Important note: The lectures will be in-person in March and June 2025 for theory and final projects. We will meet online in April and May to share codes and simulations more efficiently.

We will handle stochastic simulation techniques for biochemical systems in this course. Recently, in mathematical biology, approaches using probability processes, in addition to ordinary and partial differential equations, have been gaining attention. Basic knowledge of probability processes, as well as simulation techniques and numerical analysis, are widely used not only in mathematical biology but also in artificial intelligence and statistics, making them highly beneficial for graduate students studying applied mathematics. We will mainly study stochastic simulations for continuous-time Markov chains, Markov chain Monte Carlo, and Brownian motions. We will also study theoretical aspects of the error analysis and variance reduction related to the stochastic simulation methods.

At the end of the semester, a project will be assigned to students, where they will simulate a probability process about biochemical systems of their interest. This will provide them with an opportunity to directly explore how mathematics can be used to analyze biological phenomena, helping graduate students develop a research mindset

이번 수업에서는 수리생물학에서 주로 사용되는 확률프로세스를 수치적으로 접근하는 방법을 주로 다루려 한다. 최근 수리생물학에 사용되는 수리적 접근법에 상미분이나 편미분방정식뿐 아니라 확률프로세스를 이용한 분석이 많은 주목을 받고 있다. 확률프로세스에 관한 기본 지식, 더 나아가 시뮬레이션 기법과 수치해석은 수리생물학뿐아니라 인공지능이나 통계학에도 많이 사용되므로 현재 응용수학을 연구하는 대학원생들에게 큰 도움이 될 것이다.
우리는 주로 연속시간 마르코프체인, 마르코프체인 몬테카를로, 브라운 운동 등의 확률 프로세스를 다룰 예정이다. 그리고 이들 프로세스를 시뮬레이션 하는데에 있어 수반되는 error 해석법과 분산 줄이기등의 방법론을 다룰 것이다.

학기말에는 실제 생화학시스템를 모델링하는 확률프로세스를 직접 시뮬레이션 해보는 프로젝트를 학생들에게 부여함으로서 수학이 생명현상을 분석하는데 어떻게 사용될 수 있는지 직접 연구해보는 기회도 제공하여 대학원생들에게 연구에 대한 감각을 익히도록 하는 과정도 수반할 계획이다

Calculus.
Probability (Required but not mandatory)
Statistics (Required but not mandatory)

- Homeworks: 50%
- Final presentation (Students can pick any topics and problems related to simulating stochastic biochemical systems): 50%

1. Lecture note written by the instructor
2. Prof. J.R. Norris' Markov chains

-March: Background knowledge on continuous-time Markov chains and an introduction to papers that employ this concept.
-April: Simulation methods related to Markov chains.
-May: Simulation methods related to Brownian motion, and an introduction to numerical analysis results regarding the accuracy of these simulation methods.
-June: Introduction to the latest methodologies for reducing simulation computational costs and variation, and students' final project presentations.

-No exam.
-Grade is based on: 1. homework and 2. the final project.
- Students can use any type of computational language including Python, Matlab, Mathematica, etc.