偏微分方程约束贝叶斯反问题中近似误差的泰勒近似方差缩减
Taylor Approximation Variance Reduction for Approximation Errors in PDE-constrained Bayesian Inverse Problems
AI总结：
本文提出了一种基于泰勒展开的高维PDE约束贝叶斯反问题中近似误差方差缩减方法，通过线性化PDE求解高效计算均值和协方差，提升计算效率。
摘要：
在许多应用中，代理模型被用作替代准确的参数到观测映射，以求解由偏微分方程（PDE）支配的大规模反问题。代理模型可能是计算上更便宜的替代品，或者可能忽略额外的未知数或不确定性来源。贝叶斯近似误差（BAE）方法提供了一种手段来考虑由此引起的不确定性和近似误差（在准确的参数到观测映射与代理模型之间）。这些误差的统计特性通常在事前是未知的，因此使用蒙特卡洛采样来计算。尽管采样通常是在线下进行的，但这一过程仍可能成为计算瓶颈。在本文中，我们开发了一种可扩展的计算方法，以减少BAE方法采样阶段的成本。具体而言，我们考虑了准确和代理正向模型相对于不确定参数场的泰勒展开，无论是作为方差缩减的控制变量，还是作为高效近似近似误差均值和协方差的手段。我们提出了高效评估泰勒近似表达式均值和协方差的方法，基于线性（化）的PDE求解。此外，所提出的方法不依赖于不确定参数的维度，而是依赖于数据的内在维度，从而确保能够扩展到高维问题。所提出方法的潜在优势通过两个由PDE支配的高维反问题示例得到展示，即对于线性扩散问题中分布式Robin边界系数的估计，以及由非线性扩散问题支配的系数估计问题。
英文摘要：
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors (between the accurate parameter-to-observable mapping and the surrogate). The statistics of these errors are in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
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