Conjugate Gradients

This is the fourth post in our series on Krylov subspaces. The previous ones (i.e Arnoldi Iterations and Lanczos Algorithm were mostly focused on eigenvalue and eigenvector computations. In this post we will have a look at solving strategies for linear systems of equations.

Numerical Linear Algebra and Optimization

A collection of notes and investigations on numerical linear algebra and optimization related topics.

Designing Neural Networks in Mathematica

Implementing LeNet for MNist in Wolfram Mathematica

Variations of the Bregman Algorithm (4/4)

In the previous post in our series on the Bregman algorithm we discussed how to solve convex optimization problems. In this post we want to give reference to some variations and extensions of the Bregman algorithm.

The Bregman Algorithm (3/4)

In a previous post we discussed how to solve constrained optimization problems by using the Bregman algorithm. Here we want to extend the approach unconstrained problems. Let’s start simple. Assume we want to minimize a convex and smooth function $f\colon\mathbb{R}^{n}\to\mathbb{R}$.

The Bregman Algorithm (2/4)

In a previous post we discussed how to find a common point in a family of convex sets by using the Bregman algorithm. Actually the algorithm is capable of more. We can use it to solve constrained optimization problems.

The Bregman Algorithm (1/4)

In the 1960s Lev Meerovich Bregman developed an optimization algorithm [1] which became rather popular beginning of 2000s. It’s not my intention to present the proofs for all the algorithmic finesse, but rather the general ideas why it is so appealing.

Accoustic Source Characterisation

Let us consider a microphone array comprising $n$ microphones at known locations (see figure above). These microphones register the sound that is emitted by a number of sources with unknown locations.

Towards PDE-Based Video Compression with Optimal Masks Prolongated by Optic Flow

Lossy image compression methods based on partial differential equations have received much attention in recent years. They may yield high-quality results but rely on the computationally expensive task of finding an optimal selection of data. For the …

Photometric Stereo

We have investigated high performing optimization algorithms and matrix differential calculus technique in the context of Photometric Stereo and presented the results at the BMVC 2016 Source Code A github repository with the code is maintained by Yvain Quéau.