Abstract:
In this thesis, we will be looking at the inverse source problem for the Helmholtz equation in 1D. The application of inverse problem occurs in many fields of science such as in seismology and medical imaging. The inverse source problem is the problem of guring out the location and the shape of the source that is propagating waves when we are given the medium and the data. To find the source, we express the inverse problem as an optimisation problem with Partial Dierential Equation (PDE) constraints. Now we want to find a minimum for that problem. Since this problem is unstable we add regularisation such as the Tikhonov and the Total Variation method or reduce the problem in to a finite number of eigenfunctions which gets adapted every iteration to mitigate the instability. The novelty of my work is to find the optimal number of eigenfunctions for the inverse source problem. To do so, I needed to regularise the optimisation problem using the Adaptive Eigenspace Inversion (AEI) method. For the first time, I used techniques such as the L-curve method, the Morozov Discrepancy Principle and the General Cross Validation method to identify the optimal number of eigenfunctions in a systematic way.
