Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems
Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems
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DOI：
10.1137/140965144
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发表时间：
2015-06
期刊：
影响因子：
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通讯作者：
Liang Yan;Ling Guo
中科院分区：
文献类型：
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作者：
Liang Yan;Ling Guo
贝叶斯方法已被证明是一种从有限信息的可用数据中求解逆问题的方便框架。反问题的贝叶斯解的主要计算挑战来自于需要重复评估正演模型，正如马尔可夫链蒙特卡罗（MCMC）方法对后验抽样的要求。本文提出了一种构造随机代理模型的有效方法，以加速统计逆问题的贝叶斯推理方法。采用基于广义多项式混沌的l1 -最小化（L1-SCM）随机配置算法，在先验分布的支持下构造正解的多项式逼近。然后，这个近似定义了一个计算成本低廉的代理后验概率密度。严格的误差分析表明，后验密度的收敛速度至少是gPC扩展收敛速度的两倍。
The Bayesian methodology has proven to be a convenient framework to solve inverse problems from available data with limited information. The main computational challenges in the Bayesian solution of inverse problems arise from the need for repeated evaluations of the forward model, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. In this paper, we present an efficient technique for constructing stochastic surrogate models to accelerate the Bayesian inference approach for statistical inverse problems. The stochastic collocation algorithms using $l_1$-minimization (L1-SCM), based on generalized polynomial chaos, are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that is inexpensive to evaluate. A rigorous error analysis shows that the convergence rate of the posterior density is at least twice as fast as the convergence rate of the gPC expa...