Efficient reconstruction of functions using eigenfunctions of linear operators

使用线性算子的特征函数高效重构函数

• 批准号: 244772350
• 负责人: Professorin Dr. Gerlind Plonka-Hoch
• 金额: --
• 依托单位: Institut für Numerische und Angewandte Mathematik (NAM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/244772350?language=en
• 关键词: Efficient; reconstruction; functions; using; eigenfunctions

In this project, we aim to unify and to generalize nonlinear reconstruction methods that go back to the Prony method. For this purpose, we will use the new insight that both the classical Prony method with a lot of applications in system identification and the Ben-Or and Tiwari algorithm for multivariate M-term polynomial interpolation can be understood as reconstruction methods for M-term expansions of eigenfunctions of special linear operators. This new perspective allows us to derive new generalized reconstruction algorithms for structured functions that can be represented for example as M-term expansions of trigonometric functions, orthogonal polynomials or Bessel functions. Thereby the generalized Prony methods will be applicable in a significantly larger range of applications e.g. in signal processing as well as for approximation of solutions of partial differential equations.The project especially focusses on the construction and analysis of efficient and numerically stable reconstruction algorithms and on deriving new error estimates for the reconstructed parameters in case of noisy input data. We want to apply the generalized Prony method to nonlinear approximation of Green's functions and to the reconstruction of structured functions from Fourier data in Magnetic Resonance Imaging.

在这个项目中，我们的目标是统一和推广非线性重建方法，可以追溯到Prony方法。为此，我们将使用新的见解，即经典的Prony方法与许多应用在系统识别和Ben-Or和Tiwari算法的多元M-项多项式插值可以理解为重建方法的M-项扩展的特征函数的特殊线性算子。这种新的视角使我们能够得到新的广义重建算法的结构化功能，可以表示为例如三角函数，正交多项式或贝塞尔函数的M-项扩展。因此，广义Prony方法将适用于一个显着更大的应用范围，例如在信号处理以及近似的偏微分方程的解决方案，该项目特别侧重于建设和分析的有效和数值稳定的重建算法和推导新的误差估计的情况下，噪声输入数据的重建参数。我们希望将广义Prony方法应用于绿色函数的非线性逼近和磁共振成像中傅立叶数据的结构函数重建。

项目成果

A sparse fast Fourier algorithm for real non-negative vectors

实数非负向量的稀疏快速傅立叶算法

• DOI: 10.1016/j.cam.2017.03.019
• 发表时间: 2017
• 期刊: J. Comput. Appl. Math.
• 作者: Gerlind Plonka;Katrin Wannenwetsch
• 通讯作者: Katrin Wannenwetsch

A deterministic sparse FFT algorithm for vectors with small support

• DOI: 10.1007/s11075-015-0028-0
• 发表时间: 2015-04
• 期刊: Numerical Algorithms
• 影响因子: 2.1
• 作者: G. Plonka-Hoch;Katrin Wannenwetsch

Prony methods for recovery of structured functions

• DOI: 10.1002/gamm.201410011
• 发表时间: 2014-11
• 期刊: GAMM‐Mitteilungen
• 作者: G. Plonka;M. Tasche

The Generalized Operator Based Prony Method

• DOI: 10.1007/s00365-020-09501-6
• 发表时间: 2019-01
• 期刊: Constructive Approximation
• 影响因子: 2.7
• 作者: Kilian Stampfer;G. Plonka

Deterministic sparse FFT for M-sparse vectors

• DOI: 10.1007/s11075-017-0370-5
• 发表时间: 2017-07
• 期刊: Numerical Algorithms
• 影响因子: 2.1
• 作者: G. Plonka-Hoch;Katrin Wannenwetsch;A. Cuyt;Wen-shin Lee