Recent Advances in Monte Carlo Methods

What is it about?

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. This book will aim to discuss some important and fundamental aspects of Monte Carlo methods and will explore their use to solve a large array of problems, as we shall see. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, physical sciences, engineering and technology, computer science, numerical analysis, scientific computing, and science in general.

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Why is it important?

This book is titled Recent Advances in Monte Carlo Methods. It gives me great pleasure to introduce as well as to discuss, to learn, to solve, to teach, and to work with Monte Carlo Methods. In this book we discuss some fundamental aspects of the theory of these Monte Carlo methods and explore their use to solve a large array of problems. Therefore, we will treat in the current manuscript many topics discussing these random techniques. Moreover, Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. Furthermore, in physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. Finally, this volume will be intended to be an illustration of the use of Monte Carlo Methods when applied to solve specific problems. Thus, it will intend to be a deep treatment to this exciting, profound, and modern field of mathematics and knowledge. This book will aim to discuss some fundamental aspects of applied Monte Carlo Techniques and as we shall see. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in pure and applied mathematics, physical sciences, engineering and technology, computer science, numerical analysis, scientific computing, and science in general.

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