A model reduction approach to numerical inversion for a parabolic partial differential equation
A model reduction approach to numerical inversion for a parabolic partial differential equation
复制标题
抛物型偏微分方程数值反演的模型简化方法
DOI：
10.1088/0266-5611/30/12/125011
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发表时间：
2012
期刊：
影响因子：
2.1
通讯作者：
M. Zaslavsky
中科院分区：
文献类型：
--
作者：
L. Borcea;V. Druskin;A. Mamonov;M. Zaslavsky
提出了一种基于模型降阶的抛物型偏微分方程系数数值反演新算法。该研究的动机是受控源电磁勘探的应用，其中未知的是地下电阻率和数据是时间分辨的磁场表面测量。本文提出的算法考虑在一维和二维反演。在频率（拉普拉斯）域中采用有理插值和有理Krylov子空间投影方法得到简化模型。它相当于从未知电阻率的函数空间到简化模型的参数的小维空间的非线性映射。我们使用这个映射作为一个非线性预条件的高斯-牛顿迭代解的反问题。反演算法的优点是双重的。首先，非线性预条件子解决了问题的大部分非线性。因此，迭代不太可能陷入局部极小值，收敛速度很快。第二，反演是计算高效的，因为它避免了时域响应的重复精确模拟。我们研究了各种合理的Krylov子空间的反演算法的稳定性，并评估其性能与数值实验。
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss–Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.