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https://github.com/p-acampora98/schwarz-rearrangement

MATLAB implementation of Schwarz rearangement and rearrangement along level sets of other functions.
https://github.com/p-acampora98/schwarz-rearrangement

mathematics matlab rearrangement shape-optimization

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MATLAB implementation of Schwarz rearangement and rearrangement along level sets of other functions.

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README 

        
This is a repository containing tools to rearrange two-variables functions in MATLAB. 

![Schwarz](Schwarz.png)

# Installation

Add the folder "Code" to your MATLAB path and you are ready to go!



# Usage

Fucntions are stored as `contourLines` objects. This is a class that contains all the needed informations to rearrange.

## Constructor

Given a two-variables function `F`, you want to pass it to the constructor as a function handle.

``` 

function z = F(x,y)

	z=x+y;

end

Fsharp = contourLines(@F);

```

The line `Fsharp = contourLines(@F);` will create a *contourLines* object named Fsharp, which stores 100 levels of the function F .  The complete constructor method can be called as follows.

```

Fsharp = contourLines(@F, gridCoordinates,numberOfCells,numberOfLevels);

```

When specified, the variables should be of the following form:

- `gridCoordinates=[xmin xmax ymin ymax]` is describing the square domain in which the function is restricted to; **default** `[-1 1 -1 1]`; 

- `numberOfCells` determines how many points are constructed in the grid; **default** `20` will give a 20x20 grid;

- `numberOfLevels` determines how many levels of the function are stored; **default** `100`

### Different bounded domains

If you want to construct a function with a domain E different from a rectangle, you can pass to the constructor method a function `F` which is `-inf` outside of E.

## Schwarz rearrangement

Given a function F and the contourLines object `Fsharp` relative to it, you can call the Schwarz (spherically decreasing) rearrangement of F as the function

```

Fsharp.Schwarz(x,y);

```

with $x$ and $y$ real numbers. If the couple $(x,y)$ is outside of the ball where the Schwarz rearrangement is defined, a `NaN` will be given as an output.

## Rearrangement along another function

An example is given in the file `Rearrangement2.m`.



![SquaredRearrangement](SquaredRearrangement.png)



Given two functions `F`, `G` and the respective contourLines objects `Fsharp` and `Gsharp`,  you can call the rearrangement of F along G as the function

```

Fsharp.Schwarz(x,y,Gsharp);

```

with x and y real numbers. If the couple (x,y) is outside of the set where the Schwarz rearrangement is defined (i.e. the biggest superlevel set of G inside the initial grid), a `NaN` will be given as an output.

## Plot Level

Given a contourLines object `Fsharp`, you can plot the `i`-th level of `Fsharp` with the line

```

Fsharp.plotLevel(i)

```

The number `i` can go from 1 to `Fsharp.numberOfLevels`. Bigger the `i`, smaller the level set.

## Distribution function

 The distribution function of $F$ is the function

```math

\mu_F(t)=\lvert \set{f>t}\rvert. 

```

A distribution function is stored inside the class contourLines. If Fsharp is a contourLines object, you can call its distribution function by

```

Fsharp.distributionFunction(t);

```

where `t` is a real number.

## Radially decreasing rearrangement

The radially decreasing rearrangement of $F$ is defined as

```math

	F^*(m)=\inf \Set{t\in\mathbb{R} | \mu_F(t)\le m}.

```

The radially decreasing rearrangement function is stored in the contourLines object, and you can call it by

```

Fsharp.radDecrRearr(m);

```

where `m` is a real number.