Uncertainty quantification and data assimilation

What we can help you with

A simulation produces results for a particular set of inputs and model assumptions. In practice, material properties, initial conditions, external forcing and other quantities are often only partially known. Uncertainty quantification examines how these uncertainties affect the results. Data assimilation combines models with observations to update our description of the system and improve predictions.

We can help you:

• Represent uncertainty in model parameters, initial conditions and input data.
• Select and adapt methods for propagating uncertainty through a simulator.
• Develop efficient ensemble computations and methods based on Monte Carlo, quasi-Monte Carlo and multilevel Monte Carlo.
• Combine simulations and measurements through ensemble-based data assimilation.
• Investigate which observations provide information about states and parameters.
• Analyse and communicate uncertainty in quantities relevant to a decision.

Do you have a simulation model with uncertain inputs or observations you would like to use more effectively? Contact our Applied Computational Science research group.

From uncertain inputs to uncertain results

Uncertainty may arise from limited measurements, natural variability or model simplifications. We work on representing uncertain quantities through probability distributions, spatial fields or ensembles of possible realisations. Dependencies matter: parameters that vary together should not automatically be treated as independent.

Through simulation, we examine how variability in the inputs affects selected outputs. These may include water levels, the trajectories of drifting objects, pressure evolution or other quantities used to inform decisions. The analysis can provide expected values, measures of spread, quantiles or probabilities of exceeding specified thresholds.

Results must be interpreted in light of the uncertainty actually represented. An ensemble that varies only a few parameters does not automatically capture uncertainty arising from the model's other assumptions.

Efficient methods for uncertainty propagation

Monte Carlo methods use repeated simulations with different inputs to estimate statistical properties of the results. These methods are flexible but can require many model runs. We work on selecting and adapting sampling methods to make effective use of computational resources.

Quasi-Monte Carlo uses systematically distributed points to cover the parameter space more evenly. Multilevel Monte Carlo combines many inexpensive computations at coarse levels with fewer computations at finer levels. By exploiting the relationship between levels, it can reduce the cost of achieving a given statistical accuracy.

The effectiveness of these methods depends on factors such as problem dimension, regularity and how models are coupled across levels. We have a particular interest in problems where abrupt transitions, fronts or other features of the physical model make standard smoothness assumptions inappropriate.

Data assimilation: updating models with observations

Observations provide an incomplete and often noisy picture of a physical system. The model describes how the system evolves and how different quantities are related. Data assimilation combines these sources of information while accounting for uncertainty in both.

We work with ensemble-based methods in which a collection of model realisations represents possible states or parameter values. As new observations become available, the ensemble is updated before the simulations continue. Measurements can therefore inform estimates of quantities and regions that are not observed directly.

A central challenge is to obtain useful updates with a limited number of ensemble members. We work on relating observations to model variables, handling spatial dependencies and maintaining a realistic ensemble spread through repeated updates.

Ensemble computations for dynamic systems

In time-dependent problems, uncertainty must be tracked as the system evolves. Small differences in initial conditions or external forcing can grow, and new observations can change both the prediction and its associated uncertainty.

One example is GPU Ocean, which combines ensembles of simplified ocean models with observations to investigate uncertainty in ocean currents and drift trajectories. This work brings together numerical simulation, ensemble-based data assimilation and GPU acceleration.

Read more about applications in coastal current and storm surge modelling.

When computational cost matters

Uncertainty analysis and data assimilation often require far more simulations than a single deterministic study. Statistical methods and numerical implementations therefore need to be developed together. Model resolution, ensemble size and computational accuracy affect both cost and quality.

We combine efficient sampling methods with parallel computing and GPU-accelerated simulation and numerical methods. When data volumes become large, statistical quantities can be computed during execution rather than storing every simulation output.

Where supported by the models, sensitivity analysis can provide complementary information about which parameters have the greatest influence on the results. This can help prioritise computations and identify particularly valuable observations.

A stronger basis for assessment and decisions

Uncertainty should be described in terms of the quantities that matter for the application. An average alone can conceal substantial variability, while a wide interval can have different implications depending on the thresholds or events of interest.

We examine how results depend on the number of realisations, model resolution and assumptions about uncertainty. For data assimilation, we also assess whether updates improve predictions and whether the estimated uncertainty is consistent with observed discrepancies.

The aim is to make both the results and their limitations understandable, so that simulations provide a stronger basis for further analysis, planning and optimisation.

Further reading

• GPU Ocean: software for GPU-accelerated ensemble computations and data assimilation in ocean models.
• Coastal current and storm surge modelling: simulation and the use of observations in marine applications.