Monte Carlo

‘It will be a strange score: people will either not like it at all, or will like it enormously’, prophesied Camille Saint-Saëns a few days before the premiere of Déjanire. The opera, first performed in Monte Carlo on 14 March 1911, is based on incidental music written in 1898 for the Béziers Arena. Fascinated by the subject, the composer soon wanted to give it a second, more ambitious life. He therefore conceived a mythological epic that inspired ‘powerfully evocative music’, according to Gabriel Fauré, who was struck by the impact of the choral writing. Yet the love drama that rends the heroine’s heart engenders wildly romantic duets and culminates in the public immolation of Hercules, set ablaze by the poisoned tunic offered to him by the fallen queen. This new Déjanire received high praise from the critics, who flocked to Monaco to see it. But the modernist path that French opera was taking at the time did not allow the work to survive the upheavals of the First World War. It would have been a shame to prolong this unjustified ostracism any longer.

Rendering photorealistic images is a costly process which can take up to several days in the case of high quality images. In most cases, the task of sampling the incident radiance function to evaluate the illumination integral is responsible for an important share of the computation time. Therefore, to reach acceptable rendering times, the illumination integral must be evaluated using a limited set of samples. Such a restriction raises the question of how to obtain the most accurate approximation possible with such a limited set of samples.

Monte Carlo simulation is now a well established method for studying semiconductor devices and is particularly well suited to highlighting physical mechanisms and exploring material properties. Not surprisingly, the more completely the material properties are built into the simulation, up to and including the use of a full band structure, the more powerful is the method. Indeed, it is now becoming increasingly clear that phenomena such as reliabil­ ity related hot-electron effects in MOSFETs cannot be understood satisfac­ torily without using full band Monte Carlo. The IBM simulator DAMOCLES, therefore, represents a landmark of great significance. DAMOCLES sums up the total of Monte Carlo device modeling experience of the past, and reaches with its capabilities and opportunities into the distant future. This book, therefore, begins with a description of the IBM simulator. The second chapter gives an advanced introduction to the physical basis for Monte Carlo simulations and an outlook on why complex effects such as collisional broadening and intracollisional field effects can be important and how they can be included in the simulations. References to more basic intro­ the book. The third chapter ductory material can be found throughout describes a typical relationship of Monte Carlo simulations to experimental data and indicates a major difficulty, the vast number of deformation poten­ tials required to simulate transport throughout the entire Brillouin zone. The fourth chapter addresses possible further extensions of the Monte Carlo approach and subtleties of the electron-electron interaction.

Der Text gibt eine Einführung in die Mathematik und die Anwendungsmöglichkeiten der Monte Carlo-Methoden und verwendet dazu durchgängig die Sprache der Stochastik. Der Leser lernt die Grundprinzipien und wesentlichen Eigenschaften dieser Verfahren kennen und wird dadurch in den Stand versetzt, dieses wichtige algorithmische Werkzeug kompetent einsetzen und die Ergebnisse interpretieren zu können.

This book is an illustration of the use of Monte Carlo methods applied to solve specific problems in mathematics, engineering, physics, statistics, and science in general.

This book provides a comprehensive overview of these methods, discussing their fundamental aspects and their applications in solving a large array of problems. It is a useful reference for scholars, researchers, and students in pure and applied mathematics, physical sciences, engineering and technology, computer science, numerical analysis, scientific computing, and the general sciences.

The role of Monte Carlo methods and simulation in all of the sciences has in­ creased in importance during the past several years. These methods are at the heart of the rapidly developing subdisciplines of computational physics, compu­ tational chemistry, and the other computational sciences. The growing power of computers and the evolving simulation methodology have led to the recog­ nition of computation as a third approach for advancing the natural sciences, together with theory and traditional experimentation. Monte Carlo is also a fundamental tool of computational statistics. At the kernel of a Monte Carlo or simulation method is random number generation. Generation of random numbers is also at the heart of many standard statis­ tical methods. The random sampling required in most analyses is usually done by the computer. The computations required in Bayesian analysis have become viable because of Monte Carlo methods. This has led to much wider applications of Bayesian statistics, which, in turn, has led to development of new Monte Carlo methods and to refinement of existing procedures for random number generation.

This book provides a general introduction to Sequential Monte Carlo (SMC) methods, also known as particle filters. These methods have become a staple for the sequential analysis of data in such diverse fields as signal processing, epidemiology, machine learning, population ecology, quantitative finance, and robotics.