Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 589-611, January 2020.
Many naturally occurring models in the sciences are well approximated by simplified models using multiscale techniques. In such settings it is natural to ask about the relationship between inverse problems defined by the original problem and by the multiscale approximation. We develop an approach to this problem and exemplify it in the context of optical tomographic imaging. Optical tomographic imaging is a technique for inferring the properties of biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a medical imaging methodology. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE from boundary measurements. We study this problem in the Bayesian framework, focussing on the strong scattering regime. In this regime the forward RTE is close to the diffusion equation (DE). We study the RTE in the asymptotic regime where the forward problem approaches the DE and prove convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings. Convergence is proved by studying the distance between the two posterior distributions using the Hellinger metric and using the Kullback--Leibler divergence.

中文翻译：

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贝叶斯框架中的漫射光学层析成像

2020年1月，《多尺度建模与仿真》，第18卷，第2期，第589-611页。
科学中许多自然发生的模型都可以通过使用多尺度技术的简化模型很好地近似。在这样的设置中，自然会询问由原始问题和多尺度逼近定义的反问题之间的关系。我们开发了解决此问题的方法，并在光学断层扫描成像的背景下进行了举例说明。光学层析成像技术是一种通过测量入射和出射光强度来推断生物组织特性的技术。它可以用作医学成像方法。在数学上，光的传播是通过辐射传递方程（RTE）建模的，光学层析成像相当于根据边界测量值重建RTE中的散射和吸收系数。我们在贝叶斯框架中研究这个问题，专注于强散射机制。在这种情况下，前向RTE接近扩散方程（DE）。我们研究了渐近状态下的RTE，其中前向问题接近DE，并证明了在非线性和线性情况下逆RTE到逆DE的收敛性。通过使用Hellinger度量和使用Kullback-Leibler散度研究两个后验分布之间的距离来证明收敛性。