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2 Direct and Inverse Problem Formulation     continued...

Once the problem of nonuniqueness is addressed, there still exists the problem occurring when the solution does not depend continuously on the data. The linear algebra counterpart to the elliptic boundary value problem is often useful in discussing this problem of discontinuity. The numerical solution to all elliptic boundary value problems (such as the Poisson and Laplace problems) can be written in the form of a set of linear equations:

where z is the solution vector (the electrostatic potential on the surface of the brain or heart, for example), u is the vector of input data (the subset of voltage values on the surface of the skull or torso, for example), and A is the transfer matrix between z and u, which usually contains the geometry and physical properties (conductivities, dielectric constants, etc.) of the volume conductor. Z and U are the metric spaces where we measure z and u. The direct problem is then simply posed as solving (7) for u given z. Likewise, the inverse problem is to determine z given u.