Two of the most significant challenges in uncertainty quantification pertain to the high computational cost for simulating complex physical models and the high dimension of the random inputs. In applications of practical interest, both of these problems are encountered, and standard methods either fail or are not feasible. To overcome the current limitations, we present a generalized formulation of a Bayesian multi-fidelity Monte-Carlo (BMFMC) framework that can exploit lower-fidelity model versions in a small data regime. The goal of our analysis is an efficient and accurate estimation of the complete probabilistic response for high-fidelity models. BMFMC circumvents the curse of dimensionality by learning the relationship between the outputs of a reference high-fidelity model and potentially several lower-fidelity models. While the continuous formulation is mathematically exact and independent of the low-fidelity model's accuracy, we address challenges associated with the small data regime (i.e., only a small number of 50 to 300 high-fidelity model runs can be performed). Specifically, we complement the formulation with a set of informative input features at no extra cost. Despite the inaccurate and noisy information that some low-fidelity models provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained for uncertainty quantification problems in high stochastic dimensions, with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty quantification. We illustrate our approach by applying it to challenging numerical examples such as Navier-Stokes flow simulations and fluid-structure interaction problems.
翻译:不确定性量化方面两个最重大挑战涉及模拟复杂物理模型的计算成本高以及随机投入的高度。在实际应用中,遇到这些问题,遇到这些问题,标准方法要么失败,要么不可行。为了克服目前的局限性,我们提出了一个通用的巴伊西亚多信仰蒙特卡洛(BMFMC)框架,这个框架可以在一个小型数据系统中利用较低信仰模式版本。我们的分析目标是有效和准确地估计对高忠诚模型的全面概率反应。BMFMC在实际应用中避开了维度的诅咒,了解了高忠诚参考模型产出与可能若干较低忠诚模型之间的关系。尽管持续制定的方法在数学上准确且独立于低忠诚模型的准确性,但我们处理与小数据机制相关的挑战(也就是说,只有少量50至300种高忠诚模型可以运行 ) 。具体地说,我们补充了一套信息输入特征,没有额外的成本。尽管一些不准确和不精确的不确定性模型可以证明我们所获取的低信任度数据数量,但通过高透明度的量化方法,我们可以提供一些低信任度的准确性和高透明度模型。