From a single simulation to a range of possible outcomes
A simulation produces a result for a particular set of inputs and modelling assumptions. In practice, material properties, initial conditions, boundary conditions and external forcing are rarely known precisely. Model simplifications introduce further uncertainty. Uncertainty quantification investigates how these uncertainties affect the quantities we want to calculate.
The aim may be to estimate an expected value, variability, quantiles or the probability of exceeding a specified threshold. These results provide a better basis for comparing alternatives and assessing the robustness of conclusions drawn from simulations. Their interpretation must also reflect which sources of uncertainty and assumptions the analysis actually includes.
How we can help
- Design the computational approach: Identify the uncertain quantities and outputs of interest, and determine how uncertainty can be represented in the simulator.
- Select and adapt methods: Assess Monte Carlo, quasi-Monte Carlo and multilevel methods against the model's properties and the required accuracy.
- Reduce computational cost: Allocate simulations across different resolutions and make effective use of parallel computing and GPUs.
- Process large volumes of results: Calculate statistics and estimate probability distributions, either during simulation or through post-processing.
- Assess errors and reliability: Investigate statistical estimation error and numerical discretisation error, and clarify limitations in the uncertainty representation.
Methods adapted to the simulator
Many uncertainty quantification methods are available, but their efficiency depends on the problem. Some assume that outputs vary sufficiently smoothly with uncertain inputs. These assumptions can be difficult to verify, particularly when a model contains shocks, sharp fronts or transitions between physical regimes.
Our particular expertise includes:
- Monte Carlo: Uses random samples of the inputs to estimate statistical properties of model outputs. It requires little regularity, but may need many simulations to achieve a small statistical error.
- Quasi-Monte Carlo: Uses systematically distributed samples to cover the parameter space more evenly. This can improve convergence over standard Monte Carlo when the problem's regularity and effective dimension are favourable.
- Multilevel Monte Carlo: Combines many inexpensive simulations on coarse grids with fewer, coupled simulations on finer grids. Statistics from coarse models are corrected using differences between levels.
For multilevel methods, the interplay between discretisation, coupling of realisations and allocation of computational effort is crucial. We investigate when these methods offer a practical advantage and how they can be adapted to the simulator.
Uncertainty analysis for models with sharp fronts
In flow and transport models, small changes in the inputs can shift a front or change the timing of an event. An output measured at a particular location and time may therefore vary abruptly, even when other quantities, such as spatial averages or cumulative volumes, vary more smoothly.
We therefore assess the regularity of the quantity that actually needs to be estimated. Combining numerical analysis with knowledge of the simulator helps us select suitable methods and avoid computational approaches based on unrealistic smoothness assumptions.
Efficient computation and manageable data volumes
Uncertainty quantification can require thousands of model realisations and generate far more data than a single simulation. We combine method development with GPU-accelerated simulation and numerical methods to make these computations more feasible.
Where appropriate, we calculate statistics during the simulation rather than storing every model field from each realisation. This can reduce storage requirements and data movement. We also develop post-processing to estimate quantities such as expectations, variances, quantiles and probability densities.
Combining uncertainty quantification and data assimilation
Uncertainty quantification describes how uncertainty affects model outputs. Data assimilation uses observations to update estimates of model states and, where appropriate, parameters. The methods can be combined: after an update, the ensemble can be advanced to investigate uncertainty in new forecasts.
Work with us
Do you need to investigate uncertainty in simulation results, or is computational cost preventing the analyses you want to perform? Contact our Applied Computational Science research group to discuss method selection, simulator development and efficient computational approaches.