Stochastic Methods in Scientific Computing

From Foundations to Advanced Techniques






The book

Edited by the Taylor & Francis Group (see here for the official material from the publisher), Stochastic Methods in Advanced Computing - From Foundations to Advanced Techniques, or, in short, SMSC, provides a fresh approach to computational probability and stochastic modelling, covering the basics of the subject with applications to topics in finance, physics and biosciences, and more specialised discussions on Statistical Mechanics, Monte Carlo Methods and Lattice Field Theory. In particular, the latter is treated using a Statostical Mechanics perspective, and does not require knowledge of Quantum Field Theory. The last chapter provides an overview of recent advances at the interface between Machine Learning and Statistical Mechanics. The book has three intended audiences:

  • An undergraduate or non-specialistic audience looking for an introduction to computational probability and stochastic modelling;
  • A postgraduate or more specialised audience seeking to deepen their knowledge of the subject;
  • An expert audience aiming to broaden their perspective on the most advanced material.

The book is self-contained. Excercises are provided to stimulate the reader to reflect on the learned material.


Purpose of this site

We have published this site as a companion resource to the SMSC book. Here we provide links to supplementary material such as code for the examples discussed in the book and solutions of the exercises proposed to the reader. This material will build over time, and we aim to updating regularly these pages as new resources become available. Additionally, we report here instructions on how to provide feedback and report issues with the material published in the book.


Table of Contents

The table of contents of SMSC is as follows:

  1. Random Numbers. 1.1. Random numbers and probability distribution. 1.2. Central limit theorem. 1.3. Beyond the Normal distribution. 1.4. Exercises.
  2. Random walks. 2.1. Random walk as a Markov process. 2.2. Random walks in 1 and 2 dimensions. 2.3. Levy flight. 2.4. Random walks with potentials. 2.5. Exercises.
  3. Monte Carlo methods. 3.1. Objectives and concepts. 3.2. Monte-Carlo integration. 3.3. Markov Chain Monte-Carlo. 3.4. Advanced Error Analysis Techniques. 3.5. Error estimate in the presence of autocorrelation. 3.6. Error estimate for non-trivial estimators: The Jackknife, and the Bootstrap. 3.7. Biased Estimators. 3.8. Exercises.
  4. Statistical models. 4.1. An introduction to thermodynamics. 4.2. From thermodynamics to statistical mechanics. 4.3. Phase transitions. 4.4. The Ising model. 4.5. An overview of other models. 4.6. Exercises.
  5. Advanced Monte-Carlo simulation techniques. 5.1. Hamiltonian (Hybrid) Monte-Carlo (HMC) simulations. 5.2. Non-local Monte-Carlo update. 5.3. Micro-canonical simulations. 5.4. Flat histogram methods. 5.5. The Linear Logarithmic Relaxation (LLR) method. 5.6. Exercises.
  6. From Statistical Systems to Quantum Field Theory. 6.1. Invitation: The O(2) model. 6.2. The Bridge to QFT: the Feynman path-integral. 6.3. Gauge Theories. 6.4. Adding fermion fields. 6.5. Exercises.
  7. Current challenges in Monte-Carlo Simulations. 7.1. Sign and overlap problems. 7.2. Introduction to overlap problems. 7.3. Estimating probability density functions.
  8. Data Analytics and Statistical Systems. 8.1. Model regression - L2 norm. 8.2. Gaussian Process. 8.3. Machine learning with graphs. 8.4. Emulation of statistical systems with Machine Learning. 8.5. Categorisation in statistical physics: Naive Bayes. 8.6. Machine learning classification of phase transitions.


Supplementary material

Solutions for selected exercises and Python code for some of the examples provided in the book can be found on GitHub, both in pdf and in jupyter notebook (ipynb) format.


Citing the book

If you need to refer to the SMSC book in your own work, for LaTeX, please use the BibTeX entry

@book{smscbook,
    title={Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques},
    author={D’Elia, Massimo and Langfeld, Kurt and Lucini, Biagio},
    year={2024},
    ISBN={9781498796330},
    publisher={CRC Press},
    URL={https://smscbook.github.io},
}


Sharing feedback

We encourage feedback from readers and appreciate contributions and support to improve the content of the book. We would be very grateful if you could report any issues you find and share any feedback you would like to share with us using the relevant github issue tracker.