Materials for Review 2023
Below gives a sample of materials developed to enhance our courses, provide resources for students, or for special topics. This includes links to videos, slides, and other resources. For additional information, please see our bio-bib and our website http://atzberger.org/.
Videos
| Math 4B: Differential Equations (Winter 2022): Videos | ||
|---|---|---|
| Lectures | Topics | Links |
| 01a | Introduction to Differential Equations, Class Overview, Examples | https://bit.ly/3G1xneD |
| 01b | Direction Fields, Integral Curves, Integrating Factors, Separation of Variables | https://bit.ly/3HJ7zEs |
| 02a | Separation of Variables, Direction Field interpretation, Example: SIR Infection Model | https://bit.ly/34BM1eI |
| 02b | Example: SIR Infection Model, Integrating Factors, Separation of Variables | https://bit.ly/3qt2dHy |
| 03a | Existence and Uniqueness of Solutions to Ordinary Differential Equations (ODEs), Examples | https://bit.ly/3fTmujP |
| 03b | Existence and Uniqueness of ODEs, Numerical Approximation, Euler's Method | https://bit.ly/3nGP8Zp |
| 04ab | Numerical Approximation of ODEs, Euler's Method, Second Order Differential Equations, Solution Methods | https://bit.ly/3GepZfj |
| 05a | Second Order Differential Equations, Solution Methods, Euler's Identity and Oscillations | https://bit.ly/3JeGRUT |
| 05b | Second Order Differential Equations, Solution Methods, Examples | https://bit.ly/3uv9AAL |
| 06a | Second Order Differential Equations, Bead-Spring System Demonstration, Existence & Uniqueness | https://bit.ly/34Mc9DC |
| 06b | Bead-Spring Systems, Linear Systems, Solution Methods, Examples | https://bit.ly/3LpuyHo |
| 07ab | Non-Homogeneous Equations, Method of Undetermined Coefficients, Reduction of Order, Variation of Parameters | https://bit.ly/3gUNPlX |
| 08a | System of ODES, Higher Order ODEs, Well-Posedness, Examples | https://bit.ly/3pjHcy5 |
| 08b | Higher Order Equations, Solution Methods, Wronskian, Examples | https://bit.ly/3C0Lcc3 |
| 09a | High Order ODEs and Systems of Equations, Lipschitz Continuity, Constant Coefficient Methods, Real/Distinct Roots | https://bit.ly/35PXKXA |
| 09b | Higher Order Equations, Inhomogeneous Equations, Complex Roots, Repeated Roots | https://bit.ly/3KmvN93 |
| 10a | Higher Order Equations, Variation of Parameters, Systems of Equations | https://bit.ly/3MIFuQX |
| 10b | Systems of Equations, Soluion Techniques, Phase Portraits, Eigenvalues, 2D Behaviors, General Homogeneous Equations | https://bit.ly/3pXZWUe |
| Math 104A: Numerical Analysis (Fall 2021): Videos | ||
| Lectures | Topics | Links |
| 2 | Floating Point Arithmetic, Error Analysis, Algorithms, Complexity | https://bit.ly/46jGrZm |
| 3 | Solving Equations in One Variable, Fixed-Point Iteration, Period-Doubling / Chaos, Cob-Web Diagrams, Stability Theorems | https://bit.ly/45zyTkl |
| 4 | Fixed Point Iterations, Newton's Method, Secant Method, Error Analysis, Lagrange Interpolation | https://bit.ly/46nrBBd |
| 5 | Lagrange Interpolation, Neville's Method, Introduction to Python, Divided Differences | https://bit.ly/46liQb0 |
| 6 | Hermite Interpolation, Evaluating Hermite Polynomial Approximations | https://bit.ly/48HO1P3 |
| 7 | Spline Interpolation Motivations, Cubic Splines, Data Fitting, Gradient Decent, Bezier Curves | https://bit.ly/3PMlRd6 |
| 8 | Bezier Curves, Algorithms for Evaluation, Demonstrations, Numerical Integration | https://bit.ly/3PN3rsZ |
| 9 | Numerical Integration, Quadrature Methods, Newton-Cotes Methods, Gaussian Quadrature | https://bit.ly/46Cazir |
| 10 | Numerical Differentiation, Finite Difference Methods, Theory of ODEs | https://bit.ly/3LQytiq |
| 11 | ODEs Theory, Numerical Time-Step Methods, Euler's Method, and Other Methods | https://bit.ly/3PDY13g |
| Math 104C: Numerical Analysis (Spring 2021): Videos | ||
| 1 | Approximation and Least-Squares | https://bit.ly/46Dgu6R |
