Eventually, I can spare some time talking about the course.
The course I took for the first week, Numerical Analysis & Image Processing and Surface Computing, was delivered by Dr. Martin Robinson from the department of computer science. Since the syllabus of this course wasn’t quite detailed before on the official website, hereby I would like to give an outline for future information.
The course consists of four lectures covering seven topics:
- Lagrange Interpolation. This topic gives basic concepts including Lagrange interpolating polynomial and Hermite interpolating polynomial. Important theorems regarding existence and uniqueness as well as how to build polynomials from lower degree ones are also mentioned.
- Newton-Cotes Quadrature. This topic includes Trapezium Rule, Simpson’s Rule and evaluation of error.
- This topic covers how to fit interpolating polynomial and differentiate it. The method is illustrative but rarely used in everyday practice. A widely used method of undetermined coefficients is also introduced, together with three ways of computing difference schemes.
- Timestepping and ODEs. This topic starts with recalling ordinary differential equations and further covers methods in time discretization (Euler, Runge-Kutta, Adams-Bashforth, etc.). The fundamental theorem of finite difference methods, convergence is an equivalent of the combination of consistency and stability, is discussed combined with definition of Stiffness.
- Finite differences and PDEs. This topic explains finite differences in space and time and includes MOL approach and Fisher-KPP equation. Backward Euler and forward Euler methods are shown to illustrate local truncation error.
- PDEs in higher dimensions. This topic covers advection, heat equation and complex matrix structure. (But relax! No perplexing calculation is required!)
- Application to Image Processing. This topic shows how colorful images are generated and optimized in Matlab using previously covered methods. Some more advanced methods are also discussed in the example of spatially varying diffusion.
Since my major is biology, technically I didn’t have a strong perception of materials covering so much math. However, I tried my best to understand and get used to the logic. Learning something totally new is part of the study. Thanks so much to Dr. Robinson for being so dedicated and patient!