View source for MAT5XXX

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