Details of Grant 

EPSRC Reference:	GR/J97076/01
Title:	MULTIDIMENSIONAL NONLINEAR INVERSE PROBLEMS AND ELECTROMAGNETIC INDUCTION IN THE EARTH>
Principal Investigator:	Huppert, Professor H E
Other Investigators:	Schultz, Dr A
Researcher Co-Investigators:
Project Partners:
Department:	Applied Maths and Theoretical Physics
Organisation:	University of Cambridge
Scheme:	Standard Research (Pre-FEC)
Starts:	01 July 1995	Ends:	31 March 1999	Value (£):	124,301
EPSRC Research Topic Classifications:	Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
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Summary on Grant Application Form
The mathematics of inverse theory is concerned with determining the behaviour of large systems governed by uncertain parameters. We will investigate multidimensional nonlinear inverse problems, in particular methods for the direct reconstruction of the coefficients in partial differential equations from incomplete knowledge of the solution and its gradient at the boundary. We are interested in the cases where the boundary data are I) necessary and sufficient or ii) grossly inadequate to determine uniquely the system. The applications of this research are widespread and include operations research, engineering, and the physical sciences. Initially, we will focus on the reconstruction of the electrical conductivity of a heterogeneous Earth from surface electromagnetic data. Our approach will be based on reducing the discrete inverse problem to a system of polynomial equations. We will construct real, positive (physical) solutions using the methods of homotopy continuation and develop an efficient solver recognising the special structure of the equations. We will determine how much boundary data permits the unique restoration of the discrete Maxwell equations in multidimensions. If the data are noisy, our plan is to construct optimal solutions to the governing polynomial equations, again using homotopy. We will also investigate applications of the theory to fluid mechanics and seismology.
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Organisation Website:	http://www.cam.ac.uk