随机数值方法：学生与科学家入门指南

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"这本书《Stochastic Numerical Methods_ An Introduction for Students and Scientists》主要介绍了概率理论和随机过程概念在数值方法中的应用。它不仅适用于概率和统计领域，而且可以用于解决高维积分和求和等纯确定性问题。书中的内容源自作者在多个大学和暑期学校教授的硕士课程，特别是IFISC（跨学科物理与复杂系统研究所）的物理复杂系统硕士课程。本书面向几乎任何科学领域的研究生以及不熟悉这些特定数值方法但需要在研究中应用它们的资深科学家。书中涵盖了三大主题：蒙特卡洛抽样（第2-5章和第10章）、随机微分方程的轨迹生成（第6、7和11章）和主方程的数值解（第8和9章），并强调了这些方法之间的联系。" 在深入探讨这些知识点之前，先简单介绍下与之相关的其他书籍，如《The Complexity of Dynamical Systems》、《Stochastic Simulation and Applications in Finance with MATLAB Programs》、《WIE Probability and Stochastic Processes》、《MATLAB: An Introduction with Applications》和《Introduction to Stochastic Models》等，它们都是对随机过程和概率理论在不同领域应用的深入研究。现在我们详细讨论书中的关键知识点： 1. **蒙特卡洛方法**：这是一种基于概率统计的数值计算方法，通过大量的随机抽样来求解复杂问题。书中第2-5章和第10章详细介绍了这一方法，包括基本的随机数生成、模拟和平均值估计，以及更高级的算法如Metropolis-Hastings算法和 Importance Sampling。 2. **随机微分方程（SDEs）**：在第6、7和11章中，作者讲解了如何生成SDE的轨迹，这对于模拟物理、化学和金融等领域中随机过程的动态行为至关重要。这些章节会涵盖Itô和Stratonovich积分、Euler-Maruyama方法以及其他数值解法。 3. **主方程的数值解**：主方程是描述离散时间系统演化的重要工具，第8和9章将介绍如何数值求解这些方程，这在理解复杂系统的演化和统计特性时非常有用。 4. **多学科应用**：书中强调了这些方法的广泛应用，无论是在物理、工程、化学还是金融学，都可以找到它们的身影。例如，在金融学中，蒙特卡洛模拟常用于风险分析和衍生品定价；在物理学中，SDEs则用于描述量子系统或混沌系统的行为。 5. **交叉连接**：书中努力揭示了这些不同方法之间的内在联系，帮助读者理解它们是如何协同工作的，以解决实际问题。 6. **MATLAB编程**：虽然没有直接提及，但考虑到其他相关书籍中提到了MATLAB在随机过程和模拟中的应用，可以推测这本书可能也涉及了使用MATLAB实现这些方法的实际示例，这对于学生和研究人员来说是非常实用的。通过学习这本书，读者不仅可以掌握基础的随机数值方法，还能了解到如何将这些方法应用于实际的科学问题中，从而提升其研究能力。

XIV Preface

here and there numerous references to the different authors who have contributed

to some aspect of the methods explained in the book. We feel that a book with a

clear pedagogical orientation, as this one, is different from a review, and the reader

should not be distracted by an excess of references and we apologize to those

authors who, having contributed to the ﬁeld, do not ﬁnd their work cited here.

The basic bibliographic sites as well as suggestions for further reading have been

included at the end of each chapter.

The goal of the book is to teach the reader to program the numerical algo-

rithms to do different tasks. To this end, we have included more or less complete

pieces of computer code. They are written in Fortran but they are, in the vast

majority of cases, simple enough that they can be understood by anyone with

a basic knowledge of programming in any language. The book is not intended

to teach programming or code optimization. Sometimes we provide a full pro-

gram, sometimes just some relevant lines of code. In any event, we do consider

the lines of code to be part of the text, an important part that has to be read

and analyzed in detail. Our experience indicates that full understanding is only

achieved when one does not only know what he or she wants to do but also

how this is implemented practically, and we do recommend the reader to imple-

ment and execute the computer programs along with the reading of the text.

In particular, the section of the code that we consider absolutely essential has

been framed in a box. It is not possible to reach a good understanding of the

numerical algorithms without the understanding of the lines of code in the

boxes.

We have included some exercises to complement the theory of the algorithms

explained at the end of each chapter. We have not used the exercises to introduce

difﬁcult or more advanced topics but the exercises are, in general, of the level

of the course and are given here so that the reader can test his or her level of

comprehension of the algorithms and theory of the main text.

Some material is left for the appendices. They cover either standard material

(generation of uniform random numbers, calculation of the correlation time of a

series and discrete Fourier transforms), some more specialized topics (generation

of Gaussian ﬁelds with a given correlation function, extrapolation techniques),

or an introduction to more advanced simulation methods (collective algorithms,

multicanonical simulations).

As we have said, the book is addressed to scientists of many disciplines, and

beyond the general framework we have included only a few speciﬁc applications.

In particular, in Chapter 5, we explain how to use the Monte Carlo sampling to

derive properties of physical systems near a phase transition. The reason for the

inclusion of this chapter is twofold. Historically, the ﬁeld of phase transitions has

made extensive use of the Monte Carlo methods (to the extent that some people

might wrongly think that this is the only ﬁeld of application). Moreover, it is the

ﬁeld of expertise of the authors, and we felt more conﬁdent explaining in detail

how one can use the Monte Carlo sampling to derive precisely the equation of state

and the critical exponents of some model systems of interest in Statistical Mechan-

ics, including the Ising and scalar models, than other examples. Nevertheless,

Review of probability concepts

In this chapter, we give a brief summary of the main concepts and results on

probability and statistics which will be needed in the rest of the book. Readers who

are familiar with the theory of probability might not need to read this chapter in

detail, but we urge them to check that effectively this is the case.