| 2 | Least Squares, Orthogonal Polynomials, Power Series, Pade’ Approximation | https://bit.ly/3rOhQNj |
| 3 | Pade' Approximation, Chebyshev Rational Functions, Trigonometric Approximation | https://bit.ly/3tugqrG |
| 4 | Fourier Series and Fast Transforms | https://bit.ly/45n5cCN |
| 5 | Linear Systems, Linear Operators, Eigenvalue Methods, Gerschgorin Theorem, Power Method, Google Page-Rank | https://bit.ly/45gDG9X |
| 6 | Google Page-Rank, Eigenvalue Methods, Householder Transformations, Singular Value Decomposition | https://bit.ly/3LQgnNc |
| 7 | Singular Value Decomposition, Regression Methods, Principle Component Analysis, Eigen-Facial Recognition, | https://bit.ly/3LOMInD |
| 8 | Finite Difference Methods for PDEs, Parabolic PDEs, Elliptic PDES, Iterative Methods, Preconditioners | https://bit.ly/46HD12B |
| 9 | Elliptic PDEs, Poisson Problem, Iterative Solvers, Multigrid Methods, PDE Well-Posedness and Stability | https://bit.ly/3RK6h4j |
| 10 | Stability of Finite Difference Methods, Example Black-Scholes in Finance, Von Neumann Analysis | https://bit.ly/3Q4zsxH |
| Special Topics, Video Clips, Motivating Applications (subset) | ||
| Topics | Links | |
| Machine Learning: Statistical Learning Theory | https://vimeo.com/500198684/4a8fbcffbc?share=copy | |
| Machine Learning: Regression Part 1 | https://www.youtube.com/watch?v=08RhUj9Y5W4 | |
| Machine Learning: Regression Part 2 | https://www.youtube.com/watch?v=BlVefI1RunU | |
| SIR Model of Disease Infections in Pandemics (Part 1) | https://bit.ly/3ZJQDYO | |
| SIR Model of Disease Infections in Pandemics (Part 2) | https://bit.ly/3rMu7BT | |
| Mechanics of Bead-Spring Systems | https://bit.ly/46eDzNx | |
| Computational Methods for ODEs and Python Codes | https://bit.ly/3ZJQOTY | |
| Behaviors of Systems of ODEs: Lorenz Dynamics and Chaos | https://bit.ly/45eOJ3r | |
| Black-Scholes-Merton (Finance) | https://bit.ly/3RHXFLI | |
| LASSO Regression and Computed Tomography | https://bit.ly/3ROxFyn | |
| Google Page-Rank | https://bit.ly/48JsCFb | |
Slides and Other Resources
| Math 206D: Finite Element Methods Slides | ||
|---|---|---|
| Topics | Links | |
| Introduction to FEM and Ritz-Galerkin Approximation | [PDF] Δ [GoogleSlides] | |
| Finite Element Spaces | [PDF] [GoogleSlides] | |
| Sobolev Spaces | [PDF] [GoogleSlides] | |
| Variational Formulations and Elliptic PDEs | [PDF] [GoogleSlides] | |
| Finite Element Approximation Properties and Convergence | [PDF] [GoogleSlides] | |
| Elasticity Theory | [PDF] [GoogleSlides] | |
| Mixed Methods | [PDF] [GoogleSlides] | |
| Elasticity Theory: Numerical Example | [PDF] [GoogleSlides] | |
| Math 4B: Differential Equations Slides | ||
|---|---|---|
| Lectures | Topics | Links |
| 1 | Introduction Differential Equations | [PDF] |
| 2 | Integrating Factors, Separation of Variables | [PDF] |
| 3-4 | Separation_of_Variables Methods, Summary | [PDF] |
| 4-5 | Existence and Uniqueness, Numerical Approximation | [PDF] |
| 6-7 | Numerical Approximation, Second Order DE Distinct Roots | [PDF] |
| - | Example Python Code, Jupyter Notebook: Numerical Approximation of ODEs | [PDF] |
| 8-9 | Second Order Equations, Constant Coeffient Homogeneous | [PDF] |
| 10 | Reduction of Order, Undetermined_Coeff, Variation_of_Parameters | [PDF] |
| 11-12 | Higher-Order Differential Equations and Systems of Equations | [PDF] |
| 13-14 | Higher-Order Differential Equations, Theory and Undetermined Coefficients Method | [PDF] |
| 15 | Higher-Order Variation_of_Parameters Methods, First Order Linear Systems of Equations | [PDF] |
| 16 | First Order Linear Systems of Equations, Solution Methods, Phase Portraits | [PDF] |
Summary
The summaries above give a sample of materials developed to enhance our courses, provide resources for students, or for special topics. This includes links to videos, slides, and other resources. For additional information, please see our bio-bib and our website http://atzberger.org/.