多尺度模型的基本原理

內(nèi)容概要

　　《數(shù)學(xué)與現(xiàn)代科學(xué)技術(shù)叢書6：多尺度模型的基本原理（英文版）》系統(tǒng)介紹有關(guān)多尺度建模的基本問題，主要介紹其基本原理而非具體應(yīng)用。前四章介紹有關(guān)多尺度建模的一些背景材料，包括基本的物理模型，例如，連續(xù)統(tǒng)力學(xué)、量子力學(xué)，還包括一些多尺度問題中常用的分析工具，例如，平均方法、齊次化方法、重正規(guī)化群法、匹配漸近法等，同時，還介紹了運用多尺度思想的經(jīng)典數(shù)值方法。接下來介紹一些更前沿的內(nèi)容：多物理模型的實例，即明確使用多物理漸近的分析模型，當(dāng)宏觀經(jīng)驗?zāi)Ｐ筒蛔銜r，借助微觀模型，使用數(shù)值方法來獲取復(fù)雜系統(tǒng)的宏觀行為規(guī)律，使用數(shù)值方法將宏觀模型和微觀模型結(jié)合起來，以便更好地解決局部奇點、虧量及其他問題；最后一部分主要介紹三類具體問題：帶多尺度系數(shù)的微分方程、慢動力和快動力問題以及其他特殊問題。

書籍目錄

《數(shù)學(xué)與現(xiàn)代科學(xué)技術(shù)叢書》序
Preface
Chapter 1 Introduction
Chapter 2 Analytical Methods
Chapter 3 Classical Multiscale Algorithms
Chapter 4 The Hierarchy of Physical Models
Chapter 5 Examples of Multi-physics Models
Chapter 6 Capturing the Macroscale Behavior
Chapoter 7 Resolving Local Events or Singularities
Chapter 8 Elliptic Equations With Multiscale Coefficients
Chapter 9 Problems With Multiple Time Scales
Chapter 10 Rare Events
Chapter 11 Some Perspectives
《數(shù)學(xué)與現(xiàn)代科學(xué)技術(shù)叢書》已出版書目

章節(jié)摘錄

　　Chapter 1　　Introduction　　1.1 Examples of multiscale problems　　Whether we explicitly recognize it or not, multiscale phenomena are part of our daily lives. We organize our time in days, months and years, as a result of the multiscale dynamics of the solar system. Our society is organized in a hierarchical structure, from towns to states, countries and continents. Such a structure has its historical and political origin, but it is also a re°ection of the multiscale geo- graphical structure of the earth. Moving into the realm of modeling, an important tool for studying functions, signals or geometrical shapes is to decompose them according to their components at di?erent scales, as is done in Fourier or wavelet expansion. From the viewpoint of physics, all materials at the microscale are made up of the nuclei and the electrons, whose structure and dynamics are responsible for the macroscale behavior of the material, such as transport, wave propagation, deformation and failure.　　In fact, it is not an easy task to think of a situation that does not involve any multiscale characteristics. Therefore, speaking broadly, it is not incorrect to say that multiscale modeling encompasses almost every aspect of modeling. However, with such a broad view, it would be impossible to carry out a serious discussion in any kind of depth. Therefore we will adopt a narrower viewpoint and focus on a number of issues for which the multiscale character is the dominating issue and is exploited explicitly in the modeling process. This includes analytical and numerical techniques that exploit the disparity of scales, as well as multi-physics problems. Here \multi-physics problems"" is perhaps a misnomer, what we have in mind are problems that involve physical laws at di? erent levels of detail, such as quantum mechanics and continuum models. We will start with some simple examples.　　1.1.1 Multiscale data and their representation　　A basic multiscale phenomenon comes from the fact that signals (functions, curves, images) often contain components at disparate scales. One such example is shown in Figure 1.1 which displays an image that contains large scale edges as well as textures with small scale features. This observation motivated the decompo- sition of signals into di?erent components according to their scales. Classical examples of such decomposition include the Fourier decomposition and wavelet decomposition[20].　　……

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