The construction of efficient methods for uncertainty quantification in kinetic equations represents a challenge due to the high dimensionality of the models: often the computational costs involved become prohibitive. On the other hand, precisely because of the curse of dimensionality, the construction of simplified models capable of providing approximate solutions at a computationally reduced cost has always represented one of the main research strands in the field of kinetic equations. Approximations based on suitable closures of the moment equations or on simplified collisional models have been studied by many authors. In the context of uncertainty quantification, it is therefore natural to take advantage of such models in a multi-fidelity setting where the original kinetic equation represents the high-fidelity model, and the simplified models define the low-fidelity surrogate models. The scope of this article is to survey some recent results about multi-fidelity methods for kinetic equations that are able to accelerate the solution of the uncertainty quantification process by combining high-fidelity and low-fidelity model evaluations with particular attention to the case of compressible and incompressible hydrodynamic limits. We will focus essentially on two classes of strategies: multi-fidelity control variates methods and bi-fidelity stochastic collocation methods. The various approaches considered are analyzed in light of the different surrogate models used and the different numerical techniques adopted. Given the relevance of the specific choice of the surrogate model, an application-oriented approach has been chosen in the presentation.
翻译:由于模型具有高度的维度,为动态方程的不确定性量化构建高效方法是一项挑战:计算成本往往变得令人望而却步。另一方面,正是由于对维度的诅咒,建造能够以计算成本降低提供近似解决办法的简化模型,始终是动量方程领域的主要研究分支之一。许多作者研究了基于适当关闭瞬时方程或简化碰撞模型的近似方法。因此,在不确定性量化的背景下,在多纤维化环境中利用这些模型是很自然的,其中最初选择的动能方程代表高纤维化模型,而简化模型则界定低纤维化代谢模型。本文章的范围是调查关于动量方程中多纤维化方法的一些最新结果,这些结果能够通过将高纤维化和低纤维化方法相结合,加速不确定性量化进程的解决办法。在不确定性量化模型中,尤其注意可压缩和压缩的流力动力模型的多纤维化方法。我们将把最初选择的偏向性方程模型的选用于两种不同的展示方法。我们主要侧重于两种不同的递增性方法,即采用不同的递增性方法。