Development of time-parallel numerical integration algorithms using probabilistic methods with applications to magnetic fusion plasma.

研究背景-在许多科学分支中，从天体物理学到流行病学，数学和计算建模是至关重要的，通常涉及大型微分方程系统的数值积分。许多模拟都受到瓶颈的限制，这些瓶颈是在尝试长时间顺序地进行数值集成时出现的，导致它们在实时计算中变得难以处理。通过将计算分布到现代超级计算机中的许多处理器上，可以大大减少计算负载。虽然空间并行积分方法已经在文献中得到了很好的探索，但对时间维度的并行化的关注要少得多。该博士项目的目的是开发新的（或改进现有的）时间并行积分算法，使用随机和统计方法，以减少大规模微分方程系统的模拟时间。数字和统计之间的相互作用（称为不确定性量化）目前是一个非常开放的领域，正在开发的算法将需要足够通用，以适应正在解决的问题的复杂性。研究方法的新颖性-目前已知的大多数时间并行算法都是使用确定性方法制定，研究和分析的。在这个项目中开发的算法将包含一个随机度量，以便利用确定性生成的解决方案之间的统计差异。在此过程中，可以利用统计和机器学习方法来获得数值增益。潜在的影响，应用和好处- UKAEA可以使用成功设计的算法，以显着减少模拟运行时间，从而集成以前（计算）难以处理的磁融合等离子体时间范围。这些方法也不严格局限于数值融合问题。在气候模拟、天体物理学、药物研究等其他大规模集成问题的应用中，也可以实现同样的好处。通过开发基于统计的时间并行算法，并将其应用于磁融合等离子体，拟议的研究领域有助于EPSRC职权范围内的多个主题。即“数值分析”和“英国磁聚变研究计划”主题。研究领域;能源，数学科学外部合作伙伴- Culham聚变能源中心（英国原子能管理局的一部分）

The context of the research - In many branches of science, from astrophysics to epidemiology, mathematical and computational modelling is vital and often involves the numerical integration of large systems of differential equations. Many simulations are limited by bottlenecks that arise when attempting to numerically integrate over long time periods sequentially, causing them to become computationally intractable in real time. By distributing a calculation across the many processors present in modern supercomputers, the computational load can be reduced drastically. Whilst spatially-parallel integration methods have been well explored in the literature, much less attention has been devoted to parallelisation in the time dimension.The aims and objectives of the research - The purpose of this PhD project will be to develop new (or improve existing) time-parallel integration algorithms, using stochastic and statistical methods, in order to reduce simulation times for large-scale systems of differential equations. The interplay between numerics and statistics (known as uncertainty quantification) is currently a very open field and the algorithm(s) being developed will need to be general enough to adapt to the complexity of the problems being solved.The novelty of the research methodology - The majority of currently known time-parallel algorithms have been formulated, studied and analysed using deterministic methods. The algorithm(s) developed in this project will incorporate a measure of stochasticity in order to exploit the statistical differences between deterministically generated solutions. In doing this, statistical and machine learning approaches may be exploited for numerical gain.The potential impact, applications, and benefits - A successfully designed algorithm could be used by UKAEA in order to significantly reduce simulation run times and hence integrate over previously (computationally) intractable ranges of time for magnetic fusion plasmas. These methods are not strictly limited to numerical fusion problems either. The same benefits could be realised in applications to other large-scale integration problems in climate modelling, astrophysics, drug research and many more.How the research relates to the remit - By developing statistically-based time-parallel algorithms with applications to magnetic fusion plasma, the proposed area of research contributes to multiple themes under the EPSRC remit. Namely the 'Numerical analysis' and 'UK Magnetic Fusion Research Programme' themes.Research area; Energy, Mathematical SciencesExternal Partner - Culham Centre for Fusion Energy (part of the UK Atomic Energy Authority)