Numerical Methods using R

Mingyang Lu, Nov 2021

The purpose of this tutorial is to illustrate the use of R programming in numerical analyses, which can easily lead to a large variety of applications in science and engineering. I will cover basic elements of numerical methods and algorithms for solving different types of differential equations, running simulations, and performing optimization. Rather than just showing how to do so with existing packages, I will focus on the details of each algorithm using R and provide its usage in the context of real-world applications in the fields of biomedical engineering, bioengineering, and data science. The tutorial is organized into a series of R Markdown files. Students should be able to view the materials and play around with the provided R scripts using R/RStudio.

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Table of Contents
1. Introduction to R programming
-- 1A. Basics of R
-- 1B. Efficient R programming
-- 1C. Numerical methods
-- 1D. Exercises
2. Ordinary differential equations
-- 2A. Modeling gene circuits with rate equations
-- 2B. Numerical integration
-- 2C. Practice: modeling bacterial growth
-- 2D. Stability
-- 2E. Bifurcation
-- 2F. Exercises
3. Phase plane
-- 3A. Nulllines
-- 3B. Stability in 2D
-- 3C. Practice: modeling chemostat
-- 3D. Practice: predator-prey model
-- 3E. Bifurcation for two-variable systems
-- 3F. Separatrix
-- 3G. Effective potential revisited
-- 3H. Multi-component systems
-- 3I. Exercises
4. Systems with time delays
-- 4A. Delayed differential equations
-- 4B. Examples of systems with time delays
-- 4C. Delays from indirect interactions
-- 4D. Exercises
5. Molecular dynamics
-- 5A. Integrators for second order ODEs
-- 5B. Orbital motions
-- 5C. Modeling a box of 2D particle
-- 5D. Exercises
6. Stochastic differential equations
-- 6A. Random number generators
-- 6B. Brownian motion
-- 6B. SDE integrators
-- 6D. Stochastic state transitions
-- 6E. Exercises
7. Partial differential equations
-- 7A. Modeling diffusion
-- 7B. Reaction-diffusion systems
-- 7C. Turing instability
-- 7D. Pattern formation in Dictyostelium
-- 7E. Exercises
8. Monte Carlo Simulations
-- 8A. Monte Carlo Method
-- 8B. Metropolis algorithm
-- 8C. Particles in a box: MCMC sampling
-- 8D. Gillespie Algorithm
-- 8E. Exercises
9. Global optimization
-- 9A. MCMC optimization methods
-- 9B. Dynamic programming
-- 9C. Genetic algorithm
-- 9D. Exercises
10. High dimensional data analysis
-- 10A. Dimensionality reduction
-- 10B. Clustering
-- 10C. Network algorithms
-- 10D. Exercises