Collaborative Research: Construction, Analysis, Implementation and Application of New Efficient Exponential Integrators

随着科学和工程问题的规模和复杂性的增长，计算机模拟成为绝大多数研究工作中必要和不可或缺的一部分。为正在研究的过程创建计算机模型的能力，无论是来自物理学、经济学、生物学还是其他领域，不仅为研究节省了大量成本，还带来了仅通过实验程序无法获得的见解。我们描述感兴趣的现象的日益复杂，需要越来越复杂的计算机模型。特别是，能够在非常长的时间内模拟许多复杂的过程是很重要的，这是一项计算密集型和具有挑战性的任务。这个项目的重点是开发新的计算方法，允许模拟和研究各种科学和工程学科在长时间间隔内的时间演变现象。在这个项目中创造的数学和计算机工具将使研究人员能够研究目前可用的计算工具所不可能的规模和复杂性的问题。这个项目将在时间离散化方法的理论和实践方面推进最先进的技术。在该项目的过程中，将开发出新一代指数时间积分器的完整理论框架和高性能实现。新方法将显著提高许多重要科学和工程领域的数值模型的计算效率，并将使模拟达到目前无法实现的规模和复杂性。这项研究将通过开发和研究新的指数传播迭代(EPI)方法，如分裂、混合、分区和基于Krylov的技术，来推进核心数值分析。此外，还将针对广泛的问题设计和优化专门的高效方案。除了理论工作外，还将创建一个数学软件包，为广大科学界提供最有效的指数积分器的高质量实施。

As the scale and complexity of scientific and engineering problems grow, computer simulations become a necessary and integral part of the vast majority of research endeavors. An ability to create a computer model of a process under investigation, whether it comes from physics, economics, biology or some other field, provides not only significant cost savings for a study, but also brings insights inaccessible through experimental procedures alone. The growing complexity with which we describe phenomena of interest requires increasingly more sophisticated computer models. In particular, it is important to be able to simulate many complex processes over very long times, which is a computationally intensive and challenging task. This project is focused on developing new computational methods that allow simulating and studying time evolving phenomena from a wide range of scientific and engineering disciplines over long time intervals of interest. The mathematical and computer tools created during this project will enable researchers to study problems at a scale and complexity not possible with currently available computational tools.This project will advance the state-of-the-art in both the theory and practice of time discretization methods. In the course of the project a complete theoretical framework and high performance implementations of the new generation of exponential time integrators will be developed. The new methods will significantly improve computational efficiency of numerical models in many important areas of science and engineering, and will enable simulations at a scale and complexity that are not currently possible. The research will advance core numerical analysis through the development and study of new classes of exponential propagation iterative (EPI) methods such as split, hybrid, partitioned, and Krylov-based techniques. In addition, specialized efficient schemes will be designed and optimized for a wide range of problems. The theoretical work will be complemented by the creation of a mathematical software package that will provide high quality implementations of the most efficient exponential integrators to the broad scientific community.