Iterative updating of model error for Bayesian inversion
Iterative updating of model error for Bayesian inversion
复制标题
DOI：
10.1088/1361-6420/aaa34d
复制
发表时间：
2018-02-01
期刊：
影响因子：
2.1
通讯作者：
Stuart, Andrew
中科院分区：
文献类型：
--
作者：
Calvetti, Daniela;Dunlop, Matthew;Stuart, Andrew
在计算反问题中，通常用计算难度较小的替代模型来近似详细而准确的正演模型。模型简化可能是必要的，以满足约束条件的计算时间时，优化算法被用来找到一个单一的估计，或加快马尔可夫链蒙特卡罗（MCMC）计算的贝叶斯框架。近似模型的使用引入了差异或建模误差，这可能对不适定逆问题的解决方案产生不利影响，或者可能严重扭曲后验分布的估计。在贝叶斯范例中，建模误差可以被认为是随机变量，并且通过使用未知的概率分布的估计，可以估计建模误差的概率分布并将其并入反演中。我们引入了一种算法，它迭代这个想法来更新模型误差的分布，从而产生一系列后验分布，这些后验分布根据经验证明可以以越来越高的准确性捕获潜在的真相。由于该算法不是基于拒绝，它只需要有限的完整的模型evaluations.We分析表明，在线性高斯的情况下，该算法收敛的几何快速迭代次数时，数据是有限维的。对于更一般的模型，我们引入粒子近似的迭代生成的序列的分布，我们还证明了每个元素的序列收敛在一个简化的假设下的大粒子限制。我们数值表明，在线性的情况下，快速收敛相对于迭代次数发生。此外，我们通过计算实例表明，由该迭代算法获得的点估计优于忽略模型误差获得的点估计，具有上级性能。
In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations.We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations when the data is finite dimensional. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit under a simplifying assumption. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.