Numerical Approximation and Discretization

“One of the questions I got at the end of last week was ‘How do you deal with the loneliness?'”

Text: A First Course on Numerical Methods, by Uri M. Ascher and Chen Greif

Prof:  Dr. Jessica Bosch

Dr. Jessica Bosch is the literally the kindest professor I have ever met. She is super-receptive of feedback and cares (possibly too much) about her students’ trials and tribulations. She once spoke about her current research relating to mathematical models of tumour growth which was pretty interesting. Her slides (along with the helpful summary slides) are usually very clear and well structured. She also has in-class chapter group quizzes which are very useful in preparing students for the midterm and final.

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Difficulty

The content in the course is in some ways very simple, it is a bunch of algorithms for solving well-known math problems. However, I think this simplicity is deceiving and the course hints at a fundamental aspect of computing – how problems get transformed as they are represented in the computer, and what properties are preserved. From this perspective, the course could be very complicated. Fortunately, we do not do very complex mathematics in the course, and largely the course is quite manageable. The midterm, given the practice material, was pretty doable. The final was a bit trickier but was combined with very liberal scaling. The assignments were not bad, though Matlab can get annoying. There was also a bonus assignment which, along with the liberal final scaling, contributed to the A- average in the course.

Key Concepts

Floating point number systems

Polynomial Interpolation

Numerical Differentiation

Numerical Integration

Numerical Solutions to ODEs

Global error (Convergence) vs local error (Consistency)

Taylor Series

Hard Concepts

Butcher Tableau: Used to represent an instance of general Runge-Kutta method. Easy to get mixed up with the meaning of the different coefficients.

Barycentric weights: Used for efficient computation of Lagrange basis. Not something that would occur to you naturally.

Order of a method: Each chapter seems to have a slightly different meaning of the order and precision of a method. Good to keep track of what it means in a particular context.

Conclusion

Interesting course. A combination of annoying Matlab along with neat mathematics and clever algorithms.