ChatGPT Chats related to the course

General interest

This class uses ChatGPT in instruction (in mostly non-essential, education enhancing ways).

• Image (signal) restoration 1D - problem formulation, subgradient, comparing methods using MSE, PSNR.
• A derivation of Euler-Lagrange equation
• Commentary on a fatal (in my book) flaw of Python - GIL
• Image restoration 2D - Rudin, Osher, and Fatemi paper; derivation of the fundamental equation.
• Iterative methods for Ax=b - basic descriptions and suggestions of further inquiries.
• Hyperbolic space in dimension 2
• Geodesics derivation in Riemannian Geometry - a demonstration of tensor calculus and Euler-Lagrange approach. Includes the reparameterization argument.
• The theory of connections - a deeper dive into differential geometry. Equation to calculate parallel translation on a manifold. Explanation why Levi-Civita connection is the unique torsion-free connection satisfying natural axioms, thus the "right way" to move vectors around on a curved space. Derivation of gradient.

Helmholtz equation in 2D

• Handling Neumann boundary conditions with a circular domain - contains ideas for 'Programming Assignment 3'.
• An exact Helmholtz equation problem - a mostly failed attempt to get ChatGPT to solve $u-f-\lambda\nabla^2u=0$ subject to $\partial u/\partial n=0$, where $f(x,y):=x^2$.
• Newton and Stirling iterated finite difference formulas - a summary of useful formulas to incorporate in a difference scheme.
• 2D Newton-like difference formulas - show how to approximate in 2D using iterated differences.
• Solution of homogeneous Helmholtz with Dirichlet boundary conditions - helpful in proving necessity of the Neuman boundary problem: $(u-f)-\lambda\nabla^2u=0$, $\frac{\partial u}{\partial n}|_{\partial D}=0$ in the image smoothing problem.
• Solution to Helmholtz equation with Neumann BC - one way to solve the problem $\nabla^2u-u = x^2$.
• Series solution to a variant of Bessel equation - the method of Frobenius.
• A bump function argument - Extending $C^\infty$ function on the boundary of the unit disk to the unit disk.
• Another version of the bump function argument - This was done with ChatGPT 3.5.
• Converse of Taylor series, Whitney Extension Theorem - definining functions starting from Taylor series.
• An evaluation of a solution strategy for the Helmholtz equation on the disk - a somewhat tortured chat on the subject.

Control Theory

• Linear Quadratic Regulator (LQR) and The Algebraic Ricatti Equation (ARE) - Formulation of A derivation of ARE from HJB.
• A derivation of PMP = Pontryagin Maximum Principle - a simplified version incorporating the dynamics into the cost function.

Conjugate Gradients method

• A 2-by-2 example of CG

Broyden Good Algorithm

• Broyden" s "Good" Algorith - an example of a quasi-Newton method without explicit Jacobian calculation.
• A 2-variable example of Broyden Good Algorithm

Runge-Kutta Methods

• Runge-Kutta introduction
• RK3 and Heun, derivation
• Three steps of RK3 on \( y'=y, y(0)=1, h=0.1 \)