AI 中文总结
本文提出伪时间POD框架，将静态算子方程转为伪动态形式人工生成时间数据，证明其收敛性与基函数逼近性，经理论和数值验证可用于两类定常问题的模型降阶。
AI 中文摘要
本文提出一种新型伪时间本征正交分解（POD）框架，以实现缺乏时间快照数据的定常问题的模型降阶。通过将静态算子方程重铸为伪动态演化形式，我们在保留系统固有谱特性的同时人工生成时间数据。数学上，我们严格证明了伪时间轨迹指数收敛至精确定常解，还严格建立了生成的POD基函数的逼近特性。最后，在椭圆型逆源问题和第一类Fredholm积分方程这两类典型场景中，我们从理论和数值两方面验证了该框架的通用性与准确性。
英文摘要
This paper proposes a novel Pseudo-time Proper Orthogonal Decomposition (POD) framework to enable model reduction for stationary problems lacking temporal snapshot data. By recasting static operator equations into a pseudo-dynamic evolution form, we artificially generate temporal data while preserving the system's intrinsic spectral properties. Mathematically, we rigorously prove the exponential convergence of the pseudo-time trajectory to the exact stationary solution. Furthermore, the approximation properties of the generated POD basis functions are rigorously established. Finally, the universality and accuracy of the framework are validated both theoretically and numerically across two representative settings: elliptic inverse source problems and Fredholm integral equations of the first kind.
Comments25 pages
