  We study inverse problems for the Poisson equation with source term the
divergence of an $\mathbf{R}^3$-valued measure, that is, the potential $\Phi$
satisfies $$
  \Delta \Phi= \text{div} \boldsymbol{\mu},
  $$ and $\boldsymbol{\mu}$ is to be reconstructed knowing (a component of) the
field grad $\Phi$ on a set disjoint from the support of $\boldsymbol{\mu}$.
Such problems arise in several electro-magnetic contexts in the quasi-static
regime, for instance when recovering a remanent magnetization from measurements
of its magnetic field. We develop methods for recovering $\boldsymbol{\mu}$
based on total variation regularization. We provide sufficient conditions for
the unique recovery of $\boldsymbol{\mu}$, asymptotically when the
regularization parameter and the noise tend to zero in a combined fashion, when
it is uni-directional or when the magnetization has a support which is sparse
in the sense that it is purely 1-unrectifiable.
  Numerical examples are provided to illustrate the main theoretical results.
