一种用于扩散方程系数反演的应用降维方法
A dimension reduction method with applications for coefficient inversion of diffusion equations
AI总结：
本文提出一种降维方法，结合POD和RBF，高效求解反问题，通过梯度正则化和泛化交叉验证实现扩散方程系数反演的准确重建。
摘要：
本文提出一种降维方法，用于降低参数空间和状态空间的维度，以高效求解反问题。通过将POD和RBF结合，以变量分离的形式表示正向模型的解。该POD-RBF方法可高效评估模型输出。梯度正则化方法用于求解反问题并实现快速收敛。泛化交叉验证方法用于选择正则化参数和梯度计算的微分步长。由于正则化方法需要许多模型评估，这适合POD-RBF方法。因此，将POD-RBF方法与梯度正则化方法结合，提供了一种高效求解反问题的方法。我们专注于使用所提出的方法解决扩散方程的系数反演问题。基于不同的测量数据和不同的基函数，我们展示了几个数值例子。数值结果表明，可以高效地实现系数的准确重建。
英文摘要：
In this paper, we present a dimension reduction method to reduce the dimension of parameter space and state space and efficiently solve inverse problems. To this end, proper orthogonal decomposition (POD) and radial basis function (RBF) are combined to represent the solution of forward model with a form of variable separation. This POD-RBF method can be used to efficiently evaluate the model's output. A gradient regularization method is presented to solve the inverse problem with fast convergence. A generalized cross validation method is suggested to select the regularization parameter and differential step size for the gradient computation. Because the regularization method needs many model's evaluations. This is desirable for POD-RBF method. Thus, the POD-RBF method is integrated with the gradient regularization method to provide an efficient approach to solve inverse problems. We focus on the coefficient inversion of diffusion equations using the proposed approach. Based on different types of measurement data and different basis functions for coefficients, we present a few numerical examples for the coefficient inversion. The numerical results show that accurate reconstruction for the coefficient can be achieved efficiently.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
