论文标题
使用自适应减少订单模型的强大参数反演
Robust Parameter Inversion using Adaptive Reduced Order Models
论文作者
论文摘要
非线性参数逆问题出现在许多应用中，通常非常昂贵，尤其是在涉及许多测量的情况下。这些问题对评估目标函数或不合适的情况构成了巨大的计算挑战，需要解决大量参数化的偏微分方程，通常每个源术语一个。牛顿型算法可能是快速收敛所需的，通常需要大量伴随问题的其他解决方案。 使用参数模型的使用可能会大大减轻此问题。在[De Sturler，E.，Gugercin，S.，Kilmer，M。E.，Chaturantabut，S.，Beattie，C。和O'Connell，M。（2015年）。使用插值模型还原的非线性参数反演。 SIAM科学计算杂志，37（3）]，成功使用插值模型降低，以极大地加快弥漫性光学层析成像（DOT）的速度。但是，当在高维参数空间中使用模型降低时，在参数空间中获得误差界限通常是棘手的。在本文中，我们建议使用随机估计来解决此问题。以一个（随机）全尺度线性求的成本为每个优化步骤，我们获得了强大的算法。此外，由于我们现在可以在需要时更新模型，因此这种鲁棒性使我们能够进一步降低订购模型的顺序，从而进一步降低计算和使用成本，从而进一步降低反转成本。我们还提出了一种更新模型还原基础的方法，该方法将所需的大型线性求解数量减少了46％-98％，而不是固定的还原级模型。我们证明，这导致了一种高效且强大的反转方法。
Nonlinear parametric inverse problems appear in many applications and are typically very expensive to solve, especially if they involve many measurements. These problems pose huge computational challenges as evaluating the objective function or misfit requires the solution of a large number of parameterized partial differential equations, typically one per source term. Newton-type algorithms, which may be required for fast convergence, typically require the additional solution of a large number of adjoint problems. The use of parametric model reduction may substantially alleviate this problem. In [de Sturler, E., Gugercin, S., Kilmer, M. E., Chaturantabut, S., Beattie, C., and O'Connell, M. (2015). Nonlinear Parametric Inversion Using Interpolatory Model Reduction. SIAM Journal on Scientific Computing, 37(3)], interpolatory model reduction was successfully used to drastically speed up inversion for Diffuse Optical Tomography (DOT). However, when using model reduction in high dimensional parameter spaces, obtaining error bounds in parameter space is typically intractable. In this paper, we propose to use stochastic estimates to remedy this problem. At the cost of one (randomized) full-scale linear solve per optimization step we obtain a robust algorithm. Moreover, since we can now update the model when needed, this robustness allows us to further reduce the order of the reduced order model and hence the cost of computing and using it, further decreasing the cost of inversion. We also propose a method to update the model reduction basis that reduces the number of large linear solves required by 46%-98% compared with the fixed reduced-order model. We demonstrate that this leads to a highly efficient and robust inversion method.
