Optimal control problems in mathematical biology: Modelling, analysis and numerics

在胚胎发育中，许多模式形成的机制已经被偏微分方程系统所确定和描述。由于这些偏微分方程中的一些参数或函数可能无法在实验室条件下测量，因此我们可以使用最优控制理论，特别是偏微分方程约束优化来识别它们。在我的项目中，我正在为PDE约束优化问题中出现的PDE设计快速数值求解器。在优化问题中，模型中的未知参数或函数被设置为控制变量。使用快速数值求解器和实验数据（例如图像），我们可以识别控制以完成模型。到目前为止，我已经处理了反应扩散方程，然而，我目前正在研究涉及趋化性的更具挑战性的问题，即细胞向上或向下的化学梯度运动。

Many of the mechanisms governing pattern formation in embryonic development have been identified and described by systems of PDEs. As some of the parameters or functions in these PDEs may be impossible to measure in laboratory conditions, we can use optimal control theory, in particular PDE-constrained optimization, to identify them. In my project, I'm devising fast numerical solvers for PDEs that arise from PDE-constrained optimisation problems. In the optimisation problem, the unknown parameter or function in the model is set to be the control variable. Using fast numerical solvers and experimental data (e.g. an image), we can identify the control to complete the model. So far, I have dealt with reaction-diffusion equations, however, I'm currently working on more challenging problems that involve chemotaxis, i.e. movement of cells up or down chemical gradients.