Towards surrogate models of large scale parametric industrial flow problems

计算机模拟是一个越来越重要的工具，以支持虚拟样机在科学和工程，因为他们能够减少时间和成本的实验测试。虚拟设计旨在通过测试表征几何特征（例如，光束的长度）或材料特性（例如，流体的密度）等各个方面的几个参数值来确定系统的最佳配置。通过对基础方程进行参数化，这些量被纳入系统的数学模型中。然后对得到的参数问题进行数值求解，以确定最优构型。然而，这项任务在计算上的要求非常高。事实上，在实际的3D应用中，数值模型可能涉及数百万个未知数，并且必须对系统的每种可能配置进行多次求解。因此，需要新颖、快速、可靠的算法来降低计算成本。这是一个非常具有挑战性的研究领域，对于汽车工业中过滤系统的优化设计和空气动力学模拟等许多应用具有重要的实际意义。该项目将通过开发一种新的计算框架来促进这一领域的发展，该框架结合了两种数学方法：域分解和适当的广义分解。这些将用于将参数问题拆分为更简单的子问题集合，在考虑参数的所有重要值的情况下独立解决它们，并将局部解“粘合”到原始问题的解中。拟议的项目将有助于推动数值数学的最新发展，并为科学家和工程师开发有效的算法，通过计算建模来使用虚拟设计。

Computer simulations are an increasingly important tool to support virtual prototyping in science and engineering as they enable to reduce the time and cost of experimental testing. Virtual design aims to identify the best configuration of a system by testing several values of parameters that characterise various aspects such as geometric features (e.g., the length of a beam) or material properties (e.g., the density of a fluid). These quantities are incorporated in the mathematical model of the system by parametrising the underlying equations. The resulting parametric problem is then solved numerically to identify the optimal configurations. However, this task is computationally very demanding. In fact, in practical 3D applications, numerical models can involve millions of unknowns and must be solved multiple times for each possible configuration of the system. Novel fast and reliable algorithms are thus needed to make the computational cost affordable. This is a very challenging research area of great practical importance for many applications such as optimal design of filtration systems and aerodynamic simulations in the automotive industry. This project will contribute to this field by developing a new computational framework that combines two mathematical methods: domain decomposition and proper generalised decomposition. These will be used to split parametric problems into collections of simpler subproblems, to solve them independently accounting for all significant values of the parameters, and to 'glue' the local solutions to obtain those of the original problems. The proposed project will contribute to advance the state-of-the-art in numerical mathematics and develop effective algorithms for scientists and engineers that use virtual design by computational modelling.