SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 635-658, January 2021.
Prony's method is a standard tool exploited for solving many imaging and data analysis problems that result in parameter identification in sparse exponential sums $f(k)=\sum_{j=1}^{M}c_{j}e^{-2\pi i\langle t_{j},k\rangle}$, $k\in \mathbb{Z}^{d}$, where the parameters are pairwise different $\{ t_{j}\}_{j=1}^{M}\subset [0,1)^{d}$, and $\{ c_{j}\}_{j=1}^{M}\subset \mathbb{C}\setminus \{ 0\}$ are nonzero. The focus of our investigation is on a Prony's method variant based on a multivariate matrix pencil approach. The method constructs matrices $S_{1}$, łdots , $S_{d}$ from the sampling values, and their simultaneous diagonalization yields the parameters $\{ t_{j}\}_{j=1}^{M}$. The parameters $\{ c_{j}\}_{j=1}^{M}$ are computed as the solution of an linear least squares problem, where the matrix of the problem is determined by $\{ t_{j}\}_{j=1}^{M}$. Since the method involves independent generation and manipulation of a certain number of matrices, there is an intrinsic capacity for parallelization of the whole computational process on several levels. Hence, we propose a parallel version of the Prony's method in order to increase its efficiency. The tasks concerning the generation of matrices are divided among the block of threads of the graphics processing unit (GPU) and the central processing unit (CPU), where heavier load is put on the GPU. From the algorithmic point of view, the CPU is dedicated to the more complex tasks of computing the singular value decomposition, the eigendecomposition, and the solution of the least squares problem, while the GPU is performing matrix--matrix multiplications and summations. With careful choice of algorithms solving the subtasks, the load between CPU and GPU is balanced. Besides the parallelization techniques, we are also concerned with some numerical issues, and we provide detailed numerical analysis of the method in case of noisy input data. Finally, we performed a set of numerical tests which confirm superior efficiency of the parallel algorithm and consistency of the forward error with the results of numerical analysis.

中文翻译：

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具有多元矩阵铅笔法的平行 Prony 方法及其数值方面

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 635-658 页，2021 年 1 月。
Prony 方法是一种标准工​​具，用于解决许多成像和数据分析问题，这些问题导致稀疏指数和 $f(k)=\sum_{j=1}^{M}c_{j}e^{-2 中的参数识别\pi i\langle t_{j},k\rangle}$, $k\in \mathbb{Z}^{d}$，其中参数成对不同 $\{ t_{j}\}_{j= 1}^{M}\subset [0,1)^{d}$, 和 $\{ c_{j}\}_{j=1}^{M}\subset \mathbb{C}\setminus \{ 0\}$ 非零。我们调查的重点是基于多变量矩阵铅笔方法的 Prony 方法变体。该方法从采样值构造矩阵 $S_{1}$, łdots , $S_{d}$，并且它们的同时对角化产生参数 $\{ t_{j}\}_{j=1}^{M} $. 参数 $\{ c_{j}\}_{j=1}^{M}$ 计算为线性最小二乘问题的解，其中问题的矩阵由 $\{ t_{j}\}_{j=1}^{M}$ 决定。由于该方法涉及一定数量矩阵的独立生成和操作，因此具有在多个层次上并行化整个计算过程的内在能力。因此，我们提出了 Prony 方法的并行版本，以提高其效率。与矩阵生成有关的任务在图形处理单元 (GPU) 和中央处理单元 (CPU) 的线程块之间分配，其中 GPU 的负载较重。从算法的角度来看，CPU 专注于计算奇异值分解、特征分解和最小二乘问题的求解等更复杂的任务，而 GPU 则执行矩阵——矩阵乘法和求和。通过仔细选择解决子任务的算法，CPU 和 GPU 之间的负载是平衡的。除了并行化技术，我们还关注一些数值问题，并且在输入数据有噪声的情况下，我们提供了该方法的详细数值分析。最后，我们进行了一组数值测试，证实了并行算法的优越效率和前向误差与数值分析结果的一致性。