METRIC DEPENDENT ASPECTS OF INVERSE PROBLEMS AND FUNCTIONALS BASED ONHELICITY

The helicity of a vector field is a metric independent density. Functi onals with first order elliptic systems for Euler-Lagrange equations h ave been constructed from the helicity. The metric invariance is prese rved for finite element discretizations involving ''Whitney elements.' ' This paper relates differential geometric aspects of inverse problem s to helicity based functionals in two contexts. First, the inverse pr oblem of electrical impedance tomography in isotropic media is known t o be equivalent to determining a metric within a given conformal class from a given ''Dirichlet to Neumann'' map. This fact is related to th e helicity functional and Wexler's algorithm for recovering an isotrop ic conductivity. Second, Maxwell's equations in ''spinor form'' are sh own to be the Euler-Lagrange equations of some complexified time depen dent generalization of the helicity functional. In this case metric de pendent aspects yield insight into the ''inverse kinematic problem in seismology.'' These two examples illustrate the underlying geometric s tructure in classes of inverse problems and algorithms for their solut ion.