Extreme-scale UQ for Bayesian inverse problems governed by PDEs
Extreme-scale UQ for Bayesian inverse problems governed by PDEs
复制标题
由偏微分方程控制的贝叶斯反问题的极端规模 UQ
DOI：
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发表时间：
2012
期刊：
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通讯作者：
L. Wilcox
中科院分区：
文献类型：
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作者：
T. Bui;Carsten Burstedde;O. Ghattas;James Martin;G. Stadler;L. Wilcox
量化大规模模拟中的不确定性已成为CS&E面临的核心挑战。当模拟需要超级计算机，且不确定参数尺寸较大时，传统的UQ方法失效。在这里，我们在贝叶斯推理框架中解决大规模逆问题的不确定性量化：给定数据和模型的不确定性，找到描述参数不确定性的pdf。为了克服传统方法的维数问题，我们利用数据在参数空间的低维流形中具有典型信息的事实，通过无矩阵随机化方法构造后验概率的协方差矩阵的低秩逼近。我们得到了一种独立于前向问题维数、不确定参数维数、数据维数和核数的扩展方法。我们将该方法应用于一个三维全球地震波传播逆问题的贝叶斯解，该问题具有超过100万个不确定的地球模型参数，6.3亿个波传播未知，涉及多达262K个岩心，我们获得了问题维数减少2000以上的系数。这使得UQ对于逆问题来说很容易处理。
Quantifying uncertainties in large-scale simulations has emerged as the central challenge facing CS&E. When the simulations require supercomputers, and uncertain parameter dimensions are large, conventional UQ methods fail. Here we address uncertainty quantification for large-scale inverse problems in a Bayesian inference framework: given data and model uncertainties, find the pdf describing parameter uncertainties. To overcome the curse of dimensionality of conventional methods, we exploit the fact that the data are typically informative about low-dimensional manifolds of parameter space to construct low rank approximations of the covariance matrix of the posterior pdf via a matrix-free randomized method. We obtain a method that scales independently of the forward problem dimension, the uncertain parameter dimension, the data dimension, and the number of cores. We apply the method to the Bayesian solution of an inverse problem in 3D global seismic wave propagation with over one million uncertain earth model parameters, 630 million wave propagation unknowns, on up to 262K cores, for which we obtain a factor of over 2000 reduction in problem dimension. This makes UQ tractable for the inverse problem.