Solving Stochastic Ordinary Differntial Equation using PCE

UQ with UQLab Community Q&A and How To
1

Hi,

I would like to solve an ordinary stochastic differential equation using PCE. Is there any example to follow.

Thanks,
Ahmed

3

Hi @Ahmed_Atallah ,

Thanks for your interest in UQLab!

It depends on your stochastic ODE. Is it stochastic because one or several parameters are stochastic? In that case: The PCE module of UQLab uses the black-box approach, i.e., it is non-intrusive. Therefore, you can apply PCE in the same way that you would apply it to any other model: You specify an experimental design (in the random parameter space), compute model evaluations, and based on this data set you fit a PCE.

Is your quantity of interest a scalar or the whole time series? In the first case, it should be quite straightforward to solve. In the second case, maybe these discussions help you:

In case your ODE is stochastic because there is some stochastic process driving the evolution, your model might actually be a stochastic simulator. Then it is not straightforward to use PCE.

Hope this helps! :slight_smile: Feel free to give us more information about your model, inputs and outputs, and keep us updated about how you solve your problem.

5

Hi Nora,

Thanks so much!

I am interested in the non-intrusive approach. The problem I want to solve is the orbit problem which is an ODE. I want to propagate the uncertainty caused by the initial condition and some parameters.

Thanks,
Ahmed

6

Hi Ahmed,

It sounds like it could be done in a straightforward way by

  1. representing the initial condition and the parameters as Input object (with proper distributions),
  2. creating a model that solves the ODE based on your inputs and returns the corresponding output (scalar? else, see e.g. Nagel, Rieckermann, Sudret (2020) for how to deal with time series output),
  3. computing a PCE for this setup.

Have you figured out how to do it? If yes, could you share your solution with us?