1.1

Random Variables

In most occasions, we cannot predict with absolute certainty the outcome of an

experiment (otherwise it might not be necessary to perform the experiment). We

understand here the word ‘‘experiment’’ in a broad sense. We can count the

number of electrons emitted by a 𝛽-radioactive substance in a given time interval,

determine the time at which a projectile hits its target or a bus reaches the station,

measure an electron’s spin, toss a coin and look at the appearing side, or have a

look through the window to observe whether it rains or not. We will denote by

E the set of possible results of the experiment. For the 𝛽-radioactive substance,

E ={0, 1, 2, …} is the set of natural numbers ℕ; the hitting times of the projectile

or the arrival times of the bus (in some units) both belong to the set of real

numbers E = ℝ; the possible outcomes of a measure of an electron’s spin are

E ={−ℏ∕2,ℏ∕2}; when tossing a dice, the possible results are E ={heads, tails};

and, ﬁnally, for the rain observation the set of results is E ={yes, no}. In all these

cases, we have no way (or no effective way) of knowing a priori which one of the

possible outcomes will be observed. Hence, we abandon the deterministic point of

view and adopt a ‘‘probabilistic’’ description in which subsets of results (which are

called ‘‘events’’) are assigned a number measuring their likelihood of appearance.

The ‘‘theory of probability’’ is the branch of mathematics that allows us to perform

such an assignation in a logically consistent way and compatible with our intuition

of how this likelihood of events should behave.

It is useful for the theory to consider that the set of results contains only numbers.

In this way, we can use the rules of calculus (add, multiply, differentiate, integrate,

etc.). If the results themselves are numbers (case of counting the number of

electrons, determine the time of impact of the projectile, etc.), this requires no

Stochastic Numerical Methods: An Introduction for Students and Scientists, First Edition.

ul Toral and Pere Colet.

 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Review of probability concepts

special consideration. In other cases (to observe whether it rains or not), we need

to label each result with a number. This assignation is arbitrary, but usually it

responds to some logics of the problem under consideration. For instance, when

tossing a coin, it might be that we win

¤1 every time heads show up and we lose

¤1 when tails appear. The ‘‘natural’’ identiﬁcation is +1 for heads and −1 for tails.

This assignation of a number to the result of an experiment is called a ‘‘random

variable.’’ Random variables are, hence, an application of the set of results to the

set of real numbers. This application maps each result of the experiment 𝜉 ∈ E

into one, and only one, number. The application need not be one-to-one, but it can

be many-to-one. For instance, if the experiment is to extract cards from a shufﬂed

deck, we could assign +2 to all hearts cards, +1 to spades, and 0 to diamonds and

clubs. It is customary to denote random variables by using a ‘‘hat’’ on top of its

name, say



𝐱,or



𝐲, or whatever name we choose for it. If we choose the name



𝐱,

the number associated with the result 𝜉 is



𝐱(𝜉)∈ℝ. This distinction between the

result of the experiment and the real number associated with it is important from

the mathematical point of view, but in many cases of interest they both coincide

because the result of the experiment is already a real number, 𝜉 = x, and it is

natural to deﬁne



𝐱(x)=x in a somewhat redundant notation.

In summary, a random variable



𝐱 is a real number that is obtained as a result of

an experiment.

The next step in the theory is to assign numbers called ‘‘probabilities’’ to the

possible results of the experiment or, equivalently, to the different values of the

random variable. The assignation should match our a priori expectations (if any)

about the likelihood (expected frequency of appearance) of the different outcomes.

For instance, when tossing a coin, it is natural (but not necessarily useful or

convenient) to assign a probability equal to 1∕2 to the appearance of heads,

such that P(heads)=1∕2 or, equivalently, to the random variable



𝐱(heads) taking

the value +1 (as assigned arbitrarily before), P(



𝐱 =+1)=1∕2. The assignation

of probabilities to events might follow some physical law (as in the case of the

radioactive substance, the Boltzmann law for the distribution of energies, or

the quantum mechanical postulates), might come after some lengthy calculation

(the probability of rain tomorrow), or might follow other arguments such as

symmetry (the probability of heads is equal to 1∕2), Jayne’s principle (based on

the extremization of the information function), and so on; however, whatever its

origin, we consider the assignation to be known. A typical consequence of the

theory is the calculation of probabilities for more or less complicated events: What

is the probability of obtaining ﬁve heads in a row if we toss a coin 10 times? What

is the probability that the next emission of an electron by the radioactive substance

occurs in the next 10 ms? and so on.

In practice, the assignation of probabilities to values of the random variable

is performed differently if the random variable is continuous (i.e., it can take

continuous values in a given interval



𝐱 ∈(𝛼, 𝛽) where 𝛼 can also be −∞ or 𝛽

can be +∞) or discrete (can take only a ﬁnite or inﬁnite numerable set of values



𝐱 ∈{x

, x

, …}). For example, the random variable counting the number of

times a coin must be tossed before heads appear can take an inﬁnite numerable

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