The cookies problem forward UQ benchmark#
Overview#
This benchmark runs a forward uncertainty quantification problem for the cookies model using the Sparse Grids Matlab Kit interface to UM-Bridge. See below for full description.
Run#
docker run -it -p 4242:4242 linusseelinger/cookiebenchmark
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 for info) |
Feature |
Supported |
|---|---|
Evaluate |
True |
Gradient |
False |
ApplyJacobian |
False |
ApplyHessian |
False |
Config |
Type |
Default value |
Description |
|---|---|---|---|
None |
Mount directories#
Mount directory |
Purpose |
|---|---|
None |
Source code#
Description#
The benchmark implements a forward uncertainty quantification problem for the elliptic version of the cookies model. 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 configuration of the docker uses all config options set to their default values, see againg the cookies model page. The structure of this benchmark thus is identical to the one discussed in [Bäck et al.,2011]; however, raw numbers are different since in [Bäck et al.,2011] a different mesh was used.
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].
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 |
The script available here generates the results, using the Sparse Grids Matlab Kit [Piazzola et al.,2024] for generating sparse grids. The Grids Matlab Kit is available on Github here and a dedicated website with full resources including user manual is available here.