AI 中文总结
研究希尔伯特空间中演化方程基于残差最小化的时间离散参数逼近，考虑两种残差公式并进行统一误差分析，应用于高斯逼近流形，通过数值实验验证理论收敛速率及残差精度对守恒性质的影响。
AI 中文摘要
我们研究了基于残差最小化的希尔伯特空间中演化方程的时间离散参数逼近。解由属于低维非线性流形的参数化假设表示，并且通过在每个步骤中最小化适当定义的残差来执行时间步长。考虑了两种自然的残差公式：演化方程的离散化后参数化，以及控制参数动力学的狄拉克 - 弗伦克尔变分原理的离散化。在ζ方法族中为这两种方法开发了统一误差分析。所得界限将时间离散化的影响与残差最小化的影响分开，并在利普希茨、单边利普希茨和耗散性假设下产生一阶和二阶收敛。对于变分公式，涉及参数化映射条件的额外稳定性条件自然出现。该框架应用于高斯逼近流形，当涉及多项式算子时，残差范数和梯度允许显式封闭形式表达式。这使得无需空间离散化即可有效实现。含时薛定谔方程的数值实验说明了理论收敛速率以及残差精度对守恒性质的影响。
英文摘要
We study time-discrete parametric approximations of evolution equations in Hilbert spaces based on residual minimization. The solution is represented by a parametrized ansatz belonging to a low-dimensional nonlinear manifold, and time stepping is performed by minimizing suitably defined residuals at each step. Two natural residual formulations are considered: discretization followed by parametrization of the evolution equation, and discretization of the Dirac--Frenkel variational principle governing the parameter dynamics. A unified error analysis is developed for both approaches within the family of $ζ$-methods. The resulting bounds separate the effects of time discretization from those of residual minimization and yield first- and second-order convergence under Lipschitz, one-sided Lipschitz, and dissipativity assumptions. For the variational formulation, additional stability conditions involving the conditioning of the parametrization map arise naturally. The framework is applied to Gaussian approximation manifolds, for which residual norms and gradients admit explicit closed-form expressions when polynomial operators are involved. This enables efficient implementation without spatial discretization. Numerical experiments for time-dependent Schrödinger equations illustrate the theoretical convergence rates and the influence of residual accuracy on conservation properties.
Comments31 pages, 11 figures
