UQ of NESO

Example scripts using EasyVVUQ to quantify the uncertainty in the electrostatic potential as a function of the initial particle velocity and the particle charge and number density (NOTE: these are set to the same value).

Usage

• Run pip install -r requirements.txt
• Modify the run.sh to match your environment
• Modify python/campaign.py file as required
• run python python/campaign.py

Explanation

This particular script campaign.py uses Polynomial Chaos Expansion(PCE) to estimate the uncertainty of the electrostatic potential ($\phi$) as a function of two uniformly distributed random variables- the initial particle velocity ($v$) and the particle charge density ($\rho$) - along a line $y=\mathrm{const}$.

PCE estimates $\phi$ at each evaluation point ($x_n$) as a sum of orthogonal polynomials $\Phi$,

$$ \phi(x_n,v,\rho) \approx \sum_{k=0}^{k=N}c_k(x_n)\Phi_k(v,\rho). $$

The number of runs required to calculate the approximation is $(p + 1)^d$, $p$ is the polynomial order, and $d$ is the number of uncertain parameters. In this script $p=3$ and $d=2$, but virtually identical results are obtained with $p=2$ (see moments_2ndorder.png and sobols_2ndorder.png)

Results

• DENSITY is the particle charge density and particle number density (these are always equal in this analysis)
• VELOCITY is the initial particle velocity