讲座题目：A localized reduced-order modeling approach for PDEs with bifurcating solutions 主讲人：Max Gunzburger教授 主持人：郑海标 副教授 开始时间🤸🏿♂️：2020-07-22 20:00:00 讲座地址💅🏻：Zoom会议 房间号：924 6976 5319 密码:123456 主办单位：数学科学学院
报告人简介🚶➡️： Max Gunzburger 是佛罗里达州立大学Francis Eppes 杰出教授🛷。1966获纽约大学学士学位，1969年获柯朗研究所博士学位🚒。他的开创性研究成果包括流体控制、有限元分析、超导方程的分析与计算以及非局部问题的分析与计算🐟。 Gunzburger 还在空气动力学、材料、声学、气候变化和地下水等领域做出了卓越的贡献。Gunzburger曾任SIAM理事会主席，SINUM主编😾，SIAM/American Statistical Society Journal on Uncertainty Quantification的创始主编🧑🏽💼。2006年国际数学家大会邀请报告，2008 年获SIAM W. T. and Idalia Reed Award，首届（2009）SIAM Fellow🚴🏽♂️。
报告内容： Reduced-order models (ROMs) are low-dimensional discretizations of PDEs that more efficiently treat settings that require multiple solutions such as optimization and UQ. Although ROMs are successful in many cases, ROMs built for the efficient treatment of bifurcating solutions as input parameter values change have not received much attention. In such cases, the parameter domain can be subdivided into subregions that corresponds to a different branch of solutions. ROM approaches such as proper orthogonal decomposition (POD) results in global low-dimensional bases that do not respect the large differences in solutions corresponding to different subregions. In this work, we develop and test a new ROM specifically aimed at bifurcation problems. In the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method can detect the cluster a new parameter point belongs to, after which the local basis for that cluster is used to determine a ROM solution. Numerical examples show the effectiveness of the method both for when bifurcations cause continuous and discontinuous changes in the solution. [Joint work with Alessandro Alla, Martin Hess, Annalisa Quaini, and Gianluigi Rozza] |
