We study an inverse problem that consists in estimating the first (zero-order) moment of some ${\mathbb{R}}^2$-valued distribution $\mathbf{m}$ that is supported within a closed interval $\bar{S}\subset \mathbb{R}$, from partial knowledge of the solution to the Poisson–Laplace partial differential equation with source term equal to the divergence of $\mathbf{m}$ on another interval parallel to and located at some distance from $S$. Such a question coincides with a 2D version of an inverse magnetic “net” moment recovery question that arises in paleomagnetism, for thin rock samples. We formulate and constructively solve a best approximation problem under constraint in $L^2$ and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of $\mathbf{m}$. Numerical results obtained from the described algorithms for the net moment approximation are also furnished.

我们研究一个逆问题，该逆问题包括估计某些物体的第一（零阶）矩 ${\mathbb{[R}}^2$值分布 $\mathbf{米}$ 在封闭间隔内受支持 $\overline{\mathrm{小号}}\subset \mathbb{[R}$，从部分解到泊松-拉普拉斯偏微分方程的源项等于 $\mathbf{米}$ 在另一个平行于并位于一定距离处的区间上 $\mathrm{小号}$。对于薄岩石样品，该问题与古磁学中出现的逆磁“净”矩恢复问题的2D版本一致。我们制定并建设性地解决约束条件下的最佳逼近问题${\mathrm{大号}}^2$ 在Sobolev空间中涉及Poisson扩展的散度的限制。 $\mathbf{米}$。还提供了从描述的净矩逼近算法获得的数值结果。