一类求解离散不适定问题的截断奇异值分解方法
A Truncated Singular Value Decomposition Method for Solving Discrete Ill-Posed Problems
摘要: 奇异值分解(SVD)是矩阵计算中的核心工具。在图像处理领域，SVD被广泛用于图像压缩、去噪和特征提取经典SVD算法在处理高分辨率图像或大规模科学仿真数据时，计算复杂度高，内存开销巨大，本文提出随机SVD能保持高精度，并适合处理大规模图像矩阵和高维科学观测数据。经典SVD适用于中小规模或高精度需求的任务，而随机SVD已成为大规模图像分析、科学计算与工程应用中高效获取低秩近似的首选方案之一，本文的随机SVD与Tikhonov正则化方法结合得到的随机SVD-Tikhonov正则化方法可处理大规模不适定问题，数值实例说明了该方法的有效性。
Abstract: Singular Value Decomposition (SVD) is a core tool in matrix computing. In the field of image processing, SVD is widely used in image compression, denoising and feature extraction. The classic SVD algorithm has high computational complexity and huge memory overhead when dealing with high-resolution images or large-scale scientific simulation data. This paper proposes that random SVD can maintain high precision and is suitable for processing large-scale image matrices and high-dimensional scientific observation data. Classic SVD is suitable for tasks of medium and small scale or with high precision requirements, while random SVD has become one of the preferred solutions for efficiently obtaining low-rank approximations in large-scale image analysis, scientific computing and engineering applications. The random SVD-Tikhonov regularization method obtained by combining the random SVD in this paper with the Tikhonov regularization method can handle large-scale undetermined problems. Numerical examples illustrate the effectiveness of this method.
文章引用：吴思静, 田淼, 王忆清. 一类求解离散不适定问题的截断奇异值分解方法[J]. 应用数学进展, 2026, 15(6): 47-54. https://doi.org/10.12677/aam.2026.156264

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