Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques(Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)

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.

Table of Contents:
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.

About the Author :
Massimo D’Elia is Professor of Theoretical Physics at the Physics Department of the University of Pisa. Alumnus of the Scuola Normale Superiore in Pisa, he got his PhD in Physics from Pisa University in 1998. 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, with interests also in Quantum Computing and numerical approaches to Quantum Gravity. He has obtained various achievements, especially in the study of the Phase Diagram of strong interactions. He has served as a reviewer for major scientific journals and funding agencies. He is author of two textbooks and more than 200 papers. Kurt Langfeld is Professor of Theoretical Physics and Head of School of Mathematics at the University of Leeds, England. His work in numerical methods for simulating Quantum Field Theories and Particle Physics is widely respected, with over 180 articles published in international journals. In 1991, he was awarded a PhD in Theoretical Physics from Technical University of Munich. He went on to serve as Researcher and Lecturer at the University of T¨ubingen, Germany from 1991 to 2006; during this time he also enjoyed research visits to CEA, Saclay (Paris) and KIAS (Seoul). In 1999, he achieved the highly esteemed Venia Legendi award at the University of Tubingen. 2005 saw him become Professor for Theoretical Physics at the University of Tubingen, before moving on to Plymouth Univeristy as part of their Particle Physics group in 2006. April 2012 saw him assume full Professorship for Theoretical Physics at Plymouth and 2016 saw him become Professor and Head of Department of Mathematical Sciences at the University of Liverpool. In 2020, he took up his current role as Head of School of Mathematics at the University of Leeds. He has dedicated himself to giving back to the community; he has been an active reviewer for the Engineering and Physical Sciences Research Council (EPSRC), Austrian council FWF, Swiss National Supercomputer Center (CSCS). He is a member of the Parliament Scientific Committee, devoted trustee of the University of Liverpool Maths School (ULMaS) and part of the Steering Group opening an University sponsored Mathematics specialist 6th form in Leeds. Biagio Lucini is a Fellow of the Learned Society of Wales and a Professor of Mathematics at Swansea University. He was awarded a Ph.D. in Theoretical Particle Physics by Scuola Normale (Pisa, Italy) in 2000. 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. Fellowship and awards he has received include a Royal Society Wolfson Merit Award (2017-2022) and a Leverhulme Research Fellowship (2020-2022). His research activity is centred around Monte Carlo calculations in Statistical Mechanics and Particle Physics. In particular, his interests are in Phase Transitions and Critical Phenomena, including machine learning approaches and efficient algorithms for simulations near criticality using High Performance Computing architectures. To date, his scientific contributions have resulted in over 180 research papers.

Review :
“The collaborative work of theoretical physics professors Massimo D'Elia, Kurt Langfeld, and Biagio Lucini, "Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques" from CRC Press is exceptionally well organized and presented, making it an ideal and unreservedly recommended textbook for college/university library instructional reference collections, as well as Computer Science, Physics, and AI technology curriculums.” —Midwest Book Review