30561 - STOCHASTIC PROCESSES AND SIMULATION IN NATURAL SCIENCES

Linear algebra, calculus, probability, basic statistics Basic Python and some familiarity with numpy

The aim of the course is to introduce the theoretical and numerical tools for the analysis and simulation of stochastic and natural processes, which are ubiquitous in many of the program's other subjects.

• Discrete-time Markov chains
• Poisson processes and other continuous-time stochastic models
• Stochastic simulation and Monte Carlo methods
• Numerical methods for ordinary differential equations

• Characterize and describe Monte Carlo and Markov Chain Monte Carlo methods
• Formulate probabilistic models based on Markov chains, Poisson processes and other continuous time processes
• Analyze the above stochastic processes using probability theory and other mathematical tools
• List and explain fundamental methods to solve numerically differential equations
• Recognize numerical issues and identify workaround strategies
• Estimate the computational cost of implementing all of the above

• Translate phenomena involving randomness and uncertainty into appropriate probabilistic models
• Characterize the average and long-run behavior of a given stochastic process
• Determine whether a Monte Carlo method is appropriate for a task, and if so choose the best approach
• Develop a Markov Chain Monte Carlo algorithm for a given problem
• Simulate a process described by a set of differential equations

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

The teaching method is face-to-face lectures.

The written exam will contain theoretical questions and exercises, intended to verify that the students have acquired both the basic mathematical knowledge (about discrete and continuous time stochastic processes) and the analytical skills to relate the different techniques to given problem instances. For the written exam, students can either take the two partial exams or directly the general exam. After (optionally) taking the first partial exam, students can decide whether to take the second partial exam or to ignore the grade of the first partial exam and directly take the general exam.

The group project will consist in implementing from scratch a simulation or a numerical method for a problem that was not discussed in class. The students can demonstrate that they have internalized the theoretical aspects, that they can design a strategy and implement it in code.

The written exam (either general or partials) will form 80% of the final grade, and the group project the remaining 20%. The final grade will be the sum of the two grades.

The main teaching material for the first half of the course (dealing with stochastic processes) will be typesetted lecture notes. Additional textbook references for that part will be: "Markov Chains" by J.Norris; Cambridge University Press; "Introduction to probability models" by S.M. Ross; 12th edition, Academic Press; "Essentials of Stochastic Processes" by R.Durrett; 3rd edition, Springer.

The material for the second part of the course (dealing with numerical methods and more elaborate examples) will be determined later on.