Randomized Approach to Nonlinear Inversion Combining Simultaneous Random and Optimized Sources and Detectors
Randomized Approach to Nonlinear Inversion Combining Simultaneous Random and Optimized Sources and Detectors
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结合同时随机和优化源和探测器的非线性反演随机方法
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发表时间：
2017
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通讯作者：
M. Kilmer
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作者：
Selin S. Aslan;E. D. Sturler;M. Kilmer
在基于偏微分方程的多量测反问题中，每次求解目标函数都需要求解多个大型线性方程组。在非线性的情况下，雅可比矩阵的每一个评估都需要求解一组额外的系统。这导致了巨大的计算成本，这是迄今为止这些问题的主要成本。
几位作者已经提出通过利用随机技术来大幅减少系统解的数量[Haber等人，SIAM Optim.，22：739-757]并将该问题作为随机优化问题[Shapiro等人，随机规划讲座，SIAM，2009]。在这种方法中，目标函数的估计只使用几个随机线性组合的来源，称为同步随机源。对于雅可比矩阵，我们表明，一个类似的方法可以用来减少额外的伴随解决检测器的数量。
虽然其他人已经报告了良好的解决方案的质量，大大降低了计算成本，使用这些随机化的方法，我们的问题的兴趣，漫射光学断层扫描，该方法往往不会导致足够准确的解决方案。因此，我们取代了一些随机的同时源和检测器的同时源和检测器进行优化，以最大限度地提高采样雅可比矩阵的Frobenius范数后解决一个适度的公差。该选择受到以下启发：（1）正则化模型问题在TREGS非线性最小二乘求解器中求解[de Sturler和Kilmer，SIAM Sci.计算：33：3057-3086]，以及（2）这些优化方向对应于信息量最大的数据分量的事实。
我们的方法导致使用所有的源和探测器，但在一个大大降低了计算成本获得相同质量的解决方案。
In partial differential equations-based inverse problems with many measurements, we have to solve many large linear system for each evaluation of the objective function. In the nonlinear case, each evaluation of the Jacobian requires solving an additional set of systems. This leads to a tremendous computational cost, which is by far the dominant cost for these problems.
Several authors have proposed to drastically reduce the number of system solves by exploiting stochastic techniques [Haber et al., SIAM Optim., 22:739-757] and posing the problem as a stochastic optimization problem [Shapiro et al., Lectures on Stochastic Programming, SIAM, 2009]. In this approach, the objective function is estimated using only a few random linear combinations of the sources, referred to as simultaneous random sources. For the Jacobian, we show that a similar approach can be used to reduce the number of additional adjoint solves for the detectors.
While others have reported good solution quality at a greatly reduced computational cost using these randomized approaches, for our problem of interest, diffuse optical tomography, the approach often does not lead to sufficiently accurate solutions. Therefore, we replace a few random simultaneous sources and detectors by simultaneous sources and detectors that are optimized to maximize the Frobenius norm of the sampled Jacobian after solving to a modest tolerance. This choice is inspired by (1) the regularized model problem solves in the TREGS nonlinear least squares solver [de Sturler and Kilmer, SIAM Sci. Comput., 33:3057-3086] used for minimization in our method and (2) the fact that these optimized directions correspond to the most informative data components.
Our approach leads to solutions of the same quality as obtained using all sources and detectors but at a greatly reduced computational cost.