Parameter control in the presence of uncertainties

Abstract

To understand and to be able to forecast natural phenomena is increasingly important nowadays, as those predictions are often the basis of many decisions, whether economical or ecological. In order todo so, mathematical models are introduced to represent the reality at a specific scale, and are then implemented numerically.

However in this process of modelling, many complex phenomena occurring at a smaller scale than the one studied have to be simplified and quantified. This often leads to the introduction of additional parameters, which then need to be properly estimated. Classical methods of estimation usually involve an objective function, that measures the distance between the simulations and some observations, which is then optimised. Such an optimisation require many runs of the numerical model and possibly the computation of its gradient, thus can be expensive to evaluate computational-wise.However, some other uncertainties can also be present, which represent some uncontrollable and external factors that affect the modelling. Those variables will be qualified as environmental. By modelling them with a random variable, the objective function is then a random variable as well, that we wish to minimise in some sense. Omitting the random nature of the environmental variable can lead to localised optimisation, and thus a value of the parameters that is optimal only for the fixed nominal value.

To overcome this, the minimisation of the expected value of the objective function is often considered in the field of optimisation under uncertainty for instance.In this thesis, we focus instead on the notion of regret, that measures the deviation of the objective function from its optimal value given a realisation of the environmental variable. This regret (either additive or relative) translates a notion of robustness through its probability of exceeding a specified threshold. So, by either controlling the threshold or the probability, we can define a family of estimators based on this regret.

The regret can quickly become expensive to evaluate since it requires an optimisation of the objective for every realisation of the environmental variable. We then propose to use Gaussian Processes (GP) in order to reduce the computational burden of this evaluation. In addition to that, we propose a few adaptive methods in order to improve the estimation: the next points to evaluate are chosensequentially according to a specific criterion, in a Stepwise Uncertainty Reduction (SUR) strategy.

Finally, we will apply some of the methods introduced in this thesis on an academic problem of parameter estimation. We will study the calibration of the bottom friction of a model of the Atlantic ocean near the French coasts, while introducing some uncertainties in the forcing of the tide, and get a robust estimation of this friction parameter in a twin experiment setting.

Keywords

• Calibration
• Robust optimisation
• Relative-regret
• Shallow-water equations
• Ocean Modelling