Sensitivity Analysis of an Ordinary Differential Equation (ODE)
MADS is applied to perform sensitivity analysis of an Ordinary Differential Equation (ODE).
The analyses below are performed using examples/ode/ode.jl.
ODE
Analyzed ODE looks like this:
\[x''(t) = -\omega^2 x(t) - k x'(t)\]
Unknown ODE parameters:
- k
- omega ($\omega$)
Example ODE solution:
For model parameters:
- k = 0.1
- omega ($\omega$) = 0.2
Local sensitivity analysis
Global sensitivity analysis (using eFAST)
Probabilistic distributions of the prior parameter uncertainties are:
- k = LogUniform(0.01, 0.1)
- omega ($\omega$) = Uniform(0.1, 0.3)
Bayesian sensitivity analysis
Observations
Synthetic observations are applied to constrain the ODE parameters:
Observation errors are equal for all the sample locations with standard deviation equal to 1 (observation weight = 1 / observation standard deviation = 1 / 1 = 1)
Prior parameter uncertainties
k only
omega ($\omega$) only
Both parameters
The observation data are plotted as a solid black line.
Histograms/scatter plots of Bayesian MCMC results
Posterior parameter uncertainties
Note that now the parameter uncertainties are constrained by the observation data.
k only
omega ($\omega$) only
Both parameters
The observation data are plotted as a solid black line.