Mathematical Sciences: Inverse Problems, Impedance Potentials, and Wavelets

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This research involves the efficient and accurate numerical solution of inverse problems by means of the construction of improved potential functions and the application of wavelets. In an inverse problem one tries to reconstruct an equation from information about its solutions, and this leads to difficult problems in numerical linear algebra, since for a large system the reconstruction must be done on a computer. It may seem strange but very often one has solutions to a problem without knowing exactly what the problem is itself. Think of the case where your doctor takes X-rays of you during an examination in order to find out if anything is amiss. The data from the X-rays are essentially like the solutions, whereas the problem in this case might be an unknown malady. Your doctor tries to determine what particular condition ıequation! is producing the data ısolutions!. In much the same spirit the principal investigator will study numerical techniques for determining the equations that are producing certain types of solutions.