The goal of this project is to see if we can improve the treatment plan for proton therapy by using reduced order models and adjoint theory for proton therapy. We shall use a singular value decomposition on a dose distribution matrix to obtain the modes from which we can reconstruct every dose distribution. Using adjoint methodologies for proton therapy, we will define a response from which we can find the sensitivities, or gradient, in order to fit a Hermite interpolation polynomial on multidimensional simplices. The results show that the Hermite interpolation polynomial is a useful tool to find responses for low dimensional problems. For errors in one or two dimensions, the Hermite polynomial was able to reconstruct all dose distributions with R²=1. However, for an error in three dimensions the Hermite polynomial sometimes fails to reconstruct the dose distribution to within acceptable margins. We conclude that the combination of a ROM and adjoint method to find Hermite interpolation polynomial is a promising tool in order to further improve the proton therapy treatment plan. Further research should be done in order to determine whether Hermite interpolation can be used in every scenario, or if it fails if the grid becomes irregular. Finally, the extrapolating qualities of the Hermite polynomial should be tested.

By placing multiple Peltier elements in a linear arrangement while two water flows run past the elements, a temperature increase can be realised in one flow while the other flow is cooled down. In this study the heating of domestic hot water with Peltier elements as solid state heat pumps, and a heating network was investigated. A numerical model that solves the thermal energy balance within the Peltier elements was derived to describe the internal temperature distribution of the Peltier element, and its interaction with the domestic hot water and the heating network. The model was used to simulate the performance of 40 Peltier elements in a custom designed Peltier Heat Exchanger. Experiments were run to validate the numerical model. The numerical simulation of the temperature distribution within a Peltier Heat Exchanger and the temperature distributions observed in the experiments were not in agreement. The model input parameters Seebeck coefficient, resistance, thermal conductivity and a relation for the Nusselt number were re-evaluated using the experimental results. After the adjustment of the model input parameters, the new simulation results were able to accurately describe the temperature distribution with the Peltier Heat Exchanger. The Peltier Heat Exchanger was able to deliver domestic hot water with a COP between 1.2 and 1.8 depending on the flow speed of the domestic hot water and the heating network. The COP can potentially be increased by using Peltier elements with a higher Seebeck coefficient.