SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1163-A1193, January 2021.
We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from the posterior distribution, we integrate an adaptive and goal-oriented model reduction technique with an optimization-based Stein variational gradient descent method. We present detailed analyses for the reduced basis approximation errors of the potential and its gradient, the induced errors of the posterior distribution measured by Kullback--Leibler divergence, as well as the induced errors of the Stein variational samples. To demonstrate the computational accuracy and efficiency of SVRB, we report results of numerical experiments on a Bayesian inverse problem governed by a diffusion PDE with random parameters with both uniform and Gaussian prior distributions. Over 100X speedups can be achieved while the accuracy of the approximation of the potential and its gradient is preserved.

中文翻译：

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斯坦因变分减少基贝叶斯反演

SIAM科学计算杂志，第43卷，第2期，第A1163-A1193页，2021年1月。
我们提出并分析了斯坦因变分简化基方法（SVRB），以解决大规模PDE约束的贝叶斯逆问题。为了解决从后验分布中抽取大量需要昂贵PDE求解的样本的计算难题，我们将自适应和面向目标的模型简化技术与基于优化的Stein变分梯度下降方法相结合。我们对电势及其梯度的简化基近似误差，由Kullback-Leibler发散测量的后验分布的诱发误差以及Stein变分样本的诱发误差进行了详细分析。为了证明SVRB的计算准确性和效率，我们报告了由具有均匀和高斯先验分布的随机参数的扩散PDE控制的贝叶斯逆问题的数值实验结果。在保持电势及其梯度近似值的精度的同时，可以实现超过100倍的加速。