確率的シミュレーション

The objective of this course is to explore various aspects of Monte Carlo methods, a fundamental computational technique for simulating stochastic events and quantifying uncertainty. Beginning with an overview of sampling from probability distributions (i.e., random number generation), which is essential for "using" Monte Carlo methods, and then proceeding to methods of "improving" Monte Carlo methods for better computational efficiency (i.e., variance reduction techniques), the aim is to ensure a proper understanding of Monte Carlo methods and their skillful applications in practical use.

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授業計画

1. Basics of stochastic simulation -- Overview of stochastic simulation -- Basic statistical measures -- Estimation by Monte Carlo methods 2. Random number generation -- Pseudo-random number generation -- Inversion method -- Transformation -- Acceptance-rejection -- Markov chain Monte Carlo sampling 3. Variance reduction: Part I -- Antithetic sampling -- Control variate -- Common random numbers -- Conditional Monte Carlo -- Importance sampling 4. Multilevel Monte Carlo methods -- Bias-variance decomposition -- From control variate to 2-level Monte Carlo -- Multilevel Monte Carlo -- Application to nested expectations 5. Variance reduction: Part II -- Stratified sampling -- Latin hypercube designs -- Quasi-Monte Carlo 6. Global sensitivity analysis -- Local sensitivity analysis -- Analysis of variance and global sensitivity analysis -- Monte Carlo estimation of sensitivity measures

授業の方法

* This course is primarily delivered in English. ** While the lectures mainly focus on theoretical aspects of stochastic simulation (Monte Carlo methods), computer programming is mandatory for the final report. Any programming language is acceptable. *** The course materials corresponding to each lecture will be uploaded to UTOL in advance, so please study them beforehand. Additionally, after each lecture, it is recommended to review the course materials and your notes.

成績評価方法

Attendance and weekly short quizzes (40%) and a final report (60%)

履修上の注意

指示しない

その他

前提となる知識と項目：Probability theory and statistics (undergraduate level in engineering), and programmin