Rigorous Computations for Infinite Dimensional Nonlinear Problems

随着我们开发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速度，有效地将我们引导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，我们不断承担着越来越雄心勃勃的任务，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，虽然进展的速度加快了，但有时我们需要后退一步，提出一个问题，我们的计算到底有多可靠？ 这一提议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中定性现象的存在，例如偏微分方程组、变分方程和滞后型微分方程，这些数学模型是成功的广泛使用的数学模型，描述、解释和预测物理、化学、生物和经济等领域的现象。更准确地说，我们有兴趣回答以下每一个问题。我们能从数学上证明用纳维-斯托克斯方程或一个密切相关的流体动力学模型计算的解的可靠性吗？对于由细胞生物学问题引起的困难的、非线性的分子动力学模型，我们能承担同样的研究计划吗？更广泛地说，我们可以开发算法来严格地和自动地构造给定无限维非线性动力系统的不变集吗？最后，我们能否使用计算方法来证明数学领域中长期存在的公开问题，就像变分和延迟微分方程一样广泛？

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?