Multi-parameter identification in systems of PDEs from internal data

TL;DR

The paper develops a general framework for multi-parameter inverse problems in elliptic PDE systems from internal data by reformulating the problem as a first-order system $\nabla \boldsymbol{\mu} + \boldsymbol{B} \cdot \boldsymbol{\mu} = F$. It proves closed-range and $L^2$-stability properties, introduces the notion of $k$-conservative third-order tensor fields to characterize the null space, and derives conditions under which the inverse problem is stably solvable. A finite-element discretization is then proposed, yielding a space-dependent least-squares solution that reduces to a linear system or an eigenvalue problem; numerical tests in 2D demonstrate stable reconstructions of isotropic and fully anisotropic elastic tensors from static and time-harmonic data. The approach provides a robust alternative to traditional regularized optimization methods and has direct implications for elastography and related multi-parameter PDE identifications.

Abstract

This article aims to present a general analysis of a class of inverse problems that consists in recovering the elliptic parameter maps in systems of PDEs, such as the linear elastic system, from the knowledge of some of their solutions. This identification problem is reformulated as a first-order linear system of the form $\nabla\bmμ + \bm{B} \cdot \bmμ = F$, where $F$ and $\g B$ are tensor fields constructed from the data. A closed range property is proved, which induces $L^2$-stability estimates. We then characterize the null space by introducing the concept of conservative third-order tensor field. Finally, a discretization based on the finite element method is proposed and some numerical examples show the efficiency of this approach to recover anisotropic elastic parameters from both static and dynamic solutions of the PDE system.

Figures (17)

• Figure 1: Illustration of the proof of \ref{['prop:dimconst']}. As $\bm{v}\in E_{\bm{B}}^{x_1}$, we get all the identities in red. From that, we deduce $R_{\bm{B}}^{\bm{\gamma}_1}(\bm{w}) =R_{\bm{B}}^{\bm{\gamma}_2}(\bm{w})$ where $\bm{w}=R_{\bm{B}}^{\bm{\gamma}}(\bm{v})$.
• Figure 2: Left: the domain $\Omega$, the boundary parts $\Gamma_1$ and $\Gamma_2$. Right: the exact map $\mu$. On both, the black square $D = (-0.6,0.6)^2$ is the subdomain of interest where the inversion will be performed.
• Figure 3: Numerical solution of the forward problem with the boundary condition $\bm{g}_1$. Top: the computed displacement field. Bottom: the corresponding strain ${\cal E}(\bm{u})$. The strain is everywhere invertible as its maximum condition number is $13.8$.
• Figure 4: The exact map $\mu^\text{ex}$ and the reconstructed map $\mu$ from the data computed with the boundary condition $\bm{g}_1$. The mesh resolution is $h=0.01$ and the relative $L^2$-error is $2.91\%$.
• Figure 5: Solution of the forward problem with the boundary condition $\bm{g}_2$. Top: the computed displacement field. Bottom: the corresponding strain ${\cal E}(\bm{u})$. The strain is numerically degenerated as its maximum condition number of ${\cal E}(\bm{u})$ is $9.32.10^7$.

Theorems & Definitions (62)

• Definition 2.1
• Proposition 2.2
• proof
• Example 2.3
• Example 2.4
• Example 2.5
• Remark 2.6
• Theorem 3.1
• Corollary 3.2
• proof