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

Abstract

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.

Introduction

The present study concerns the practical issue of estimating the “net” moment (or the mean value) of some ${\mathbb{R}}^2$-valued distribution (or integrable, or square-integrable function) supported on an interval of the real line, from the partial knowledge of the divergence of its Poisson extension to the upper half-plane, on some interval located on a parallel line at distance $h>0$ from the first one.

This is a two dimensional (2D) formulation of a three dimensional (3D) inverse magnetic moment recovery problem in paleomagnetism, for thin rock samples. There, the magnetization is assumed to be some ${\mathbb{R}}^3$-valued distribution (or integrable function) supported on a planar sample (square), of which we aim at estimating the net moment, given measurements of the normal component of the generated magnetic field on another planar measurements set taken to be a square parallel to the sample and located at some distance $h$ to it. Moment and more general magnetization recovery issues are considered in [4], [5], [7]. More about the data acquisition process and the use of scanning SQUID (Superconducting Quantum Interference Device) microscopy devices for measuring a component of the magnetic field produced by weakly magnetized pieces of rocks can be found in the introductory sections of these references.

In both situations, the partial differential equation (PDE) model that drives the behaviour of the magnetic field derives from Maxwell equations in magnetostatics, see [14]. They ensure that the magnetic field $\mathbf{B}$ derives from a scalar magnetic potential $U$ which is solution to a Poisson–Laplace elliptic PDE with right-hand side of divergence form, that relates the Laplacian of that potential to the divergence of the magnetization distribution $\mathbf{m}$:

$$\mathbf{B}=-{\mu }_0\text{grad}U,\Delta U=\text{div}\mathbf{m}\text{ in }{\mathbb{R}}^n,$$

with $n=2,3$, where both $\mathbf{B}$, $\mathbf{m}$ are ${\mathbb{R}}^n$-valued quantities, and ${\mu }_0$ is the permeability of the free space. We stick to situations where $\mathbf{m}$ has a compact support contained in the hyperplane $\{x_n=0\}$ and measurements of the vertical component $b_n$ of the field $\mathbf{B}$ are available on another compact subset of $\{x_n=h\}$ for some $h>0$.

The determination of the magnetic moment (mean value of the magnetization) provides useful preliminary information for the full inversion, i.e. for magnetization estimation, in particular in unidirectional situations where its components are proportional one to the others (the unknown quantity being a $\mathbb{R}$-valued distribution, the direction/orientation vector being fixed).

In 3D situation, these issues were efficiently analyzed with tools from harmonic analysis, specifically Poisson kernel and Riesz transforms [25], in [4], [5], [7], see also references therein, using their links with Hardy spaces of harmonic gradients. The existence of silent sources in 3D, elements of the non-zero kernel of the non injective magnetization-to-field operator (the operator that maps the magnetization to the measured component of the magnetic field) was established together with their characterization in [7]. Actually, in an analogous 3D setting, sources with vanishing third (normal) component and whose two first components are divergence free are the silent ones, and they do not necessarily vanish, see [5, Prop. 2]. This makes non unique the solution to the inverse magnetization issue from the corresponding field data. The mean value of the magnetization however is uniquely determined by the field data.

In the present 2D case, we will use similar tools, Poisson kernel, Hilbert transform and harmonic conjugation, links with Hardy spaces of functions of the complex variable [12], [13], [21]; see also [19], [23] for related issues. Considering square-summable magnetizations $\mathbf{m}$ supported on an interval $S$ of the real line, we will establish that the magnetization-to-field operator $\mathbf{m}\mapsto b_2\left[\mathbf{m}\right]$ is injective, whence there are no silent sources, and that the mean value $〈\mathbf{m}〉$ of the magnetization is yet uniquely determined by the field data $b_2\left[\mathbf{m}\right]$, provided by values of the vertical component on the measurement interval $K$.

