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

反应-扩散-平流方程在工程和自然科学的许多领域都被用来模拟时空过程。由于这些数学模型的参数往往是未知的，它们必须根据现有的实验数据来确定。在这里，第一步通常是使用最优化来确定参数值，以产生模型预测和实验数据的最佳匹配。在第二步中，分析这些参数值的不确定性，以确定模型的预测能力。在这两个步骤中，都必须解决约束优化问题。为此，需要可靠的优化算法来稳健地收敛。然而，现有的方法不能满足各种模型的可靠性要求。本项目的目的是开发一种新的基于模拟的反应-扩散-平流方程的优化方法，该方法通过利用优化问题的结构来获得更可靠的结果。利用控制工程和最优化理论的方法，我们将建立一个以优化问题的最优解为平衡点的常微分方程组和偏微分方程组的耦合系统。这使得能够使用自适应数值方法来解决具有PDE约束的优化问题。这种基于模拟的方法将允许比现有优化方法中使用的简单步长控制更稳健的收敛。对于ODE约束问题，我们已经开发了类似的优化方法，我们已经能够证明这些改进的性质。项目中开发的优化方法将用于确定最优参数值(步骤1)和使用轮廓似然进行不确定性分析(步骤2)。轮廓概率主要是通过重复优化来计算的，但是这个过程需要大量的计算。我们修改了用于优化的耦合的ODE-PDE系统，以便它们沿着单独的轮廓进化。为了评估和改进所开发的优化和不确定性分析方法，我们将把它们与我们在其他项目中使用的最新算法(如IPOT、NLPQLP和MATLAB例程fmincon)进行比较。我们将使用这些方法来研究斑马鱼的侧线形成。这一过程被描述为一个高度非线性的反应-扩散-平流耦合方程组，现有的优化方法存在严重的收敛问题。因此，这个例子非常适合于评估已开发的方法。除了纯粹的方法开发，这个项目可以为侧线形成过程中复杂神经元结构的发展提供新的见解。

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.