We study inverse problems for the Poisson equation with source term the divergence of an \({{\mathbb {R}}}^3\)-valued measure, that is, the potential \(\varPhi \) satisfies

and \({{\varvec{\mu }}}\) is to be reconstructed knowing (a component of) the field \(\, {\mathrm{grad}}\,\varPhi \) on a set disjoint from the support of \({{\varvec{\mu }}}\). 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 investigate methods for recovering \({{\varvec{\mu }}}\) by penalizing the measure theoretic total variation norm \(\Vert {{\varvec{\mu }}}\Vert _{\mathrm{TV}}\). We provide sufficient conditions for the unique recovery of \({{\varvec{\mu }}}\), 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.

我们研究泊松方程的反问题，其源项为\（{{\ mathbb {R}}} ^ 3 \）值测度的散度，即势（\ varPhi \）满足

和\（{{\ varvec {\ mu}}} \\）将被重新构造，但要知道字段\（\，{\ mathrm {grad}} \，\ varPhi \）（的一部分）与支持\（{{\ varvec {\ mu}}} \）。在准静态状态下的几种电磁环境中会出现此类问题，例如，从磁场的测量中恢复剩余磁化强度时。我们通过惩罚度量理论上的总变化范数\（\ Vert {{\ varvec {\ mu}}} \ Vert _ {\ mathrm {TV}，调查恢复\（{{\ varvec {\ mu}}} \\）的方法} \）。我们为\（{{\ varvec {\ mu}}} \）的唯一恢复提供了充分的条件当正则化参数和噪声以组合方式趋于零，单向或当磁化具有稀疏的支撑（从纯粹意义上讲是1不可整流的）时，渐近地出现。数值例子说明了主要的理论结果。