Title: Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity Show Chinese title

Title: 基于注册的参数化PDEs的时空自适应模型降阶方法

Abstract: We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions. Show Chinese abstract

Abstract: 我们提出了一种自动非线性模型降阶和网格自适应框架，用于快速可靠地求解参数化对流主导问题，重点在于可压缩流。 我们的方法的关键特性有三点：(i) 一种基于度量的网格自适应技术，用于为一系列参数生成精确的网格，(ii) 一种通用的（即与基础方程无关）配准过程，用于计算跟踪解场移动特征的映射$\Phi$，以及 (iii) 一种超降阶最小二乘 Petrov-Galerkin 降阶模型，用于快速可靠地估计映射后的解。 我们讨论了一个通用的范式——模仿网格自适应中考虑的细化循环——以同时构建高保真度和降阶近似，并讨论了加速离线阶段的操作策略。 我们针对一个准一维喷管问题和一个二维无粘流过高斯凸起的问题进行了广泛的数值研究，以展示该方法的诸多特性，并评估具有不连续解的问题的性能。