The Monte Carlo Methods - Recent Advances, New Perspectives and Applications

What is it about?

This book is titled The Monte Carlo Methods - Recent Advances, New Perspectives and Applications. It illustrates the famous Monte Carlo Methods and the computer simulation of random experiments in different areas of science. As such, the book will be of interest to all scholars, researchers, and undergraduate and graduate students in mathematics and science in general. In applied mathematics, the name Monte Carlo is given to the method of solving problems by means of experiments with random numbers. This name, after the casino at Monaco, was first applied around 1944 to the method of solving deterministic problems by reformulating them in terms of a problem with random elements which could then be solved by large-scale sampling. But, by extension, the term has come to mean any simulation that uses random numbers. The development and proliferation of computers has led to the widespread use of Monte Carlo methods in virtually all branches of science, ranging from nuclear physics (where computer-aided Monte Carlo was first applied) to astrophysics, biology, engineering, medicine, operations research, and the social sciences. The Monte Carlo Method of solving problems by using random numbers in a computer – either by direct simulation of physical or statistical problems or by reformulating deterministic problems in terms of one incorporating randomness – has become one of the most important tools of applied mathematics and computer science. A significant proportion of articles in technical journals in such fields as physics, chemistry, and statistics contain articles reporting results of Monte Carlo simulations or suggestions on how they might be applied. Some journals are devoted almost entirely to Monte Carlo problems in their fields. Studies in the formation of the universe or of stars and their planetary systems use Monte Carlo techniques. Studies in genetics, the biochemistry of DNA, and the random configuration and knotting of biological molecules are studied by Monte Carlo methods. In number theory, Monte Carlo methods play an important role in determining primality or factoring of very large integers far beyond the range of deterministic methods. Several important new statistical techniques such as “bootstrapping” and “jackknifing” are based on Monte Carlo methods. Hence, the role of Monte Carlo methods and simulation in all of the sciences has increased in importance during the past several years. These methods play a central role in the rapidly developing subdisciplines of the computational physical sciences, the computational life sciences, and the other computational sciences. Therefore, the growing power of computers and the evolving simulation methodology have led to the recognition of computation as a third approach for advancing the natural sciences, together with theory and traditional experimentation. Knowing that, at the kernel of Monte Carlo simulation is random number generation. Moreover, the book develops methods for simulating simple or complicated processes or phenomena. If the computer can be made to imitate an experiment or a process, then by repeating the computer simulation with different data, we can draw statistical conclusions. Thus, a simulation of a spectrum of mathematical processes on computers was done. The result and accuracy of all the algorithms are truly amazing and delightful; hence, this confirms two complementary accomplishments: first the triumphs of the theoretical calculations already established using different theorems and second the power and success of modern computers to verify them. Additionally, each time I work on the field of mathematical probability and Monte Carlo methods I find the pleasure to tackle the knowledge, the theorems, the proofs, and the applications of the theory. In fact, each problem is like a riddle to be solved, a conquest to be won, and I become relieved and extremely happy when I reach the end of the solution. This verily proves two important facts: firstly, the power of mathematics and its models to deal with such kind of problems and secondly the power of the human mind that is able to understand such class of problems and to tame such a wild concept that is randomness, probability, stochasticity, uncertainty, chaos, chance, nondeterminism. Sincerely, I am truly astonished by the power of probability and these random techniques to deal with random data and phenomena, and this feeling and impression never left me from the first time I was introduced to this branch of science and mathematics. I hope that in the present book I will convey and share this feeling with the reader. I hope also that he will discover and learn about the concepts and applications of the probabilistic and Monte Carlo paradigm.

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Why is it important?

In applied mathematics, the name Monte Carlo is given to the method of solving problems by means of experiments with random numbers. This name, after the casino at Monaco, was first applied around 1944 to the method of solving deterministic problems by reformulating them in terms of a problem with random elements which could then be solved by large-scale sampling. But, by extension, the term has come to mean any simulation that uses random numbers. In the twentieth century and present time, Monte Carlo methods have become among the fundamental techniques of simulation in modern science. This was accomplished after a long history of efforts done by prominent and distinguished mathematicians and scientists. This book is an illustration of the use of Monte Carlo methods when applied to solve specific problems in mathematics, engineering, physics, statistics, or science in general.

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