Computational Linear Algebra for Partial Differential Equations: An LMS Durham Research Symposium

数学研究的许多领域仅仅受到该学科研究人员思想的限制。数值分析和科学计算是完全不同的，因为这些学科的研究也受到当前可用计算机的限制。然而，这一技术障碍正在迅速消退，因此，数值分析研究界不得不不断更新其目标，以应对处理器速度和内存容量的不可阻挡的增长。来自工程应用的压力总是“解决一个更大的问题”。这不仅体现在希望使用更精细的网格以获得更高的精度方面，而且体现在解决全耦合偏微分方程（PDEs）系统的日益现实的可能性方面，而不是将问题粗糙地解耦为在迭代框架中依次解决的组件子问题。目前的研究正转向逆向或“设计”问题的研究，其中一些最优性准则需要在约束条件的限制下得到满足，这些约束条件本身就是与设计参数相关的PDE问题。这个前沿的数学研究领域，通常被称为PDE优化，是受到计算机作为工程设计工具而不是模拟工具的启发。例如，航空工程师现在可以使用计算流体动力学（用于求解Navier-Stokes方程）来设计具有理想/最佳性能的涡轮叶片。这种模式的转变提供了本提案的主要动机，以PDE优化为中心主题举行数值分析研讨会。总的来说，英国在偏微分方程数值分析方面的地位非常高，在各所大学都有重要的研究小组。特别是，英国数值分析社区在国际上被认为是在计算线性代数的重要方面工作的个人和团体的研究，适用于这方面。将这些领域的顶尖英国和国际研究人员聚集在一起，在LMS达勒姆研讨会提供的非正式氛围中，似乎是及时和适当的。这将为思想交流和研究专长的交叉融合提供机会。我们相信，如果这一建议取得成果，将实现令人兴奋的新的国际合作。研讨会将有一个整洁的演讲计划，有充足的机会进行讨论、合作和非正式会谈。我们的目标是确保所有人都了解我们的重点领域正在出现的新技术和新挑战。通过汇集广泛的专家，这次会议将对这些丰富研究领域的发展产生重大影响。会议的结果将首先通过一个积极管理和易于访问的网站分发，该网站将载有与会者名单、会谈摘要和其他相关信息。会议的主要内容将用摄像机记录下来（征得发言者的同意），并在会议结束后的一个月内上传到网站上。课堂讲稿和/或幻灯片也将放在网站上，所有这些材料都将免费提供给所有人。会议的更详细记录将通过在研究期刊上发表受会议主题启发的前沿文章来提供。本次会议有望为世界范围内的PDE优化理论研究提供一个重要而持久的推动力。

Many areas of mathematical research are constrained only by a limitation on the ideas of researchers in the subject. Numerical analysis and scientific computing are altogether different, since research in these disciplines is also constrained by the limits of the computers currently available. This technological barrier is, however, rapidly receding and as a result, the numerical analysis research community is having to continually update its aims in response to the inexorable increases in processor speed and memory capability. The pressure from engineering applications is always to ``solve a bigger problem''. This is true not only in terms of a desire to use finer meshes for greater accuracy, but also in terms of the increasingly realistic possibility of solving fully coupled systems of partial differential equations (PDEs), as opposed to crudely decoupling the problem into component subproblems that are successively solved in an iterative framework.Research is now moving towards the study of inverse or ``design'' problems, where some optimality criterion is desired to be satisfied within the restriction of constraints which are themselves PDE problems linked to the design parameters. This cutting edge mathematical research area, commonly known as PDE optimization, is inspired by the use of computers as an engineering design tool as opposed to a simulation tool. For example, aeronautical engineers can now use computational fluid dynamics (for solving the Navier-Stokes equations) to design turbine blades with desired/optimal properties. This paradigm shift provides the primary motivation for this proposal to hold a numerical analysis symposium with PDE optimization as its central theme. Broadly speaking, the UK standing in the numerical analysis of PDEs is very high, with significant research groups at various universities. In particular, the UK numerical analysis community is internationally regarded for the research of individuals and groups working on important aspects of computational linear algebra which are applicable in this regard. It seems to be timely and appropriate to bring the top UK and international researchers in these fields together in the informal atmosphere provided by an LMS Durham Symposium. This will give the opportunity for the exchange of ideas and cross-fertilisation of research expertise. We are confident that exciting new international collaborations will be realised if this proposal comes to fruition. The symposium will have an uncluttered lecture programme, with ample opportunities for discussions, collaborations and informal talks. Our aim is to ensure that all are aware of new techniques and the new challenges that are arising for our areas of focus. By bringing together a broad range of experts, this meeting will have a significant impact on the development of these rich areas of research. The results of the meeting will be distributed initially via an actively-managed and easily accessible website, which will contain a list of participants, abstracts of talks and other relevant information. The main talks will be recorded (with permission of the speakers) with a camcorder, and will be placed on the website within one month of the finish of the meeting. Lecture notes and/or slides will also be put on the site, and all this material will be made freely available to all.A more detailed record of the meeting will be provided by the publication in research journals of cutting-edge articles inspired by the subjects covered in the meeting. The conference is expected to provide a major lasting fillip to world research in the theory of PDE optimization.