Multilevel preconditioning and adaptive sparse solution of inverse problems
Multilevel preconditioning and adaptive sparse solution of inverse problems
复制标题
反问题的多级预处理和自适应稀疏解
DOI：
10.1090/s0025-5718-2011-02507-x
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
T. Raasch
中科院分区：
文献类型：
--
作者：
S. Dahlke;M. Fornasier;T. Raasch
我们关注涉及稀疏约束的希尔伯特空间中最小化问题的有效数值解。这些优化例如在逆问题的背景下出现。在这项工作中，我们分析了针对大维甚至无限维问题的著名迭代软收缩算法的有效变体。该算法按以下方式修改。我们没有规定固定的阈值参数，而是使用递减的阈值策略。此外，我们使用 Cohen、Dahmen 和 DeVore 导出的自适应方案的适当变体来逼近无限矩阵向量乘积。我们推导了一种块多尺度预处理技术，该技术允许对基础矩阵进行局部良好调节，并将受限等距属性的概念扩展到无限标记矩阵。这些成分的组合产生了一个数值方案，该方案保证以指数速率收敛，并且允许迭代支持大小的受控膨胀。我们还提出了数值实验，证实了我们的方法的适用性，该方法将概念从压缩感知扩展到大规模模拟。
We are concerned with the efficient numerical solution of minimization problems in Hilbert spaces involving sparsity constraints. These optimizations arise, e.g., in the context of inverse problems. In this work we analyze an efficient variant of the well-known iterative soft-shrinkage algorithm for large or even infinite dimensional problems. This algorithm is modified in the following way. Instead of prescribing a fixed thresholding parameter, we use a decreasing thresholding strategy. Moreover, we use suitable variants of the adaptive schemes derived by Cohen, Dahmen and DeVore for the approximation of the infinite matrix-vector products. We derive a block multiscale preconditioning technique which allows for local well-conditioning of the underlying matrices and for extending the concept of restricted isometry property to infinitely labelled matrices. The combination of these ingredients gives rise to a numerical scheme that is guaranteed to converge with exponential rate, and which allows for a controlled inflation of the support size of the iterations. We also present numerical experiments that confirm the applicability of our approach which extends concepts from compressed sensing to large scale simulation.
影响因子：
1.7
作者：
Thomas Bonesky;S. Dahlke;P. Maass;T. Raasch
通讯作者：
Thomas Bonesky;S. Dahlke;P. Maass;T. Raasch
影响因子：
2.1
作者：
R. Ramlau;Gerd Teschke;M. Zhariy
通讯作者：
R. Ramlau;Gerd Teschke;M. Zhariy