的收敛速度，并且我们还进行了模拟研究，表明 NEED-D 估计器在几乎所有情况下都优于其他标准算法。 AMS 2000 科目分类：初级 62G05、62G20；次级 65J20。
We provide a new algorithm for the treatment of inverse prob- lems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and mul- tiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the con- struction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobi- type" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the de- convolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the L2 case, we obtain interesting rates of convergence for other Lp norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations. AMS 2000 subject classifications: Primary 62G05, 62G20; secondary 65J20.