In this chapter we examine reduced order techniques for geometrical parametrized heat exchange systems, Poisson, and flows based on Stokes, steady and unsteady incompressible Navier-Stokes and Cahn-Hilliard problems. The full order finite element methods, employed in an embedded and/or immersed geometry framework, are the Shifted Boundary (SBM) and the Cut elements (CutFEM) methodologies, with applications mainly focused in fluids. We start by introducing the Nitsche's method, for both SBM/CutFEM and parametrized physical problems as well as the high fidelity approximation. We continue with the full order parameterized Nitsche shifted boundary variational weak formulation, and the reduced order modeling ideas based on a Proper Orthogonal Decomposition Galerkin method and geometrical parametrization, quoting the main differences and advantages with respect to a reference domain approach used for classical finite element methods, while stability issues may overcome employing supremizer enrichment methodologies. Numerical experiments verify the efficiency of the introduced ``hello world'' problems considering reduced order results in several cases for one, two, three and four dimensional geometrical kind of parametrization. We investigate execution times, and we illustrate transport methods and improvements. A list of important references related to unfitted methods and reduced order modeling are [11, 8, 9, 10, 7, 6, 12].

翻译：本章研究基于几何参数化的热交换系统、泊松问题以及斯托克斯、稳态与非稳态不可压纳维-斯托克斯和卡恩-希利亚德问题的降阶技术。在全阶有限元方法中，采用嵌入/浸入几何框架的移位边界法（SBM）和切割元法（CutFEM），主要应用于流体领域。我们首先介绍适用于SBM/CutFEM和参数化物理问题的尼采方法，以及高保真近似。随后阐述全阶参数化尼采移位边界变分弱形式，并基于本征正交分解伽辽金方法和几何参数化提出降阶建模思路，指出其与经典有限元方法参考域方法的主要差异与优势，同时通过上确界富集方法解决稳定性问题。数值实验验证了所引入的"hello world"问题的有效性，涵盖一维至四维几何参数化多种场景的降阶结果。我们分析执行时间并阐述传输方法及其改进。与未拟合方法及降阶建模相关的重要参考文献见[11,8,9,10,7,6,12]。