Rigorous Computations for Infinite Dimensional Nonlinear Problems

无限维非线性问题的严格计算

• 批准号: 418634-2012
• 负责人: Lessard, JeanPhilippe
• 金额: $ 1.89万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635660
• 关键词: Rigorous; Computations; Infinite; Dimensional; Nonlinear

As we develop evermore powerful computational tools, so it is incumbent upon scientists to exploit newly-developed computational techniques to maintain the pace of research progress, effectively leading us to the point where high-performance computing must be mastered to progress. As a society we continuously undertake more and more ambitious tasks, from whole genome sequencing, to whole-ocean numerical simulation and weather prediction to molecular-level simulation of nanofluids and onwards. However, while the pace of progress increases, sometimes we need to take a step back and pose the question, just how reliable are our computations? The goal of this proposal is to develop rigorous computational tools to prove, in a constructive mathematical way, the existence of qualitative phenomena in infinite dimensional nonlinear problems such as partial differential equations, variational equations and delay differential equations, which are successful widely used mathematical models that describe, explain and predict phenomena in areas as broad as physics, chemistry, biology and economics. More precisely, we are interested in answering each of the following questions. Can we mathematically demonstrate the reliability of the solutions computed using the Navier-Stokes equations or a closely related fluid dynamical model? Can we undertake the same program of research for the difficult, nonlinear molecular dynamics models thrown up by cell biological problems? More generally, can we develop algorithms to construct rigorously and automatically invariant sets of a given infinite dimensional nonlinear dynamical system? Finally, can we use computational methods to prove long-standing open problems in mathematical fields as broad as the calculus of variations and delay differential equations?

随着我们开发出越来越强大的计算工具，科学家们有责任利用新开发的计算技术来保持研究进展的步伐，从而有效地引导我们达到必须掌握高性能计算才能取得进步的地步。作为一个社会，我们不断承担越来越多雄心勃勃的任务，从全基因组测序，到全海洋数值模拟和天气预报，再到纳米流体的分子水平模拟等等。然而，虽然进展速度加快，有时我们需要退一步提出问题，我们的计算到底有多可靠？该提案的目标是开发严格的计算工具，以建设性的数学方式证明无限维非线性问题（例如偏微分方程、变分方程和延迟微分方程）中定性现象的存在，这些数学模型成功地广泛使用，描述、解释和预测物理、化学、生物学和经济学等广泛领域的现象。更准确地说，我们有兴趣回答以下每个问题。我们能否在数学上证明使用纳维-斯托克斯方程或密切相关的流体动力学模型计算的解的可靠性？我们能否对细胞生物学问题引发的困难的非线性分子动力学模型进行相同的研究计划？更一般地说，我们可以开发算法来构造给定无限维非线性动力系统的严格且自动的不变集吗？最后，我们能否使用计算方法来证明像变分法和时滞微分方程这样广泛的数学领域中长期存在的开放性问题？