This is a follow-up to the guide on how to quickly set up a prototype. Part 1 was about implementing a Node.js based back-end and can be read here. We will follow up on this and use the same repository, which you can find here. You are best to clone it or view remotely from within vs-code (see this plugin) during reading:

git clone https://github.com/applied-math-coding/prototype.git
cd prototype
npm install
cd client
npm install

I will start my writing style by assuming you were to setup the project from scratch. …


As many of us already have suffered, sometimes things must move quickly. In business good and new ideas usually appear as a rare event. But when such an idea appears, people typically cannot wait to see things in action. So, all eyes are turned onto the developer. But since the idea is new, often there is a lack of experience how things would integrate into the existing system or even how that idea finally will look like.

For these aforementioned scenarios an often applied approach is to deliver a quick prototype. As such it intends to focus and implement only…


If you are a developer who uses JavaScript and want to look for reasons to use or learn Typescript, this small account is suited for you.

First of all: What is Typescript?

Typescript is a kind of wrapper around JavaScript, which allows to annotate function declarations and variables with types. So if you look at the following JavaScript code, there are no types:

let car;function getDistanceDriven(carInstance) {
return carInstance.speed * carInstance.deltaTime;
}
console.log(getDistanceDriven({speed:5, deltaTime:10}))

The variable ‘car’ probably is referring to some object which describes a car. And ‘carInstance’ sounds like another instance of a car. In other words:


In this article we look in some detail at so called fractal trees. In addition it introduces a small online application which lets you produce your own fractal trees and play around with various parameters. The application has some interest on its own and shows how one can use the browser to create computational applications.

Fractal Trees are objects of especially aesthetic shape. The idea of their construction is adopted from nature and is based on iterating very simple shapes in order to obtain a global complicated shape.

Such a simple shape can be like this:


When modeling phenomena by means of physical laws like it is the case in many applications arising from engineering or numerical weather forecasting, the targeted result often not only depends on time but in addition on space.

These models typically lead to partial differential equations which themself are notorious hard to deal with and in particular when they are nonlinear. These harder to solve equations arise quickly. An example is the famous Navier-Stokes equation which intends to describe the evolution of a velocity field. It has the form:

Before continuing let us roughly look at how this equation is derived…


As you probably know or have heard of, differential equations are everywhere. A typical example is the law of Newton mechanics or evolution equations arising in biology.

Differential equations is a huge and active field in mathematics and this article aims by no means to introduce into this field. My intention is rather to give an overview of methods and ideas used for solving ordinary differential equations numerically.

So if you already use solvers as provided in various libraries, you may find it interesting to understand in a little more detail on which ideas they are built on. …


Coordinate descent is an optimization algorithm which is leaned very much at the idea of gradient descent but comes out without the computation of the gradient. This at first seems like a big advantage but be aware that for the algorithm convergence only can be assured in case the function is differentiable. In other words, gradient descent could have been used as well. Though, depending on the problem, coordinate descent sometimes turns out to be the faster variant.

In order to better understand this article you might be best when having read already my former article on line search: (but…


This article provides a follow-up of my series about basic nonlinear optimization techniques with sample implementations. Again, we don’t aim to cover all deep aspects, but focus on giving an overview and introduction.

This account introduces another very popular, if not the most popular, algorithm for finding minima of real-valued functions in several variables — the Gradient Descent.

Our main task can be formulated as:

The main idea behind the algorithm is to start at an arbitrary point and to move a small step into a direction which deems promising for the function to fall very quickly. …


This article aims to be a follow-up of my introduction into methods of non-linear optimization. The recent was giving an overview about heuristic methods which you can read here

Now we will turn ourselves towards very often used standard algorithms which you can find in action in many applications … especially in our so much loved subject ‘machine learning’.

Let’s start easy:

Line Search

The main task of line search is this:

In words, we are given a function f which is continuous and maps an interval of the real-line into the real-line. …


Machine Learning … probably is the biggest tech and research trend of this century. This article is not about Machine Learning, but it treats one specific subject which is of immense importance to the former, that is Optimization.

One must know, almost every Machine Learning model is relying at the very end at some sort of optimization which intends to calibrate the model parameter.

There exists a huge number of different optimization techniques which apply on different problem settings. …

applied.math.coding

I am a mathematician and software developer who likes writing advanced code and do research in applied and pure mathematics.

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