MAT4XXX

Mathematical Physics II - MAT4XXX/5XXX

Course description

The course intends to be a basic introduction to the mathematical and computational techniques in applied mathematics, computational science & engineer�ing, and data science & machine learning. This course will stress then how the methods of mathematical modeling in the STEM disciplines have transitioned from the analytical (as in Theoretical Physics) to the numerical (as in traditional methods in Computational Science and Engineering) and more recently to Data-based methods (as in current developments in Data Science and Machine Learning). The student will acquire the basic skills needed broadly in Computational Science and Engineering, of which Computational Physics, Data Science, Machine Learning, and Numerical Modeling in the Mathematical Sciences are a subset.

Catalog entry

Prerequisite: Calculus III MAT2214 and Differential Equations I MAT3613 with a letter grade of C- or better, or successful completion of at least three credits of equivalent courses.

Content: 1. Computational Science, Engineering, and Mathematics (a) Linear Algebra and Computational Science & Engineering (b) Applied Math and Computational Science & Engineering (c) Fourier Series and Integrals (d) Laplace Transform and Spectral Methods (e) Initial Value Problems (f) Conjugate Gradients and Krylov Subspaces (g) Minimum Principles 2. Data Science and Machine Learning: a Mathematical Perspective (a) Principal Components and the Best Low Rank Matrix (b) Randomized Linear Algebra (c) Low Rank and Compressed Sensing (d) Markov Chains (e) Stochastic Gradient Descent and ADAM (f) Introduction to Machine Learning: Neural Networks

Textbooks:

• Strang, G. Computational Science & Engineering. USA, Wellesley-Cambridge, 2007.
• Strang, G. Linear Algebra and Learning from Data. Wellesley-Cambridge Press, 2019.

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1		Strang's 4 special matrices
Week 2		Differences, Derivatives, BC. Gradient, Divergence. Laplace equation.
Week 3		Inverses. Positive Definite Matrices
Week 4		Stiffness Matrices. Oscillations & Newton's Laws.
Week 5		Graph Models. Networks. Clustering and k-means.
Week 6		Fourier Series. Chebyshev, Legendre, and Bessel
Week 7		Fast Fourier Transform (FFT). Convolution and Signal Processing.
Week 8		Fourier Integrals. Deconvolution, Integral Equations. Wavelets, Signal Processing.
Week 9		Computational implementation of Laplace and z- Transforms. Spectral Methods.
Week 10		Finite Difference for ODEs. Accuracy & Stability. Conservation Laws, diffusion, fluids
Week 11		Elimination with reordering, multigrid methods, conjugate gradients, Krylov subspaces
Week 12		Regular. least sq. Linear programming. Adjoint. Stoch. Gradient Descent. ADAM.
Week 13		Matrix-matrix Multiplication. 4 Fundamental Subspaces. Orthogonal Matrices. Best low rank matrix. Rayleigh quotients. Factoring matrices and tensors.
Week 14		Randomized Linear Algebra. Low rank signals. Singular values. Compressed sensing. Covariance Matrices. Multivariate Gaussian. Weighted least squares. Markov chains. Neural Networks. Backpropagation. Machine Learning.