Simulation-based Parameter Optimisation and Uncertainty Analysis Methods for Reaction-Diffusion-Advection Equations

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Reaction-diffusion-advection equations are used in many fields of engineering and natural sciences to model spatio-temporal processes. As the parameters of these mathematical models are often unknown, they have to be determined from the available experimental data. Here, the first step is usually to employ optimisation to determine the parameter values yielding the best match of the model prediction and the experimental data. In the second step, the uncertainty of these parameter values is analysed to determine the predictive power of the model. In both steps constraint optimisation problems have to be solved. For this reliable optimisation algorithms are required which converge robustly. Available methods however fail to meet these reliability requirements for a variety of models.The aim of this project is to develop a novel simulation-based optimisation approach for reaction-diffusion-advection equations, which is considerable more reliable by exploiting the structure of the optimisation problem. Using methods from control engineering and optimisation theory, we will formulate a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs), which has the optima of the optimisation problem as equilibrium points. This enables the use of adaptive numerical methods for solving optimisation problems with PDE constraints. This simulation-based approach will allow for more robust convergence than simple step-size controls used in existing optimisation methods. For ODE constrained problems, for which we developed a similar optimisation approach, we were already able to demonstrate these improved properties.The optimisation approaches developed in the project will be employed to determine the optimal parameter values (Step 1) and to perform uncertainty analysis using profile likelihoods (Step 2). Profile likelihoods are mostly calculated by repeated optimisation, this process is however computationally demanding. We modify the coupled ODE-PDE systems used for optimisation, such that they evolve along the individual profiles. Accordingly, the simulation of these reformulated coupled ODE-PDE systems will replace the repeated optimisation and reduce the computation time.To evaluate and improve the developed optimisation and uncertainty analysis approaches, we will compare them with state-of-the-art algorithms we are using in other projects (e.g. Ipopt, NLPQLP and the MATLAB routine fmincon). We will use the methods to study lateral line formation in zebrafish. This process is described by a highly non-linear system of coupled reaction-diffusion-advection equations and existing optimisation methods have severe convergence problems. Therefore, this example is very well suited for the evaluation of developed approaches. Beyond the pure method development, this project could provide new insights into the development of complex neuronal structures during lateral line formation.