Can we make long-term predictions?

该项目将包括证明统一的时间误差界的近似随机微分方程的解决方案，从那些通过多尺度方法获得。为了做到这一点，在马尔可夫过程，马尔可夫半群理论，随机分析，数值方法和概率的常用工具将被利用。在模拟生物过程和多尺度方法时，多尺度方法特别重要。了解我们可以在近似值上获得的误差范围将允许更好地量化不确定性，以及更严格的预测。在时间上统一的边界将使我们能够理解何时我们的近似值可以在所有时间保持良好，何时它们会随着时间的推移而恶化。也就是说，我们已经证明了许多过程和模型的近似是一个很好的，直到某个时间T。该项目的新颖性在于获得与时间无关的边界，以允许长期近似和更大的预测确定性。该项目的总体目标是产生一种新的理论，包括实用标准，以了解何时可以通过数值方案或其他程序近似给定的随机动态，并且误差不会随时间增加。该项目包括两个“分项目”。一个考虑通过数值方案产生的近似值，另一个考虑通过其他程序产生的近似值;在后一种情况下，它将集中在平均或均匀化程序。该项目将不专注于具体的应用程序，但这些问题的灵感来自于数学生物学的应用程序，特别是群集，在工程，材料科学，物理学等方面的应用程序。

The project will consist of proving uniform in time error bounds on approximations to the solutions of Stochastic differential equations, beginning with those obtained through multiscale methods. In order to do this, common tools in Markov processes, Markov semigroup theory, stochastic analysis, numerical methods, and probability will be utilised. Multiscale methods are particularly relevant when modelling biological processes and multiscale methods. Understanding the error bounds that we can obtain on our approximations will allow better quantification of uncertainty, as well as tighter predictions. The bounds being uniform in time will allow an understanding of when we can expect our approximations to hold well for all time, and when they will deteriorate over time.Current error bounds often hold for finite-time windows. That is, we have proven for many processes and models that the approximation is a good one up until some time T. The novel nature of the project is in obtaining bounds that are independent of time, to allow for long term approximations and greater certainty of predictions.The overarching goal of this project is to produce a novel theory, including practical criteria, to understand when a given random dynamics can be approximated - either via numerical schemes or via other procedures - with an error which does not increase in time. This project contains two "sub-projects". One considering approximations produced via numerical schemes and one considering approximations produced via other procedures; in the latter case it will in concentrate on averaging or homogenization procedures.The project will not be focussing on applications specifically but these problems are inspired by applications to mathematical biology, swarming in particular, with a number of applications in engineering, material science, physics etc.