用于半线性发展方程的 POD 降阶模型的参数敏感性及其在 G 方程中的应用
Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations
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中文总结 AI 辅助
研究针对含粘性 G 方程等的参数化半线性发展方程的 POD 降阶模型，基于非线性算子条件及连续性模进行敏感性分析，证明参考参数处 POD 基用于附近参数时误差受相应参数模控制，数值实验验证了该结论及基重用的稳健性。
中文摘要 AI 辅助
本研究聚焦于参数化问题中 POD 降阶模型的参数敏感性。恰当正交分解（POD）能从解快照构建演化方程的低维替代模型，但在参数化问题里，某一参数值下计算的基在其他参数值处未必准确，且为多查询研究中每个参数重新计算基成本高昂。我们针对一类参数化半线性发展方程的 POD 降阶模型开展敏感性分析，该类方程包括粘性 G 方程及其无粘形式的粘性应变 G 方程。分析基于非线性算子的跨空间 Lipschitz 条件及量化双线性形式和非线性的参数依赖性的连续性模。证明了在参考参数处构建的 POD 基应用于附近查询参数时，误差变化受相应参数模控制，且常数与 POD 维度、时间步数和扰动幅度无关。数值实验与预测的模依赖敏感性行为一致，说明了基对附近参数重用的实际稳健性。
英文摘要
Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.
