# Solving inverse problems with Adaptive Eigenspace Inversion method

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Dr M Graff Applications accepted all year round Self-Funded PhD Students Only

Parameter estimation is a fundamental task in various areas of engineering and science such as medical imaging and seismic exploration. In both applications we wish to provide an image of the characteristics of a medium from indirect measurements of a wavefield as it passes through the medium. Even though technologies already exist for imaging, there is still a need for improved resolution while preserving reasonable computational cost and storage size.
Solving an imaging problem often consists of expressing an optimisation problem with partial differential equation constraints. The aim is to choose a reconstruction of the image that minimises an appropriate residual function of the measured data. In essence, this is an inverse problem and it is generally ill-posed in the sense of Hadamard: we cannot ensure the existence, uniqueness and stability of a solution. Hence, one must stabilise the inverse problem, that is control the solution of the inverse problem and direct it to a unique optimum. Such regularisation is usually achieved by a penalty term, such as the well-known Tikhonov regularisation technique.
In this project, we will design and analyse new and improved high-resolution imaging techniques, namely, generalisations of the adaptive eigenspace inversion (AEI) method. The proposed research will establish a rigorous theory for AEI and explore a probabilistic counterpart of AEI that allows for better imaging methods along with a quantification of uncertainties.

What we are looking for in a successful applicant
• A good knowledge of partial differential equations and linear algebra.
• Progamming experience is essential.
• Knowledge in optimisation, inverse problems and statistics is a plus.

Objective
• Implement the inverse source problem in 1D for the Helmholtz equation and systematically study the concentration properties of the eigenfunctions.
• Study the geometry of the eigenfunctions in 2D and combine with frequency stepping.
• Derive the Bayesian AEI framework and implement it for the inverse source problem. Identify an optimal approach to update prior measures in the Bayesian AEI framework.

## Funding Notes

The projects are intended for self-funded PhD students and students who are eligible for the general scholarships offered by the University of Auckland; see

International students are also encouraged to explore funding opportunities in their home countries for studying abroad.

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