Domain decomposition methods based on proper generalized decomposition for parametric heterogeneous problems

非均匀(或多物理)问题在工程和科学应用中非常常见。它们通常在感兴趣领域的两个或多个子区域中发生不同的现象时出现，例如，在地球物理或工业应用中流体通过多孔介质的过滤、在生物医学中的组织灌流、在流体与弹性结构之间的相互作用中。在这种情况下，必须在每个子区域定义至少两组不同的方程(例如，不可压缩流体方程和弹性方程)，并且必须适当地将它们耦合到一个全局非均匀问题中，以正确地描述物理系统。由于需要精确地近似所有不同的涉及的物理现象，因此数值求解这些问题是非常困难的。当这些问题发生时，例如在虚拟设计中，为了优化目的而必须多次解决这些问题时，计算复杂性甚至进一步增加。实际上，优化需要确定用于描述系统的各种特征的几个参数的最佳值，例如几何特征(例如，结构元件的尺寸)、材料属性(例如，多孔介质的渗透性)或工艺参数(例如，过滤设备中的流入压力)。这通常是通过测试大量可能的构型来完成的，这大大增加了数值模拟的计算成本，并限制了其实际应用。在本项目中，我们将研究一种新的数学框架，通过结合两种数学方法：区域分解(DD)和适当的广义分解(PGD)，使得数值处理参数异质问题的成本更低。新方法使用区域分解技术将多参数异质问题分解成性质相同、参数较少的更简单的子问题。这些局部子问题的解可以用PGD来计算，它提供了一种有效的策略来统一处理各种性质的参数。最后，数据挖掘可以“合成”局部解，以获得原始问题的全局“通解”，该全局“通解”考虑了参数的所有有效值。寻找有效和健壮的方法来合成局部解并不是一件容易的事情，特别是在异质问题的情况下，这是一个开放的具有挑战性的研究问题，在PGD的背景下，我们在这个项目中解决。我们将为异质问题的DD-PGD方法奠定基础，并开发算法，使我们能够解决在各种应用中的多物理多参数系统的虚拟设计中遇到的计算挑战，例如膜过滤过程。

Heterogeneous (or multi-physics) problems are very common in engineering and scientific applications. They typically arise when different phenomena occur in two or more subregions of the domain of interest such as, e.g., in the filtration of fluids through porous media in geophysical or industrial applications, in tissue perfusion in biomedicine, in the interactions between fluids and elastic structures. In such cases, at least two different sets of equations (e.g., incompressible fluid equations and elasticity equations) must be defined in each subregion and they must be suitably coupled into a global heterogeneous problem to correctly describe the physical system.Solving these problems numerically is computationally demanding due to the need to accurately approximate all the different involved physical phenomena. The computational complexity increases even further when these problems must be solved several times for optimisation purposes as it occurs, e.g., in virtual design. Indeed, optimisation requires identifying the optimal values of several parameters used to describe various characteristics of the system such as geometrical features (e.g., the dimension of a structural element), material properties (e.g., the permeability of a porous medium) or process parameters (e.g., the inflow pressure in a filtering device). This is typically done by testing a large number of possible configurations, which dramatically increases the computational cost of numerical simulations and limits their practical applicability.In this project, we will study a novel mathematical framework to make the numerical treatment of parametric heterogeneous problems more affordable by combining two mathematical methods: Domain Decomposition (DD) and Proper Generalized Decomposition (PGD).The new method uses DD techniques to split multi-parametric heterogeneous problems into families of simpler subproblems of the same nature and with a reduced number of parameters. The solutions of these local subproblems can be computed by PGD that provides an efficient strategy to handle parameters of various nature in a unified manner. Finally, DD can 'compose' the local solutions to obtain the global 'general solution' of the original problem that accounts for all significant values of the parameters. Identifying effective and robust ways of 'composing' local solutions is not an easy task especially in the case of heterogeneous problems and it constitutes an open challenging research question in the PGD context that we address in this project.We will lay the foundation of the DD-PGD method for heterogeneous problems and develop algorithms that will allow us to tackle the computational challenges encountered in the virtual design of multi-physics multi-parameter systems in various applications, e.g., membrane filtration processes.