线性与非线性反问题的贝叶斯计算新进展
New Advances in Bayesian Calculation for Linear and Nonlinear Inverse Problems
摘要：
贝叶斯方法已被证明是一个连贯的方法来处理不适定的反问题。然而，贝叶斯计算需要优化或积分计算。最大后验概率（MAP）估计要求最小化复合准则，该准则一般由两部分组成：数据拟合部分和先验部分。在许多情况下，最小化的标准变成了多模态。基于模拟退火（SA）的技术的成本通常是巨大的反问题。最近的确定性优化技术，基于渐变非凸性（GNC），已被提出来克服这个困难。本文的目的是显示这种技术的两个具体实现以下情况：-线性逆问题的解决方案是作为一个分段连续函数建模。的非凸性的标准，然后是由于特殊的选择的前; --一个非线性逆问题中出现的逆散射的非凸性的标准是由于可能性的一部分。关键词：反问题，正则化，贝叶斯计算，全局优化，渐变非凸性。
英文摘要：
The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires the minimization of a compound criterion which, in general, has two parts: a data fitting part and a prior part. In many situations the criterion to be minimized becomes multimodal. The cost of the Simulated Annealing (SA) based techniques is in general huge for inverse problems. Recently a deterministic optimization technique, based on Graduated Non Convexity (GNC), have been proposed to overcome this difficulty. The objective of this paper is to show two specific implementations of this technique for the following situations: -- Linear inverse problems where the solution is modeled as a piecewise continuous function. The non convexity of the criterion is then due to the special choice of the prior; -- A nonlinear inverse problem which arises in inverse scattering where the non convexity of the criterion is due to the likelihood part. Keywords: Inverse problems, Regularization, Bayesian calculation, Global optimization, Graduated Non Convexity.
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