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## Overview | ||
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This benchmark runs a forward uncertainty quantification problem for the [cookies model](https://github.com/UM-Bridge/benchmarks/tree/main/models/fenics-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. | ||
This benchmark runs a forward uncertainty quantification problem for the [cookies model](https://um-bridge-benchmarks.readthedocs.io/en/docs/models/fenics-cookies-problem.html) using the [Sparse Grids Matlab Kit](https://github.com/lorenzo-tamellini/sparse-grids-matlab-kit) interface to UM-Bridge. See below for full description. | ||
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## Authors | ||
- [Benjamin Kent]([email protected]) | ||
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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/fenics-cookies-problem/README.md) for info) | ||
output | \[1\] | The integral of the solution over the central subdomain (see definition of $\Psi$ at [cookies model](https://um-bridge-benchmarks.readthedocs.io/en/docs/models/fenics-cookies-problem.html) for info) | ||
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Feature | Supported | ||
--- |--- | ||
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## Source code | ||
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[Benchmark sources available at this folder.](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/cookies-problem-propagation) | ||
[Model sources here..](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/cookies-problem-propagation) | ||
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## Description | ||
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![cookies-problem](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/models/fenics-cookies-problem/cookies_domain.png "geometry of the cookies problem") | ||
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The benchmark implements a forward uncertainty quantification problem for the elliptic version of the [cookies model](https://github.com/UM-Bridge/benchmarks/tree/main/models/fenics-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 benchmark implements a forward uncertainty quantification problem for the elliptic version of the [cookies model](https://um-bridge-benchmarks.readthedocs.io/en/docs/models/fenics-cookies-problem.html). 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 $F$. | ||
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The benchmark configuration of the docker uses all config options set to their default values, see againg the [cookies model page](https://github.com/UM-Bridge/benchmarks/tree/main/models/fenics-cookies-problem/README.md). The structure of this benchmark thus 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) a different mesh was used. | ||
The benchmark configuration of the docker uses all config options set to their default values, see againg the [cookies model page](https://um-bridge-benchmarks.readthedocs.io/en/docs/models/fenics-cookies-problem.html). The structure of this benchmark thus 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) a different mesh was used. | ||
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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 levels $$w=0,1,\ldots,5$$, see e.g. [[Piazzola et al.,2024]](https://doi.org/10.1145/3630023). | ||
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 levels $w=0,1,\ldots,5$, see e.g. [[Piazzola et al.,2024]](https://doi.org/10.1145/3630023). | ||
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Sparse grid $$w$$ | number of collocation points | Estimate of $$\Psi$$ | ||
---------------- |-------------------------------- |------------------- | ||
0 | 1 | 0.062255257529767 | ||
1 | 17 | 0.064176316082952 | ||
2 | 145 | 0.064206407272061 | ||
3 | 849 | 0.064202639076811 | ||
4 | 3937 | 0.064202350667514 | ||
5 | 15713 | 0.064202367186117 | ||
Sparse grid $w$ | Number of collocation points | Estimate of $\Psi$ | ||
----------------|---------------------------------|------------------- | ||
0 | 1 | 0.062255257529767 | ||
1 | 17 | 0.064176316082952 | ||
2 | 145 | 0.064206407272061 | ||
3 | 849 | 0.064202639076811 | ||
4 | 3937 | 0.064202350667514 | ||
5 | 15713 | 0.064202367186117 | ||
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The script available [here](https://github.com/UM-Bridge/benchmarks/tree/main/benchmarks/cookies-problem-propagation/run_forward_benchmark_in_matlab.m) generates the results, using the Sparse Grids Matlab Kit [[Piazzola et al.,2024]](https://doi.org/10.1145/3630023) 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). |