Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques introduces the reader to advanced concepts in stochastic modelling, rooted in an intuitive yet rigorous presentation of the underlying mathematical concepts. A particular emphasis is placed on illuminating the underpinning Mathematics, and yet have the practical applications in mind. The reader will find valuable insights into topics ranging from Social Sciences and Particle Physics to modern-day Computer Science with Machine Learning and AI in focus. The book also covers recent specialised techniques for notorious issues in the field of stochastic simulations, providing a valuable reference for advanced readers with an active interest in the field.

Features

• Self-contained, starting from the theoretical foundations and advancing to the most recent developments in the field
• Suitable as a reference for post-graduates and researchers or as supplementary reading for courses in numerical methods, scientific computing, and beyond
• Interdisciplinary, laying a solid ground for field-specific applications in finance, physics and biosciences on common theoretical foundations
• Replete with practical examples of applications to classic and current research problems in various fields.

Kurt Langfeld is Professor of Theoretical Physics and Director of the research Centre for Mathematical Sciences at Plymouth University. His fields of expertise are the numerical methods for simulating Quantum Field Theories and for Particle Physics. In 1999, he completed his "Habilitation" and was awarded the Venia Legendi. In 2005, he became Professor for Theoretical Physics at the University of Tübingen. In 2006, he joined the Particle Physics group at Plymouth University, where, in April 2012, he became full Professor for Theoretical Physics. He has served as a reviewer for the Engineering and Physical Sciences Research Council (EPSRC), the Austrian council FWF and for the Swiss National Supercomputing Center (CSCS). He regularly reviews manuscripts submitted to the high-profile journals in particle physics and has published about 100 papers in these journals.

Biagio Lucini is Professor and Head of the Mathematics Department at Swansea University. Before joining Swansea University in 2005, he was a Postdoctoral Fellow at the Rudolf Peierls Centre for Theoretical Physics (Oxford University, UK) from 2000 to 2003 and at the Theoretical Physics Institute of ETH (Zurich, Switzerland) from 2003 to 2005. He has been a Marie Curie Fellow (2001-2003) and a Royal Society University Research Fellow (2005-2013). His fields of activity are Monte Carlo calculations in Statistical Mechanics and Particle Physics. In particular, his interests are in Phase Transitions and Critical Phenomena, including efficient algorithms for simulations near criticality using High Performance Computing architectures. He is referee for various high-profile international journals and funding agencies and author of more than 100 papers, with around 3000 citations.

Massimo D'Elia is currently Associate Professor in Theoretical Physics at the Physics Department of the University of Pisa. He has been Postdoctoral Fellow at the University of Cyprus and at ETH, Zurich, and Assistant Professor at the Physics Department of the University of Genoa. He is an expert in Lattice Gauge Theories and their implementation on HPC infrastructures. He has obtained various achievements in the study of QCD at finite temperature and density and in external backgrounds, developing innovative computational approaches to the study of QCD in extreme conditions. He has served as a reviewer for major scientific journals in the field and for national funding programs (NSF). He is author of one textbook (with C.M. Becchi, "Introduction to the Basic Concepts of Modern Physics", Springer, 2007, 2010, 2016) and of around 180 papers, with more than 4000 citations

Stochastic phenomena play a central role in various scientific disciplines and underpin applications in popular industrial sectors. The purpose of the book is to introduce the reader to advanced concepts in the analysis of stochastic models starting from a detailed, intuitive and yet rigorous presentation of basic concepts. A special emphasis will be placed on problem solving and numerical implementations, with detailed solutions to all of the results and source code in the C programming language provided. The book will also cover recent specialised techniques for popular problems, providing a valuable reference for advanced readers with an active interest in the field.

Introduction. Random numbers and probability distribution. Central limit theorem. Beyond the Normal distribution. Exercises. Random walks. Random walk as a Markov process. Random walks in 1 and 2 dimensions. Levy flight. Random walks with potentials. Exercises. Monte Carlo methods. Objectives and concepts. Monte-Carlo integration. Statistical models. Introduction to Statistical Mechanics. The 2D Ising model. The 3-state Potts model in 3D: Finite size scaling. Gauge theories as statistical systems. Advanced MC simulation techniques. Langevin simulations. Hybrid Monte-Carlo simulations. Non-local Monte-Carlo update. Multicanonical simulations. Micro-canonical simulations. Density-of-States techniques. Exercises and computer experiments. From Statistical Systems to Quantum Field Theory. The O(2) model. SU(2) Yang-Mills theory. SU(N) gauge theories. Current challenges in Monte-Carlo Simulations. Sign- and overlap problems. Theories with sign problems. Topological modes and ergodicity problems. Appendix: High-Performance Computing