Our purpose is to establish the existence and to build linear estimators for the mean values $〈m_i〉$ on $S$ of the components $m_i$ of $\mathbf{m}$ for $i=1,2$, as was done in [4] for the 3D case. Indeed, though an academic version of the related physical issue, the present 2D situation possesses its own mathematical interest and specificity. These linear estimators consist in square summable functions ${\varphi }_i$ such that their effect (scalar product) against the data on $K$ is as close as possible to the moment components $〈m_i〉$, for all $\mathbf{m}$ bounded by some $L^2$ norm (the functions being built once and for all). With ${\mathbf{e}}_1=({\chi }_S,0)$, ${\mathbf{e}}_2=(0,{\chi }_S)$, observe that: $〈m_i〉={〈\mathbf{m},{\mathbf{e}}_i〉}_{L^2(S,{\mathbb{R}}^2)}$, whence, if $b_2^{\ast }$ denotes the adjoint operator to $b_2$, we have:

$$|{〈b_2\left[\mathbf{m}\right],{\varphi }_i〉}_{L^2\left(K\right)}-〈m_i〉|\le {\Vert \mathbf{m}\Vert }_{L^2(S,{\mathbb{R}}^2)}{\Vert b_2^{\ast }\left[{\varphi }_i\right]-{\mathbf{e}}_i\Vert }_{L^2(S,{\mathbb{R}}^2)},$$

where ${\varphi }_i$ is a square integrable function on $K$. We will see that the above minimization problem is still ill-posed, even if uniqueness is granted. Specifically, there exist a sequence of functions such that their scalar product on $K$ against $b_2\left[\mathbf{m}\right]$ converges to $〈m_i〉$ as $n\to \infty$. Their quadratic norm however diverge, which reflects an unstable behaviour, as is classical in such inverse problems, see [1]. Regularization is thus needed (of Tykhonov type), see [9], [18] in order to set up and to solve a well-posed problem.

In order to construct such a numerical magnetometer, we then face the best constrained approximation issue (bounded extremal problem, BEP) of finding the function ${\varphi }_i$ on $K$ satisfying some norm constraint there, such that its scalar product against $b_2\left[\mathbf{m}\right]$ is as close as possible to $〈m_i〉$.

Such a norm constraint will be considered both in the Lebesgue space $L^2\left(K\right)$ and in the Sobolev space $W_0^{1,2}\left(K\right)\subset W^{1,2}\left(K\right)$. In those Hilbert spaces, the above problems will be shown to be well-posed, the approximation subsets being closed and convex. The results obtained in $L^2\left(K\right)$ and in $W_0^{1,2}\left(K\right)$ are different. In particular, the solutions in $W_0^{1,2}\left(K\right)$ demonstrate less oscillations at the endpoints of $K$ than the ones in $L^2\left(K\right)$, and allow to incorporate vanishing boundary conditions. The results for moment estimation will be compared between them.

Preliminary numerical computations in the 3D setting, with planar squared support and measurement set, were run on appropriate finite elements bases, see [4]; the 3D computations turn out to be computationally expensive. In the present 2D setting and on intervals, they are of course much less costly and we perform some of them using expansions on the Fourier bases, in Lebesgue and Sobolev spaces.

The overview of the present work is as follows. In Section 2, we introduce some notation, recall definitions and properties, concerning Lebesgue, Sobolev and Hardy spaces of functions, together with integral Poisson and Hilbert transforms. We then study in Section 3 the properties of the operator $b_2:\mathbf{m}\to b_2\left[\mathbf{m}\right]$, which maps the magnetization to the second (vertical) component of the produced magnetic field, and of its adjoint. Section 4 is devoted to the bounded extremal problems of which we consider two versions, in $L^2\left(K\right)$ and in $W_0^{1,2}\left(K\right)$, establishing their well-posedness and characterizing their solutions. Computational algorithms together with results of preliminary numerical simulations are provided in Section 5, with figures in an Appendix. We finally discuss in Section 6 some concluding remarks and perspectives.