# The cookies model

## Overview
This model implements the so-called 'cookies problem' or 'cookies in the oven problem' (see for reference [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3), [[Ballani et al.,2015]](https://doi.org/10.1137/140960980), [[Kressner et al., 2011]](https://doi.org/10.1137/100799010)), i.e., a simplified thermal equation in which the conductivity coefficient is uncertain in 8 circular subdomains ('the cookies'), whereas it is known (and constant) in the remaining of the domain ('the oven'). The PDE is solved by an isogeometric solver with maximum continuity splines, whose degree can be set by the user. See below for full description. 


## Authors
- [Massimiliano Martinelli](mailto:martinelli@imati.cnr.it)
- [Lorenzo Tamellini](mailto:tamellini@imati.cnr.it)

## Run
```
docker run -it -p 4242:4242 linusseelinger/cookies-problem
```

## Properties

Model     | Description
---       | ---
forward   | Forward evaluation of the cookies model, all config options can be modified by the user (see below)
benchmark | Sets the config options for the forward UQ benchmark [(see benchmark page)](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/cookies-problem/README.md)

### forward

Mapping | Dimensions | Description
---     |---         |---
input   | [8]        | These values modify the conductivity coefficient in the 8 cookies, each of them must be greater than -1 (software does not check that input values are valid)  
output  | [1]        | The integral of the solution over the central subdomain (see definition of $\Psi$ below)

Feature       | Supported
---           |---
Evaluate      | True
Gradient      | False
ApplyJacobian | False
ApplyHessian  | False

Config        | Type    | Default | Description
---           |---      |---      |---
NumThreads    | integer | 1       | Number of physical cores to be used by the solver
BasisDegree   | integer | 4       | Default degree of spline basis (must be a positive integer)
Fidelity      | integer | 2       | Controls the number of mesh elements (must be a positive integer, see below for details)


## Mount directories
Mount directory | Purpose
---             |---
None            | 

## Source code

[Model sources here.](https://github.com/UM-Bridge/benchmarks/tree/main/models/cookies-problem)

## Description

![cookies-problem](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/models/cookies-problem/cookies_domain.png "geometry of the cookies problem")

The model implements the version of the cookies problem in [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3), see also e.g. [[Ballani et al.,2015]](https://doi.org/10.1137/140960980), [[Kressner et al., 2011]](https://doi.org/10.1137/100799010) for slightly different versions. With reference to the computational domain $D=[0,1]^2$ in the figure above, the cookies model consists in the thermal diffusion problem below, where $\mathbf{y}$ are the uncertain parameters discussed in the following and $\mathrm{x}$ are physical coordinates 

$$-\mathrm{div}\Big[ a(\mathbf{x},\mathbf{y}) \nabla u(\mathbf{x},\mathbf{y}) \Big] = f(\mathrm{x}), \quad \mathbf{x}\in D$$

with homogeneous Dirichlet boundary conditions and forcing term defined as

$$f(\mathrm{x}) = \begin{cases} 
100 &\text{if } \,  \mathrm{x} \in F \\
0 &\text{otherwise} 
\end{cases}$$

where $F$ is the square $[0.4, 0.6]^2$. The 8 subdomains with uncertain diffusion coefficient (the cookies) are circles with radius 0.13 and the following center coordinates:

cookie | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   |
--     | --  | --  | --  | --  | --  | --  | --  | --  |
x      | 0.2 | 0.5 | 0.8 | 0.2 | 0.8 | 0.2 | 0.5 | 0.8 |
y      | 0.2 | 0.2 | 0.2 | 0.5 | 0.5 | 0.8 | 0.8 | 0.8 |

The uncertain diffusion coefficient is defined as

$$a = 1 + \sum_{i=1}^8 y_n \chi_n(\mathrm{x})$$ 

where $y_n>-1$ and 

$$\chi_n(\mathrm{x}) = \begin{cases} 1 &\text{inside the n-th cookie} \\ 0 &\text{otherwise} \end{cases}$$


The output of the simulation is the integral of the solution over $F$, i.e. $\Psi = \int_F u(\mathrm{x}) d \mathrm{x}$


The PDE is solved with an IGA solver (see e.g. [[Da Vaiga et al., 2014]](https://doi.org/10.1017/S096249291400004X)) that uses as basis splines of degree $p$ (tunable by the user, default $p=4$) of maximal regularity, i.e. of continuity $p-1$. The computational mesh is an $N\times N$ quadrilateral mesh (cartesian product of knot lines) with square elements, with $N=100 \times \mathrm{Fidelity}$. The implementation is done using the C++ library IGATools [[Pauletti et al., 2015]](https://doi.org/10.1137/140955252), available at [gitlab.com/max.martinelli/igatools](gitlab.com/max.martinelli/igatools).