Active-dissipative nonlinear spatially extended media: Complexity, coarse-graining, multiscale analysis and numerical methods

空间扩展系统(SES)，即在大的或无界的区域上用偏微分方程描述的确定性或随机性的无限维动力系统，其特征长度和时间尺度的存在往往导致复杂的时空行为。SE经常作为各种自然现象和技术应用的数学模型出现。SES及其动力学的复杂性使得直接分析它们是非常困难的，如果不是不可能的话，无论是数学上的分析，还是在某些情况下，数字分析。因此，必须寻求对SES的低维描述，即产生能够捕捉特定应用的大部分(如果不是全部)基本动态特征并且更易于分析和数值研究的粗粒度模型。这项研究的主要目的是发展最先进的有效方法，对确定性和随机性的SES进行模式约简和粗粒化。

Spatially extended systems (SES), i.e. infinite dimensional dynamical systems described through partial differential equations deterministic or stochastic in large or unbounded domains, are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatio-temporal behavior. SES arise frequently as mathematical models of a large variety of natural phenomena and technological applications. The complexity of SES and their dynamics is such that it is very difficult, if not impossible to analyze them directly, either mathematically or, in several cases, numerically. It is imperative, therefore, to seek a low-dimensional description of SES, i.e. to produce coarse grained models that capture most, if not all of the essential dynamic features of the particular applications and which are much easier to study analytically and numerically. The primary aim of the proposed research is the development of state-of-the-art efficient methods for mode reduction and coarse-graining of SES, both deterministic and stochastic.