### Archive

Archive for the ‘Numerical Analysis’ Category

## Project Euler Problem 285

Another quite nice problem from Project Euler is number 285. The result of the problem depends on the computation of a certain probability, which in turn is related to the computation of a certain area. Below is an illustration of the computation for k equal to 10.

To save you some time, here’s a picture of the case k=1 which I ignored and spent quite a lot of time debugging… Plus, it only affects the last three digits or so after the decimal point…

Here’s a Matlab code which can construct the pictures above and can compute the result for low cases. To solve the problem, you should compute explicitly all these areas.


function problem285(k)

N = 100000;

a = rand(1,N);
b = rand(1,N);

ind = find(abs(sqrt((k*a+1).^2+(k*b+1).^2)-k)<0.5);

plot(a(ind),b(ind),'.');
axis equal

M = k;
pl = 1;

for k=1:M
if mod(k,100)==0
k
end
r1 = (k+0.5)/k;
r2 = (k-0.5)/k;

f1 = @(x) (x<=(-1/k+r1)).*(x>=(-1/k-r1)).*(sqrt(r1^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f1 = @(x) f1(x).*(f1(x)>=0);
f2 = @(x) (x<=(-1/k+r2)).*(x>=(-1/k-r2)).*(sqrt(r2^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f2 = @(x) f2(x).*(f2(x)>=0);

if k == pl
thetas = linspace(0,pi/2,200);
hold on
plot(-1/k+r1*cos(thetas),-1/k+r1*sin(thetas),'r','LineWidth',2);
plot(-1/k+r2*cos(thetas),-1/k+r2*sin(thetas),'r','LineWidth',2);
plot([0 1 1 0 0],[0 0 1 1 0],'k','LineWidth',3);
hold off
axis off
end

A(k) = integral(@(x) f1(x)-f2(x),0,1);

end

xs = xlim;
ys = ylim;

w = 0.01;
axis([xs(1)-w xs(2)+w ys(1)-w ys(2)+w]);

sum((1:k).*A)



## FreeFem to Matlab – fast mesh import

I recently wrote a brief introduction to FreeFem++ in this post. FreeFem is a software designed for the numerical study of partial differential equations. It has the advantage of being able to easily define the geometry of the domain, construct and modify meshes, finite element spaces and solve problems on these meshes.

I use Matlab very often for numerical computations. Most of the numerical stuff I’ve done (take a look here if you want) was based on finite differences methods, fundamental solutions and other classical techniques different from finite elements methods. Once I started using finite elements I quickly realized that Matlab is not that easy to work with if you want some automated quality meshing. PDEtool is good, but defining the geometry is not easy. There is also a simple tool: distmesh which performs a simple mesh construction for simple to state geometries. Nevertheless, once you have to define finite element spaces and solve problems things are not easy…

This brings us to the topic of this post: is it possible to interface Matlab and FreeFem? First, why would someone like to do this? Matlab is easier to code and use than FreeFem (for one who’s not a C expert…), but FreeFem deals better with meshes and solving PDE with finite elements. Since FreeFem can be called using system commands, it is possible to call a static program from Matlab. FreeFem can save meshes and variables to files. Let’s see how can we recover them in Matlab.

There is a tool called “FreeFem to Matlab” developed by Julien Dambrine (link on Mathworks). There’s also a longer explanation in this blog post. I recently tried to use the tool and I quickly found that it is not appropriate for large meshes. It probably scans the mesh file line by line which makes the loading process lengthy for high quality meshes. Fortunately there’s a way to speed up things and I present it below. I will not cover the import of the data (other than meshes) since the function importdata from the FreeFem to Matlab tool is fast enough for this.

## FreeFem++ Tutorial – Part 1

February 24, 2016 1 comment

First of all, FreeFem is a numerical computing software which allows a fast and automatized treatment of a variety of problems related to partial differential equations. Its name, FreeFem, speaks for itself: it is free and it uses the finite element method. Here are a few reasons for which you may choose to use FreeFem for a certain task:

1. It allows the user to easily define 2D (and 3D) geometries and it does all the work regarding the construction of meshes on these domains.
2. The problems you want to solve can be easily written in the program once we know their weak forms.
3. Once we have variables defined on meshes or solutions to some PDE, we can easily compute all sorts of quantities like integral energies, etc.

Before showing a first example, you need to install FreeFem. If you are not familiar with command line work or you just want to get to work, like me, you can install the visual version of FreeFem which is available here. Of course, you can find example programs in the FreeFem manual or by making a brief search on the internet.

I’ll present some basic stuff, which will allow us in the end to solve the Laplace equation in a circular domain. Once we have the structure of the program, it is possible to change the shape of the domain in no time.

