用于可分重建问题的多尺度矩阵束
Multiscale matrix pencils for separable reconstruction problems
AI总结：
针对指数数据拟合等可分非线性逆问题，将矩阵束法的膨胀平移范式推广到周期、多项式等多类函数，构建对应结构化矩阵束以通过特征值和线性系统求解参数，并给出方法示例。
摘要：
指数数据拟合这一非线性逆问题具有可分性，因为拟合函数是参数化指数函数的线性组合，因此可将线性系数与非线性参数分开求解。矩阵束法将问题表述转化为求解非线性参数的广义特征值问题，以及求解线性参数的结构化线性系统，通常被认为是计算上求解该问题更稳定的方法。第2节总结了与经典复指数拟合或稀疏插值问题相关的矩阵束，并引入了膨胀和平移的概念，以得到不同尺度下的矩阵束。\n 指数分析此前已被推广到多种多项式基函数和一些算子特征函数的应用中，但在大多数推广形式中，缺乏基于特征值问题的计算方案。在随后的第3-6节中，包含膨胀和平移范式的矩阵束公式被推广到更多函数类。这些周期函数、多项式函数或特殊函数类中的每一类都需要定制化的方法，即最优利用所研究的稀疏插值问题中使用的参数化初等函数或特殊函数的性质。每一种推广都对应一个结构化线性矩阵束，可直接得到非线性参数和线性参数的计算方案，分别通过广义特征值问题和一个或多个结构化线性系统求解。\n 最后，在第7节中我们对新方法进行了举例说明。
英文摘要：
The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3--6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods.
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