TU CE GSC CE People at the Graduate School CE Alumni Bettina Schieche Research Project of Bettina Schieche

Unsteady adaptive stochastic collocation methods in Uncertainty Quantification

A mathematical model that models some general physical phenomenon needs to be adjusted to every concrete simulation. That is, there are many parameters which have to be determined in order to map a certain situation to this model. Here, there are considered partial differential equations (PDEs) and input parameters might be:

- Topological information of the considered system
- Material or reaction properties
- Initial or boundary conditions

The arising problem is that they might not be explicitly known. Due to natural or human-made fluctuations and errors, one has to deal with uncertainties. Here, uncertain parameters are modeled by means of random variables or correlated random fields, which results in additional dimensions within the governing equation. This type of problem is often called "PDE with random parameters".

One rather new efficient approach for solving such problems are stochastic collocation methods on sparse grids:

- Discretise deterministic problem
- Discretise random parameter space by a set of collocation points
- Solve a deterministic PDE in each collocation point
- Interpolate all solutions and calculate statistics (e.g. expected value, variance, density...)

The main challenge is the choice of collocation points. One has to ensure an accurate resolution of the random space without too many deterministic PDEs that have to be solved. Therefore, efficient sparse grids based on Smolyak's algorithm are used.

The aim of this work is to derive error estimates for adaptive strategies in order to gain efficiency for the more and more complex problems arising in Computational Engineering. We want to combine stochastic collocation with an adjoint approach in order to estimate and control the error of the random solution field.

Multi-Physics; Numerics of partial differential equations; Uncertainties

Bettina Schieche

Dipl.-Math.

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