Boundary Element Method for an Inverse Problem in Magnetic Resonance Imaging Gradient Coils

We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils for use in magnetic resonance imaging (MRI) and is formulated using its corresponding integral repre- sentation according to potential theory. A novel boundary element method (BEM) which employs linear interpolationon quadratic surfaces and also satisfies the continuity equation for the current density, i.e. a divergence-free BEM, is presented. Since thediscretised BEM systemis ill-posedand hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularization method is employed in order to retrieve accurate and physically correct