## Even small errors can be fatal

Machines and computers represent numbers as sequences of zeros and ones called bits. The reason for doing this is the simplicity of constructing circuits dealing with two states. This fact coupled with limited memory capacities means that from the start we cannot represent all numbers in machine code. It is true that we can get as close as we want to any number with great memory cost when precision is important, but in fact we always use a fixed precision. In applications this precision is fixed to a number of bits (16, 32, 64) which correspond to significant digits in computations. Doing math operations to numbers represented on bits may lead to loss of information. Consider the following addition using ${15}$ significant digits:

$\displaystyle 5.00000000000002+6.00000000000003 = 11.0000000000000,$

notice that using only the first ${15}$ significant digits we have made an error of ${5\times 10^{-14}}$. This may seem small, but if we do not pay attention and we let errors of this kind accumulate we may have a final error which is unacceptable. This is what happened in the case of the Patriot missile case which I’ll discuss briefly below.

## Identifying edges and boundary points – 2D Mesh – Matlab

April 21, 2015 1 comment

A triangulation algorithm often gives as output a list of points, and a list of triangle. Each triangle is represented by the indexes of the points which form it. Sometimes we need extra information on the structure of the triangulation, like the list of edges, or the list of boundary points. I will present below two fast algorithms for doing this.

Finding the list of edges is not hard. The idea is to go through each triangle, and extract all edges. The algorithm proposed below creates the adjacency matrix in the following way: for each triangle ${[i,j,k]}$ we set the elements ${a_{ij},a_{jk},a_{ik}}$ (and their symmetric correspondents) to be equal to ${1}$.

In order to find the points on the boundary (in two dimensions), it is enough to look for the edges which are sides to only one triangle. We can do this using the adjacency matrix. Note that if ${A}$ is the adjacency matrix, then ${A^2=(b_{ik})}$ stores the number of paths of length ${2}$ (two sides) between two points of the triangulation. Note that any edge which is not on the boundary will contain the starting and ending point of two paths of length ${2}$. If ${[i,j,k]}$ is a triangle such that points ${i,j}$ are on the boundary, then ${b_{i,j}=1}$ (there is one path of length ${2}$ going through ${i,k,j}$. We also have ${a_{i,j} = 1}$. Conversely, if ${a_{i,j} = 1}$ and ${b_{i,j} = 1}$ then ${i,j}$ are connected, and there is a unique path of length ${2}$ going from ${i}$ to ${j}$. Thus, ${i,j}$ is an edge on the boundary. Therefore, we just need to identify the indexes ${i}$ such that there exists ${j}$ with ${a_{i,j} b_{i,j} = 1}$.

Below are two short Matlab codes doing these two algorithms. I guess they are close to being optimal, since only sparse and vectorized operations are used.


%p is the list of points
%T is the list of triangles, ap is the number of points
A = min(sparse(T(:,1),T(:,2),1,ap,ap)+sparse(T(:,2),T(:,3),1,ap,ap)+sparse(T(:,3),T(:,1),1,ap,ap),1);
A = min(A+A',1);
% this finds the boundary points, whose indexes are stored in Ibord
B = A^2.*A==1;
Ibord = find(sum(B,2)>0);


## Solving Poisson’s equation on a general shape using finite differences

One of the questions I received in the comments to my old post on solving Laplace equation (in fact this is Poisson’s equation) using finite differences was how to apply this procedure on arbitrary domains. It is not possible to use this method directly on general domains. The main problem is the fact that, unlike in the case of a square of rectangular domain, when we have a general shape, the boudary can have any orientation, not only the orientation of the coordinate axes. One way to avoid approach this problem would be using the Finite Element Method. Briefly, you discretize the boundary, you consider a triangulation of the domain with respect to this discretization, then you consider functions which are polynomial and have support in a few number of triangles. Thus the problem is reduced to a finite dimensional one, which can be written as a matrix problem. The implementation is not straightforward, since you need to conceive algorithms for doing the discretization and triangulation of your domain.

One other approach is to consider a rectangle ${D}$ which contains the shape ${\Omega}$ and add a penalization on the exterior of your domain ${\Omega}$. The problem to solve becomes something like:

$\displaystyle (-\Delta +\mu I) u = 1 \text{ on }D$

where ${\mu}$ is defined by

$\displaystyle \mu(x) = \begin{cases} 0 & x \in \Omega \\ + \infty & x \notin \Omega\end{cases}. \ \ \ \ \ (1)$

Note that doing this we do not need to impose the boundary condition ${u=0}$ on ${\partial \Omega}$. This is already imposed by ${\mu}$, and the fact that ${u}$ is forced to be zero outside ${\Omega}$.

## Two more mind games solved using Matlab

March 31, 2014 1 comment

After doing this, solving the Sudoku puzzle with Matlab, I wondered what else can we do using integer programming? Many of the mind games you can find on the Internet can be modelled as matrix problems with linear constraints. I will present two of them, as well as their solutions, below. The game ideas are taken from www.brainbashers.com. Note that in order to use the pieces of code I wrote you need to install the YALMIP toolbox.

1. Three in a row

You have a ${2n \times 2n}$ table ${(n \geq 3)}$ and on each row and column you need to have ${n}$ squares coloured with color ${A}$ and ${n}$ squares coloured with color ${B}$ such that no three squares taken vertically or horizontally have the same color. In the begining you are given the colors of some of the square such that it leads to a unique solution satisfying the given properties. The goal is to find the colors corresponding to each small square starting from the ones given such that the end configuration satisfies the given properties.