# The cookies problem forward UQ benchmark

## Overview

This benchmark runs a forward uncertainty quantification problem for the [cookies model](https://github.com/UM-Bridge/benchmarks/tree/main/models/cookies-problem/README.md) using the [Sparse Grids Matlab Kit]((https://github.com/lorenzo-tamellini/sparse-grids-matlab-kit)) interface to UM-Bridge. 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
---       | ---
benchmark | Sets the config options for the forward UQ benchmark (see below)

### Benchmark configuration

Mapping | Dimensions   | Description
---     |---           |---
input   | [8]          | These values modify the conductivity coefficient in the 8 cookies. They are i.i.d. uniform random variables in the range [-0.99, -0.2] (software does not check that inputs are within the bound) 
output  | \[1\]        | The integral of the solution over the central subdomain (see definition of $\Psi$ at [cookies model](https://github.com/UM-Bridge/benchmarks/tree/main/models/cookies-problem/README.md) for info)

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

Config        | Type    | Default value   	| Can be changed in benchmark model	| Description
---           |---      |---      		|---		  			| ---	
NumThreads    | integer | 1     		| yes					| Number of physical cores to be used by the solver 
BasisDegree   | integer | 4       		| no					| Default degree of spline basis (must be a positive integer)
Fidelity      | integer | 2       		| no					| Controls the number of mesh elements (must be a positive integer, see below for details)


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

## Source code

[Benchmark sources available at this folder.](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/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 benchmark implements a forward uncertainty quantification problem for the [cookies model](https://github.com/UM-Bridge/benchmarks/tree/main/models/cookies-problem/README.md). More specifically, we assume that the uncertain parameters $y_n$ appearing in the definition of the diffusion coefficient are uniform i.i.d. random variables on the range $[-0.99, -0.2]$ and we aim at computing the expected value of the quantity of interest (i.e., output of the model) $\Psi$, which is defined as the integral of the solution over $F$.

The PDE is solved with an IGA solver that uses as basis splines of degree $p=4$ and maximal regularity, i.e. of continuity $3$, and the mesh has $200 \times 200$ elements (i.e., the fidelity config parameter is set to $2$). The structure of this benchmark is identical to the one discussed in [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3); however, raw numbers are different since in [[Bäck et al.,2011]](https://doi.org/10.1007/978-3-642-15337-2_3) the PDE solver employed was different (standard FEM with piecewise linear basis) and the mesh was also different.


As a reference value, we provide the approximation of the expected value computed with a standard Smolyak sparse grid, based on Clenshaw--Curtis points, for level $w=5$, see e.g. [[Piazzola et al.,2023]](https://doi.org/10.48550/arXiv.2203.09314). The resulting sparse grid has 15713 points, and the corresponding approximation of the expected value is $0.064196096847169$.


The script available [here](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/cookies-problem/run_forward_benchmark_in_matlab.m) generates the results, using the Sparse Grids Matlab Kit [[Piazzola et al.,2023]](https://doi.org/10.48550/arXiv.2203.09314) for generating sparse grids. The Grids Matlab Kit is available on Github [here](https://github.com/lorenzo-tamellini/sparse-grids-matlab-kit) and a dedicated website with full resources including user manual is available [here](https://sites.google.com/view/sparse-grids-kit).