GEOM2057 Lectures 6 and 7

This repository contains the Julia teaching scripts, lecture PDFs, and supporting assets for the second part of the GEOM2057 course, centred on Lectures 6 and 7.

Topics Covered in the Second Part

Lecture 6 focuses on deterministic inversion and optimisation:

• Problem setup for a non-linear inverse problem.
• Synthetic data generation.
• Newton method inversion.
• Gauss-Newton inversion.
• A harder non-linear comparison case.
• Local versus global search behaviour.
• Particle Swarm Optimisation (PSO).
• Very Fast Simulated Annealing (VFSA).

Lecture 7 focuses on uncertainty and Bayesian inversion:

• Probability review and probability density functions.
• Joint, marginal, and conditional probability.
• Bayes' rule and posterior probability.
• Grid-search Bayesian inversion.
• Monte Carlo sampling.
• Markov chains.
• Metropolis-Hastings MCMC.
• Reading traces, marginals, and predictive uncertainty.
• Geophysical inversion examples.
• Implicit neural representations and future directions.

Repository Contents

Lecture 6 includes:

• L6/01-problem-setup-derivatives-etc.jl: shared definitions and derivatives for the baseline non-linear inverse problem.
• L6/02-data-generation.jl: synthetic data generation for the baseline inverse problem.
• L6/03-inversion-NewtonMethod.jl: a Newton-method inversion example.
• L6/04-inversion-GaussNewton.jl: a Gauss-Newton inversion example.
• L6/05-problem-setup-hard.jl: shared setup for the harder, more non-linear comparison case.
• L6/06-data-generation-hard.jl: shared noisy synthetic data for the hard comparison case.
• L6/07-inversion-GaussNewton-hard.jl: local Gauss-Newton on the hard comparison case.
• L6/08-inversion-PSO-hard.jl: Particle Swarm Optimisation on the hard comparison case.
• L6/09-inversion-VFSA-hard.jl: Very Fast Simulated Annealing on the hard comparison case.

Lecture 7 includes:

• L7/01-probability-review.jl: introductory probability and PDF concepts for the lecture.
• L7/02-bayesian-grid-search.jl: grid-based Bayesian inversion demonstration.
• L7/03-markov-chain-demo.jl: simple Markov chain demonstration.
• L7/04-metropolis-hastings-geophysical.jl: MCMC demonstration on a geophysical inverse problem.
• L7/05-inr-geophysical-inversion.jl: high-level implicit neural representation inversion example.
• L7/L7.pdf: Lecture 7 slides.
• L7/Figures/: generated lecture figures and supporting images.

Getting Started with Julia

The Julia scripts in Lecture 6 only use Julia standard libraries such as LinearAlgebra, Printf, and Random, so you do not need to install extra Julia packages to run the .jl files. Lecture 7 examples may additionally rely on plotting packages already set up in the teaching environment.

1. Install Julia

Download and install Julia from the official site:

https://julialang.org/downloads/

After installation, confirm Julia is available from a terminal:

julia --version

If that command is not found, restart VS Code or add Julia to your system PATH.

2. Open the lecture code folder

From VS Code or a terminal, work inside:

GEOM2057

3. Run the Julia scripts

For Lecture 6, move into GEOM2057/L6 and run the scripts below.

Run the baseline problem setup:

julia .\01-problem-setup-derivatives-etc.jl

Run the Newton inversion example:

julia .\03-inversion-NewtonMethod.jl

Run the Gauss-Newton inversion example:

julia .\04-inversion-GaussNewton.jl

Run the shared hard-problem setup:

julia .\05-problem-setup-hard.jl

Run the shared noisy-data generation example:

julia .\06-data-generation-hard.jl

Run the hard-case Gauss-Newton comparison:

julia .\07-inversion-GaussNewton-hard.jl

Run the hard-case PSO global-optimisation example:

julia .\08-inversion-PSO-hard.jl

Run the hard-case VFSA global-optimisation example:

julia .\09-inversion-VFSA-hard.jl

For Lecture 7, move into GEOM2057/L7 and run the lecture demos or compile the slides as needed.

Example:

julia .\04-metropolis-hastings-geophysical.jl

Each script prints the main intermediate or final results in the terminal.

Suggested Workflow for Students

1. Run 01-problem-setup-derivatives-etc.jl to understand the toy inverse problem setup.
2. Run 03-inversion-NewtonMethod.jl and inspect the printed iteration history.
3. Run 04-inversion-GaussNewton.jl and compare it against Newton's method.
4. Run 05-problem-setup-hard.jl and 06-data-generation-hard.jl to inspect the shared hard case and the noisy observations used by all three comparison methods.
5. Run 07-inversion-GaussNewton-hard.jl, 08-inversion-PSO-hard.jl, and 09-inversion-VFSA-hard.jl to compare local and global search on the same hard inverse problem.
6. Move to L7 to review the probability, Bayesian inversion, Markov chain, MCMC, and INR examples that build the uncertainty-analysis part of the course.