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Adaptive Multiscale Methods for Approximation and Preconditioning

用于逼近和预处理的自适应多尺度方法

基本信息

批准号：
EP/H043519/1
负责人：
Ivan Graham
金额：
$ 2.04万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH043519%2F1
关键词：
Adaptive Multiscale Methods Approximation Preconditioning

项目摘要

Multiscale modelling is ubiquitous in science, technology and social science. Models with multiple time or length scales arise, for example, in modelling pollutant transport in groundwater, turbulent fluid flow in reactor cooling systems, high-frequency waves in radar and sonar and atomistic-continuum models in material science. Because of the highly varying scales involved in multiscale problems, the accurate modelling of all scales is often outside the reach of even the largest supercomputers. A suitable goal of computation is then to compute a solution on the finest computationally affordable grid, in such a way that the accuracy is not polluted by the fine scales which remain unresolved. More precisely, the aim is to solve the multiscale problem on a coarse mesh in such a way that the error is of the same order (with respect to the number of degrees of freedom) as if the problem were smooth, and moreover, the accuracy does not degrade if the finest scale present in the model decreases. Such computational methods are often called ``robust''. There are two paradigms for the construction of robust numerical methods for multiscale problems. The first is to replace the multiscale problem with a (nearby) smooth problem and then solve the latter numerically. Examples of this approach include upscaling in porous medium flow and transport or the use of geometric theory of diffraction and ray-tracing in high-frequency wave propagation problems. The basic difficulty with this approach is that these approximations tend to be valid only when the fine scale small parameter is sufficiently small (in order to make some sort of averaging valid), and, moreover, their rigorous analysis requires simplifying assumptions (such as periodicity and scale separation in homogenization theory).An alternative approach is to devise problem-adapted numerical methods which are targeted to the type of multiscale behaviour arising in the particular application, and are capable of resolving it robustly on a coarse mesh. This usually involves replacing the (piecewise) polynomial approximations at the heart of classical numerical methods with problem-adapted bases which are better able to reflect the solution behaviour on coarse meshes. Examples of this type of approach in application areas include sub-grid scale modelling in large eddy simulation, and the modelling of localised convective storms in large-scale weather prediction software. This is a new collaboration between the PI and the proposed VF which will not take place without the requested EPSRC support. We will produce new results on methods of the second type. Our methods will be adaptive (i.e. the non-polynomial bases will be computed automatically, rather than designed in detail by the practitioner) and they will work well both in the presence of small lengthscale (small wavelength of data) as well as large contrast (large amplitude of data). We will test our methods on systems arising from problems with random data with small lengthscale and large variance (leading to small wavelength and large amplitude). We will also investigate the application of the same ideas in the design of robust preconditioners for conventional discretisations of multiscale problems including those which approximate equations describing high frequency wave phenomena.

多尺度建模在科学、技术和社会科学中普遍存在。例如，在模拟地下水中的污染物传输、反应堆冷却系统中的湍流流动、雷达和声纳中的高频波以及材料科学中的原子-连续模型时，出现了具有多个时间或长度尺度的模型。由于多尺度问题涉及的尺度千差万别，所有尺度的准确建模往往超出了最大的超级计算机的能力范围。一个合适的计算目标是在最精细的计算负担得起的网格上计算一个解，这样精度就不会被仍未解决的精细尺度所污染。更准确地说，目标是以这样一种方式解决粗网格上的多尺度问题，即误差具有相同的阶数(相对于自由度)，就好像问题是光滑的一样，而且，如果模型中存在的最细尺度减少，精度也不会降低。这种计算方法通常被称为“稳健的”。构造多尺度问题的稳健数值方法有两种范例。第一种方法是将多尺度问题转化为(邻近)光滑问题，然后对后者进行数值求解。这种方法的例子包括在多孔介质流动和传输中的放大，或者在高频波传播问题中使用几何绕射理论和光线追踪。这种方法的基本困难在于，只有当细尺度小参数足够小时，这些近似才是有效的(为了使某种平均有效)，而且，它们的严格分析需要简化假设(如齐化理论中的周期性和尺度分离)。另一种方法是设计针对特定应用中出现的多尺度行为的问题适应的数值方法，并能够在粗网格上稳健地解决它。这通常涉及用能够更好地反映粗网格上的解的行为的适合问题的基来代替经典数值方法的核心(分段)多项式近似。这类方法在应用领域的例子包括大涡模拟中的亚网格尺度模拟，以及大型天气预报软件中的局部对流风暴模拟。这是PI和提议的VF之间的新协作，如果没有所请求的EPSRC支持，这种协作将不会发生。我们将在第二类方法上产生新的结果。我们的方法将是自适应的(即非多项式基将自动计算，而不是由从业者详细设计)，并且它们将在小长度尺度(数据的小波长)以及大对比度(数据的大幅度)的存在下很好地工作。我们将在具有小长度尺度和大方差(导致小波长和大幅度)的随机数据问题产生的系统上测试我们的方法。我们还将研究同样的思想在多尺度问题的常规离散化中的鲁棒预条件设计中的应用，包括那些近似描述高频波现象的方程的问题。