Multiscale multilevel iterative substructuring

PI将开发新的算法，可用于模拟现实世界油藏模型中的多孔介质流动。多孔介质中流动的模拟在许多领域都有应用，如水管理、石油和天然气回收、二氧化碳(CO2)封存和核废料处理等。由于几个方面的原因，底层的数学模型和高效的数值模拟是具有挑战性的。储集层通常非常大，所以离散化的数学模型会产生具有数亿个未知数的方程组，它们具有不规则的结构，这使得模型的几何形状变得复杂，并且它们由地质特性显著不同的材料组成，这在模型中转化为系数在几个数量级上的跳跃。此外，地质构造还经常含有改变有效渗透率的裂缝，因此需要准确地将其纳入数值模型。例如，花岗岩中的水流是由具有不同拓扑结构和大小的洞穴、洞穴和裂隙组成的复杂系统传导的，花岗岩是适合存放核废料的场所之一。或者，裂缝可能是由工程活动造成的，例如用于开采天然气的水力压裂(也称为水力压裂)。PI将开发新的算法来求解鞍点线性系统，将数值提升技术与并行区域分解迭代求解器相结合。多尺度方法和区域分解方法有很多方面已经被很好地理解，但现有方法的主要缺点是没有充分利用它们的潜力，在求解器的设计中利用多尺度现象，从而导致它们的效率低下。多尺度方法实际上也常常只由两个尺度组成，而在多孔介质中通常有许多尺度。与此同时，多核体系结构、网络、高端计算机和大型数据存储的进步正在引领一个高性能并行和分布式仿真的新时代。当然，随着这些新功能的出现，计算和系统建模也面临着新的挑战。该项目的目标是开辟解决这些问题的新途径。特别是，PI将开发允许多个尺度的多尺度方法，并使用提升尺度算法来为迭代求解器构建多级预条件。因此，该方法的组件被循环使用，从而显著降低了计算成本。此外，这种方法可以递归应用，因此自然提供了多层次的多尺度潜力，而不像许多传统的多尺度方法实际上只包括两个尺度。对超大型问题的多尺度和多层次方法设计问题的理解最终将有助于开发适合在未来亿级超级计算机上实现的下一代并行迭代求解器。

The PI will develop novel algorithms that can be used for the simulation of flow through porous media in real-world reservoir models. The simulation of flow in porous media finds applications in a number of areas, such as water management, oil and gas recovery, carbon dioxide (CO2) sequestration, and nuclear waste disposal, to name a few. The underlying mathematical models and efficient numerical simulation is challenging due to several aspects. The reservoirs are typically very large, so the discretized mathematical model leads to systems of equations with hundreds of millions of unknowns, they have irregular structure, which complicates the model geometry, and they consist of materials that significantly differ in geological properties, which translates in the model to jumps in coefficients over several orders of magnitude. Moreover, the geological formations quite often also contain fractures that alter the effective permeabilities, and therefore need to be be accurately incorporated into the numerical model. For example, the flow of water in granite rock, which represents one of the suitable sites for nuclear waste deposit, is conducted by the complex system of vugs, cavities and fractures with various topology and sizes. Alternatively, the fractures might result from the engineering activities, for example hydraulic fracturing (also known as fracking) used for the extraction of natural gas.The PI will develop novel algorithms for solving saddle-point linear systems combining numerical upscaling techniques with parallel, domain decomposition iterative solvers. There are many aspects of multiscale and domain decomposition methods that are quite well understood, but the major drawback of current methodologies is that they do not take full advantage of their potential by using the multiscale phenomena in the design of the solvers, which results in their inefficiency. Multiscale methods also frequently consist in fact only of two scales, whereas in a porous medium there are typically many scales. At the same time, advances in multicore architectures, networking, high end computers, and large data stores, are ushering in a new era of high performance parallel and distributed simulations. Naturally, with these new capabilities come new challenges in computing and system modeling. The goal of this project is to open new avenues to tackle these issues. In particular, the PI will develop multiscale methods that allows for a multiple of scales, and uses the upscaling algorithm to build a multilevel preconditioner for the iterative solver. The components of the method are thus recycled, which significantly decreases the computational cost. Moreover, this approach can be applied recursively and thus naturally offers a multilevel multiscale potential, unlike many traditional multiscale approaches that consist in fact of only two scales. It is expected that understanding of the issues related to design of multiscale and multilevel methods for extremely large problems will ultimately contribute to development of the next generation of parallel iterative solvers suitable for implementation on future exascale supercomputers.