Week 3

Lecture 6: Numerical Methods for Ordinary Differential Equations

Summary

This lecture is given by MIT Applied Math Instructor, Dr. Christopher Rackauckas. In this lecture we will discuss the current state of software in differential equations and see how the continued advances in computer science and numerical methods are likely to impact our software in the near future. Issues such as efficiency improvements for stiff and non-stiff differential equations will be addressed from a numerical analysis standpoint but backed with recent benchmarking results. Newer mathematical topics like random ordinary differential equations, jump diffusion equations, and adaptivity for stochastic differential equations will be introduced and the successes and limitations in current automatic software solutions will be discussed. We will close with a discussion on how recent computational advancements have been influencing the software implementations, specifically showing the effects of generic typing over abstract algorithms and implicit parallelism.

Further Reading

Lecture 7: The SVD, its Applications, and Condition Numbers

Summary

This lecture is given by guest lecturer, Prof. Alan Edelman. In this lecture we discussed SVD, relationship to L2 norms and condition numbers, as well as applications (e.g. principal components analysis).

Further Reading

  • Read “Lectures 4 and 5” in the textbook Numerical Linear Algebra.

Lecture 8: Linear Regression and the Generalized SVD

Summary

This lecture is given by guest lecturer, Prof. Alan Edelman. In this lecture we discussed generalized SVD (GSVD), least-square problems (via QR or SVD) and different viewpoints on linear regression: linear algebra, optimization, statistics, and machine learning.

Further Reading

Course Info

Learning Resource Types

assignment_turned_in Problem Sets with Solutions
grading Exams with Solutions
notes Lecture Notes