统计有限元方法的误差分析
Error analysis for a statistical finite element method
AI总结：
针对统计有限元（statFEM）方法的误差问题，基于高斯过程先验开展概率误差分析，推导了依赖源项与模型光滑性的多项式收敛速率估计，并通过泊松方程数值算例验证理论结果。
摘要：
近期提出的统计有限元（statFEM）方法将测量数据与有限元模型相结合，可对未知的真实系统响应进行预测。我们针对基于高斯过程先验的典型statFEM架构开展了概率误差分析，假设含噪测量数据由确定性真实系统响应函数生成，该函数满足对应于未知真实源项的二阶椭圆偏微分方程。在某些情况下，源项的光滑性等属性可能会被高斯过程模型错误设定。我们推导的误差估计针对真实系统响应与statFEM后验均值之差的$L^2$范数关于测量噪声的期望。这些估计表明，误差随测量点数量和有限元基函数数量呈多项式速率收敛，且依赖于真实源项和高斯过程模型的Sobolev光滑性。我们通过泊松方程的数值算例验证了这些理论结果。
英文摘要：
The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the unknown true system response. We provide a probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior under the assumption that the noisy measurement data are generated by a deterministic true system response function that satisfies a second-order elliptic partial differential equation for an unknown true source term. In certain cases, properties such as the smoothness of the source term may be misspecified by the Gaussian process model. The error estimates we derive are for the expectation with respect to the measurement noise of the $L^2$-norm of the difference between the true system response and the mean of the statFEM posterior. The estimates imply polynomial rates of convergence in the numbers of measurement points and finite element basis functions and depend on the Sobolev smoothness of the true source term and the Gaussian process model. A numerical example for Poisson's equation is used to illustrate these theoretical results.
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