Below are a few links to some short notes on various mathematical tricks and topics that I find interesting.
Introduction to the Boundary Element Method
Getting the computer to understand functions: Describes the parametrization of family of functions.
What is an element?: Describes how elements describes both the geometry and serves as a basis for our parametrization of functions.
Quadrature: Describes how elements can be used to approximate part of the boundary integral.
The Boundary Element Method: Combines the above and shows how the Kirchhoff-Helmholtz integral can be discretized.
The Fast Multipole Method for BEM: Describes how the Fast Multipole Method can be used to speed up the BEM computations.
Hierarchical Matrices for BEM: Describes ho Hierarchical matrices can be used to speed up the BEM computations.
Line Elements: Description of continuous and discontinuous line elements.
Shape Functions: Old note on local descriptions of line and surface elements.
BEM200: A simple implementation of the 2D BEM in less than 200 lines of code.
Reduced Order Series Expansion Boundary Element Method: How to speed-up multifrequency problems
Acoustical BEM with Visco-thermal losses: Some intuition behind viscous and thermal losses. Includes a short introduction to the Kirchoff's decomposition and the sparsity patterns of the resulting system of equations.
Boundary Layer Impedance: A quick description of the BLI approximation of losses.
The Burton-Miller Method: Removing irregular frequencies using Burton-Miller regularization.
From the Wave Equation to an Integral Equation: A description harmonic time dependency and its influence on the Integral Equation.
Green's function derivative: A short derivation of derivative of the Green's function.
Quadrature: Introduction to quadrature scheme.
Introduction for FEM for the Helmholtz equation: Examples in 1D and 2D.
Sherman-Morrison Identity: Two brief derivations.
Sherman-Morrison-Woodbury Identity: Two brief derivations.
General Sherman-Morrison-Woodbury Identity: Some derivations.
The Matrix Determinant Lemma: Determinants of Block matrices.
Low Rank Approximations
The Eckart-Young-Mirsky Theorem: Short sketches of proofs.
Adaptive Cross Approximation: Short description of the ACA algorithm.
Hierarchical Matrices
Symmetric Extended Generator Representable Semiseparable Matrices: Smoothing splines and Gaussian processes. (For the interested: My master's thesis and SymSemiseparableMatrices.jl).
Sparse Factorization: An example on the importance of permuting
The Broyden–Fletcher–Goldfarb–Shanno (BFGS) Algorithm: Some intuition behind the BFGS update using the Sherman-Morrison-Woodbury formula.
The Method of Moving Asymptotes (MMA): A short description.