论文标题
统计有限元法的错误分析
Error analysis for a statistical finite element method
论文作者
论文摘要
最近提出的统计有限元（STATFEM）方法与有限元模型合成测量数据，并允许对未知的真实系统响应进行预测。在假设噪声测量数据是由确定性的真实系统响应函数生成的假设，我们为基于高斯过程的原型statFem设置提供了概率误差分析，该函数满足了未知的真实源术语的二阶椭圆形偏差方程。在某些情况下，高斯工艺模型诸如源术语的平滑度之类的属性可能会误解。我们得出的误差估计是针对$ l^2 $ norm的测量噪声的期望，而真实系统响应与STATFEM后部平均值之间的差异。这些估计意味着在测量点和有限元函数的数量中收敛的多项式速率，并取决于真实源项和高斯过程模型的Sobolev平滑度。泊松方程的数值示例用于说明这些理论结果。
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.
