This project was aimed at developing and analyzing efficient numerical tools for the solution of large scale inverse problems. The focus was towards careful investigation of the underyling matrix operators that define the forward problems, for linear problems, and on extending these approaches in the context of nonlinear problems.

Intellectual Merit

New algorithms for the inversion of the gravity and magnetic potential field data sets were designed and implemented. These algorithms were also used for investigating practical data sets, for example in a study of kimberlites in Botswana and in initial studies for the identification of thermal sources in Algeria. The latter is particularly important for green energy. 

A main contribution of the research, that fed also into the later research, is an open source code for efficiently evaluating the forward operators that describe  gravity and magnetic  potential fields. These operators exhibit a specific structure which is advantageous for efficient forward and adjoint operations using fast Fourier transforms, and avoids storage of the matrix entries. For standard large scale geophysics problems, this might require many GB memory and makes the inversion of data infeasible. With the structured formulation, this opens the field to the efficient solution of large scale three-dimensional problems, and in particular the practical studies for large data sets were only feasible in this framework. Moreover, with the efficient storage and implementation it is now viable to use joint inversion of the two data sets, leading to improved identification of subsurface structures with dominant density and magnetic susceptiblity profiles. 

In a complementary direction, this research project also addressed unsupervised machine learning for windowed spectral regularization for image restoration in the presence of noise and blur. There, matrix structure is also paramount, and provides efficient implementation. Combining information from multiple images leads to more robust of restoration using spectral windowing. This is early work that will be extended further for three dimensional projection problems. 

Broader Impacts

The project provided support for an undergraduate research experience for one female student. This student was part of an undergraduate team who were supported with other sources of funding, and has lead to one minority female student continuing on the image restoration project, after her transfer from the Community College to the university Moreover, two graduate students (one minority male) have completed their doctoral dissertations in applied mathematics with some support of the funding, one who focused mainly on the geophysics and one on the machine learning problem. The former has progressed to the postdoctoral  level.The second is  investigating positions in industry. The potential application to the geophysics problems has the potential for further impact on thermal energy studies. A  geophyscis doctoral student from Algeria visited the US to participate in this research, and is responsible for the collection of the gravity potential field data used for this investigation.