Adjoint Method For Inverse Problems Of Coupled Groundwater Models

In modeling of groundwater, compared to traditional methods, coupled models such as CCPF model and Stokes-Darcy system, could simulate the true situation better. These models consider the complicated case in practice and coupled different equations in different regions with interface conditions. Existence and regularity of the solution of the models has been established [8,19,33], based on the result of CCPF model, Lu etc [28]proposed two inverse problems, determining the exchange rate function and shape of the conduit with the knowledge of boundary cauchy data.For inverse problem, we recast the problem as a PDE-constraint optimization prob-lem. The optimal variant is the exchange rate function or the shape of the domain. We find the optimal variant that minimizes a cost function that quantifies the difference be-tween the measured and expected signals. The measured and expected signals satisfy the CCPF model and Stokes-Darcy system. Gradient-based algorithm could solve this optimization problem. At every step of the algorithm, the derivative of the cost function with respect to the exchange rate function or the parameters that describe the shape of object is calculated. The total cost of the method is equal to one forward solution and one adjoint solution.In the thesis, an efficient method based on the adjoint equations to calculate the derivative is developed and implemented in CCPF model. Numerical examples show the efficacy. Similarly, an inverse problem for determining the exchange rate function in Stokes-Darcy system is proposed. Finally, a framework of calculating the shape derivative for shape optimization problem is given and a special shape optimization problem for1-D case is established.