Scalable Parallel Algorithms for Partial Differential Equations

摘要偏微分方程的可扩展并行算法在计算科学与工程中占有重要地位。随着当今超级计算机处理器数量的快速增长，可扩展性成为算法设计中最重要的问题之一。在过去几年中，这一研究领域取得了巨大的进展。会议汇集了来自世界各地的该领域的专家，讨论可扩展算法的最新发展，并向初级研究人员介绍这个令人兴奋的研究领域。会议还支持研究生的参与，并为他们提供一个独特的机会，让他们的研究向大量听众展示，并向专家学习。主要的焦点是领域分解方法，它包含了迄今为止在大规模并行分布式分层存储计算机上进行大规模模拟的最常见的可扩展范例。在领域分解中，一个大问题被简化为一组小问题，每个小问题都比未分解的问题更容易计算解决，并且大多数或全部问题都可以独立并发地解决。通常，有必要在较小的问题集合上进行迭代，并且对领域分解算法的许多理论兴趣在于确保所需的迭代次数非常小。实际上，最好的域分解方法与其同类多网格方法共享这样的特性：总计算量与输入数据的大小成线性比例，或者所需的迭代次数在单个子域的自由度中最多为对数。在这种情况下，其工作要求在输入数据的大小上是线性的算法被认为是最优的。对于科学和工程中出现的许多重要类别的问题，现在已知的最优域分解算法，但肯定不是全部。当前领域分解算法的研究兴趣主要在于扩展已知最优算法的问题类别。域分解算法可以根据物理系统的属性（反映在数学运算符、可用处理器的数量，甚至特定的体系结构参数，如缓存大小和内存带宽与浮点处理速率的比率）进行定制。

Scalable Parallel Algorithms for Partial Differential EquationsAbstractScalable algorithms for partial differential equations play a key role in Computational Science and Engineering. As the number of processors in today's supercomputers grows rapidly, scalability becomes one of the most important issues in algorithm design. Tremendous progress has been made in this research area in the past few years. The conference brings together experts in this field from all over the world to discuss the latest development of scalable algorithms, and to introduce junior researchers to this exciting research field. The conference also supports the participation of graduate students and provides them with a unique opportunity for exposing their research to a large audience and for learning from experts.The primary focus is on domain decomposition METHODS which COMPRISE by far the most common scalable paradigm for large-scale simulation on massively parallel distributed, hierarchical memory computers. In domain decomposition, a large problem is reduced to a collection of smaller problems, each of which is easier to solve computationally than the un-decomposed problem, and most or all of which can be solved independently and concurrently. Typically, it is necessary to iterate over the collection of smaller problems, and much of the theoretical interest in domain decomposition algorithms lies in ensuring that the number of iterations required is very small. Indeed, the best domain decomposition methods share with their cousins, multigrid methods, the property that the total computational work is linearly proportional to the size of the input data, or that the number of iterations required is at most logarithmic in the number of degrees of freedom of individual subdomains. Algorithms whose work requirements are linear in the size of the input data in this context are said to be optimal. Optimal domain decomposition algorithms are now known for many, but certainly not all, important classes of problems that arise in science and engineering. Much of the current research interest in domain decomposition algorithms lies in extending the classes of problems for which optimal algorithms are known. Domain decomposition algorithms can be tailored to the properties of the physical system as reflected in the mathematical operators, the number of processors available, and even to specific architectural parameters, such as cache size and the ratio of memory bandwidth to floating point processing rate.