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

许多来自物理、化学、生物或工程科学的应用问题都是用偏微分方程（PDEs）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）。相应的基于偏微分方程的模型的数学处理对于有效解决实际问题至关重要。在学术界以及工商界，对具有该领域研究生教育和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。研究训练小组的主要目标是：(1)将前沿研究课题转移到博士生的教育中；(2)开发和研究涉及偏微分方程和具有非光滑结构的问题的数值处理和优化的新方法。研究训练组是下述四所大学的联合项目：Universität m<e:1>琛工业大学、Universität德国联邦国防军m<e:1>琛工业大学、Karl-Franzens-Universität格拉茨工业大学和Universität格拉茨工业大学。慕尼黑和格拉茨的研究机构通过其在应用数学方面的国际可见的科学实力为联合研究计划做出了贡献：慕尼黑工业大学在非线性现象的分析和数值处理方面拥有长期的经验，而格拉茨的研究机构则贡献了他们在优化，变分微积分和数值分析方面的专业知识。专业知识的结合将产生高度的协同效应，国际研究培训小组应突出和利用这一效应。从适应性和非平滑优化到处理界面和形状的方法方法将列入国际研究训练组的研究方案。

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.