Lectures on Monte Carlo Theory

2025
Buch

Verfasst von: Paweł Lorek; Tomasz Rolski
Buchreihe: Probability Theory and Stochastic Modelling
Verlag: Springer Nature Switzerland
Enthalten in: Springer Professional "Wirtschaft+Technik"; Springer Professional "Wirtschaft"

Über dieses Buch

Dieses Buch stellt eine breite Palette von Berechnungstechniken vor, die auf wiederholten Zufallsstichproben beruhen, weithin bekannt als Monte-Carlo-Methoden und manchmal als stochastische Simulation. Diese Methoden führen Ideen aus Wahrscheinlichkeitstheorie, Statistik, Informatik und statistischer Physik zusammen und bieten Werkzeuge zur Lösung von Problemen in Bereichen wie Operationsforschung, Biotechnologie und Finanzwesen. Zu den Themen gehören die Erzeugung und Analyse von Pseudorandom-Zahlen (die echte Zufallszahlen auf einem Computer nachahmen sollen), das Design und die Rechtfertigung von Monte-Carlo-Algorithmen sowie fortgeschrittene Ansätze wie die Markov-Kette Monte Carlo und stochastische Optimierung. Im Gegensatz zu deterministischen numerischen Methoden ist das Ergebnis eines Monte-Carlo-Algorithmus selbst zufällig - und man braucht die Werkzeuge der Wahrscheinlichkeit und Statistik, um diese Ergebnisse sinnvoll zu interpretieren. Die theoretischen Grundlagen, insbesondere das Gesetz der großen Zahlen und der zentrale Grenzwertsatz, werden mit praktischen Algorithmen kombiniert, die sowohl die Stärken als auch die Feinheiten der stochastischen Simulation aufzeigen. Das Buch enthält zahlreiche theoretische und rechnerische Übungen. Jedes Kapitel enthält Schritt-für-Schritt-Algorithmen, illustrierte Beispiele und Ergebnisse, die durch numerische Berechnungen, Tabellen und eine Vielzahl von Diagrammen und Abbildungen dargestellt werden. Sämtlicher Python-Code, der zur Erstellung dieser Ergebnisse verwendet wurde, ist öffentlich zugänglich, was es dem Leser ermöglicht, Simulationen selbst zu reproduzieren und zu erforschen. Die in erster Linie für Doktoranden und Forscher konzipierte Ausstellung konzentriert sich auf Kernkonzepte und intuitives Verständnis und vermeidet übermäßigen Formalismus. Das Buch eignet sich sowohl zum Selbststudium als auch als Kurstext und bietet einen klaren Weg von grundlegenden Prinzipien zu modernen Anwendungen.

Mit KI übersetzt

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Monte Carlo methods are a broad class of computational techniques based on repeated random sampling—known as replications—to compute numerical results. This chapter introduces both Monte Carlo and quasi-Monte Carlo methods, comparing their convergence rates and practical uses. Through fundamental examples such as the random walk, travelling salesman problem, and Monte Carlo integration, readers are guided from the basic principles to hands-on computational techniques. Key mathematical ideas—such as the law of large numbers and central limit theorem—are briefly discussed to illustrate the behavior of random walks and simulation results. This chapter sets the groundwork for the advanced topics explored in later chapters.

Paweł Lorek, Tomasz Rolski

Chapter 2. The Theory of Generators

Abstract

Randomness is an important, sometimes crucial, element of computational science, enabling simulations, cryptographic protocols, and statistical modeling across diverse fields. While physical randomness (e.g., from quantum processes) is theoretically ideal, it is rarely practical in computational settings.

Paweł Lorek, Tomasz Rolski

Chapter 3. Generating Random Variables

This chapter surveys methods for generating random variables with a given distribution function using random numbers. Both classical and advanced techniques are presented—including the inverse transform, acceptance-rejection, and composition methods—for a variety of basic distributions such as the normal, Poisson, and Gamma distributions, as well as normal vectors.

The chapter also discusses practical algorithms for generating uniform distributions on various geometric objects, such as subsets of \(\mathbb {R}^d\), spheres, ellipses, d-balls, and ellipsoids. In addition, methods for generating copulas are introduced, enabling the construction of dependent random vectors with prescribed marginal distributions.

Throughout, step-by-step algorithms, Python code, and illustrative figures and tables are provided to highlight both the theoretical background and practical aspects of simulation. By the end of the chapter, readers will be equipped with tools for simulating a wide range of random variables needed in Monte Carlo methods.

Paweł Lorek, Tomasz Rolski

Chapter 4. Simulation Output Analysis: Independent Replications

Abstract

This chapter is devoted to planning and analyzing the results of simulation experiments based on independent replications. A fundamental formula—derived from the central limit theorem—relates the number of replications required to achieve a desired precision with a given confidence level and the variance of the estimator. The chapter introduces the concepts of pilot and actual simulations, and discusses their role in effective simulation planning.

Key topics include the estimation of means, probabilities, and quantiles, construction of confidence intervals, and the assessment of bias, variance, and mean square error in simulation estimators. Special attention is given to evaluating the accuracy of simulation results, including a detailed discussion of the distinction between absolute and relative error and their practical implications for interpreting outcomes. Probabilistic tools such as the Chernoff bound are introduced for quantifying errors and supporting decision-making.

