Difference between revisions of "MAT4XXX"

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(Created page with "=Mathematical Physics - MAT4XXX/5XXX= ==Course description== The course intends to be a basic introduction to the mathematical and computational techniques in applied mathema...")
 
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Strang's 4 special matrices (Part 1)
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Differences, Derivatives, BC.  
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Differences, Derivatives, BC. Gradient, Divergence. Laplace equation.
 
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Inverses.  
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Inverses. Positive Definite Matrices
 
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Stiffness Matrices.  
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Stiffness Matrices. Oscillations & Newton's Laws.
 
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Graph Models. Networks.  
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Graph Models. Networks. Clustering and k-means.
 
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Fourier Series.  
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Fourier Series. Chebyshev, Legendre, and Bessel
 
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Fast Fourier Transform (FFT).  
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Fast Fourier Transform (FFT). Convolution and Signal Processing.
 
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Fourier Integrals. Deconvolution, Integral Equations.  
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Computational implementation of Laplace and z- Transforms.  
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Computational implementation of Laplace and z- Transforms. Spectral Methods.  
 
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Finite Difference for ODEs.  
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Finite Difference for ODEs. Accuracy & Stability. Conservation Laws, diffusion, fluids
 
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Elimination with reordering, multigrid methods, conjugate gradients
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Elimination with reordering, multigrid methods, conjugate gradients, Krylov subspaces
 
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Regular. least sq. Linear programming.  
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Regular. least sq. Linear programming. Adjoint. Stoch. Gradient Descent. ADAM.
 
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Matrix-matrix Multiplication. 4 Fundamental Subspaces. Orthogonal Matrices.
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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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Covariance Matrices. Multivariate Gaussian. Weighted least squares. Markov chains.
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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 (Convolutional, Deep). Backpropagation. Machine Learning.
 
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Revision as of 14:16, 24 January 2025

Mathematical Physics - 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 (Convolutional, Deep). Backpropagation. Machine Learning.
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