# Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation

## TL;DR

This paper addresses a 2D inverse problem of estimating a distribution's first moment from PDE solutions, with applications in paleomagnetism, by formulating and solving a constrained best approximation problem.

## Contribution

It introduces a novel constrained approximation framework for inverse moment estimation in 2D, with constructive solutions and numerical validation.

## Key findings

- Effective algorithms for net moment approximation demonstrated
- Numerical results confirm the approach's accuracy
- Applicable to paleomagnetic inverse problems

## Abstract

We study an inverse problem that consists in estimating the first (zero-order) moment of some R2-valued distribution m supported within a closed interval S $\subset$ R, from partial knowledge of the solution to the Poisson-Laplace partial differential equation with source term equal to the divergence of 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 L2 and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of m. Numerical results obtained from the described algorithms for the net moment approximation are also furnished.