GRK 1754: Optimisation and Numerical Analysis for Partial Differential Equations with Non-smooth Structures

物理、化学、生物或工程科学中的许多应用问题都是用涉及偏微分方程的数学模型来描述的。这类问题通常涉及不同类型的非光滑结构(奇点、界面、不等式约束等)。相应的偏微分方程模型的数学处理是有效解决实际问题的关键。学术界和工商界对受过研究生教育并具有这方面经验的高素质初级科学家的需求很大。国际研究培训小组打算满足这一需求。研究培训小组的主要目标是：(1)将前沿研究课题转移到博士生教育中，(2)开发和研究数值处理的新方法，并对涉及偏微分方程和具有非光滑结构的问题进行优化。研究培训小组是由门兴理工学院、明尼苏达联邦理工学院、格拉茨理工学院和格拉茨理工学院四所大学组成的联合项目。慕尼黑和格拉茨的机构通过其在应用数学方面在国际上举世瞩目的科学优势为联合研究计划做出贡献：德国明臣大学在非线性现象的分析和数值处理方面拥有长期经验，而格拉茨的研究所则贡献了他们在优化、变分和数值分析方面的专业知识。这些专业知识的结合将产生高度的协同效应，这一点应由这个国际研究培训小组加以突出和利用。从适应性和非流畅的最优化到界面和形状的处理等方法学方法将进入国际研究培训小组的研究和学习方案。

Many application problems coming from physics, chemistry, biology or engineering sciences are described by mathematical models involving partial differential equations (PDEs). Often such problems involve different kinds of non-smooth structures (singularities, interfaces, inequality constraints etc.). The mathematical treatment of the corresponding PDE-based models is crucial for the efficient solution of practical problems. There is a significant demand for highly qualified junior scientists with postgraduate education and experience in this area, in academia as well as in industry and commerce. The International Research Training Group intends to meet this demand. The major goals of the Research Training Group are: (1) transfer of cutting-edge research topics to the education of doctoral students, (2) development and investigation of novel methods in numerical treatment and optimisation for problems involving PDEs and possessing non-smooth structures. The Research Training Group is a joint project of the four universities Technische Universität München, Universität der Bundeswehr München, Karl-Franzens-Universität Graz and Technische Universität Graz. The institutions in Munich and Graz contribute to the joint research programme through their internationally visible scientific strengths in applied mathematics: the TU München has long-range experience in the analytical and numerical treatment of non-linear phenomena, whereas the institutes in Graz contribute their expertise in optimisation, variational calculus and numerical analysis. The combination of the expertise will result in high synergy effects, which shall be highlighted and exploited by this International Research Training Group. Methodological approaches ranging from adaptivity and non-smooth optimisation to the treatment of interfaces and shapes will enter in the research and study programme of the International Research Training Group.