Throughout, practical examples, algorithms, and Python code are provided to illustrate key concepts and support reproducibility. By the end of the chapter, readers will have a solid foundation for evaluating the reliability and precision of Monte Carlo simulation outputs based on independent replications.

Paweł Lorek, Tomasz Rolski

Chapter 5. Variance Reduction Techniques

This chapter presents a comprehensive overview of variance reduction techniques, which are essential for improving the efficiency and accuracy of Monte Carlo simulations. Key methods such as stratified sampling, antithetic variates, common random numbers, control variates, conditional Monte Carlo, and importance sampling are explained and illustrated with practical examples.

Special attention is given to the cross-entropy method and its applications in rare event simulation. The chapter discusses the theoretical motivation behind each technique, their implementation in Python, and the impact on estimator variance and computational cost. Applications in fields such as finance and actuarial science are highlighted, demonstrating the practical benefits of variance reduction in real-world problems.

Throughout, readers are provided with step-by-step algorithms, illustrative figures and tables, and publicly available Python code, enabling them to implement and compare variance reduction methods in their own simulation studies.

Paweł Lorek, Tomasz Rolski

Chapter 6. Markov Chain Monte Carlo Methods

This chapter introduces Markov chain Monte Carlo (MCMC) methods—a powerful class of algorithms for sampling from complex probability distributions by constructing Markov chains with the desired stationary distribution. After reviewing the fundamental concepts of Markov chains, including stationarity, irreducibility, and detailed balance, the chapter presents the Metropolis, Metropolis–Hastings, and Gibbs sampling algorithms with practical guidance for their construction and implementation.

Applications to classical problems such as graph coloring, hard-core models, the Ising model, and approximate counting in combinatorial problems are discussed, along with an introduction to perfect simulation techniques and an example from computational biology: the motif-finding problem. The chapter also addresses stable simulation schemes and practical criteria for assessing convergence in simulations.

Throughout, step-by-step algorithms, illustrative figures, and publicly available Python code are provided to support understanding and practical use. By the end of the chapter, readers will be equipped with both the theoretical insight and practical tools to apply MCMC methods in a range of scientific and statistical applications.

Paweł Lorek, Tomasz Rolski

Chapter 7. Stochastic Optimization

Abstract

This chapter builds on the foundations of Markov chain Monte Carlo methods from the previous chapter, introducing key modifications and enhancements for solving complex discrete optimization problems. The focus is on simulation-based algorithms for combinatorial optimization, with applications to substitution ciphers, the travelling salesman problem, and the knapsack problem.

Special attention is given to simulated annealing—particularly for the travelling salesman problem—and to the cross-entropy method for rare event estimation and optimization. The chapter also introduces heuristic methods based on locally informed proposals, which represent a significant advance in efficiently exploring complex state spaces in combinatorial problems.

Throughout, practical implementation is emphasized, with algorithmic steps, illustrative examples, and publicly available Python code provided to support hands-on learning. By the end of the chapter, readers will understand how techniques such as simulated annealing, locally informed proposals, and cross-entropy-based algorithms can substantially enhance the effectiveness of stochastic optimization in challenging discrete settings.

Paweł Lorek, Tomasz Rolski

Chapter 8. Simulation of Queues and Related Models

Abstract

This chapter introduces the simulation of queueing systems and related stochastic models, providing both theoretical foundations and practical methods for analyzing complex systems where analytical solutions are often unavailable. Key applications include classical models such as the repairman problem, Markovian queues (M/M/1, M/M/c, M/G/\(\infty \)), GI/G/1 and GI/G/c systems, as well as telecommunication networks and random access protocols like ALOHA.

The chapter covers the modeling and simulation of arrival processes—focusing on the Poisson process (homogeneous, non-homogeneous, and on general state spaces)—and renewal processes as input models for queueing systems. The principles of discrete-event simulation are explained and illustrated with examples, highlighting event-driven methods and their efficiency in simulating the evolution of systems such as queues and the repairman problem.

Emphasis is placed on the study of system characteristics under steady-state conditions, including average queue length, waiting time, system utilization, and other key performance measures. Both stable and unstable systems are discussed, with practical guidance on output analysis, confidence intervals, and error assessment. The concept of asymptotic variance is introduced as the natural analogue to the variance estimated from independent replications when analyzing processes with continuous sample paths.

By the end of the chapter, readers will be equipped to simulate, analyze, and interpret results for a broad class of queueing and related models, gaining insight into the interplay between theoretical models and empirical, simulation-based solutions.

Paweł Lorek, Tomasz Rolski

Titel: Lectures on Monte Carlo Theory
Verfasst von: Paweł Lorek
Tomasz Rolski
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-032-01190-9
Print ISBN: 978-3-032-01189-3
DOI: https://doi.org/10.1007/978-3-032-01190-9

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. The Theory of Generators

Chapter 3. Generating Random Variables

Chapter 4. Simulation Output Analysis: Independent Replications

Chapter 5. Variance Reduction Techniques

Chapter 6. Markov Chain Monte Carlo Methods

Chapter 7. Stochastic Optimization

Chapter 8. Simulation of Queues and Related Models

Backmatter