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Stability analysis of an inverse coefficients problem in a system of partial differential equations

Houcine Meftahi, Chayma Nssibi

TL;DR

The paper tackles the inverse coefficients problem in linear elasticity by aiming to recover $\lambda(x)$, $\mu(x)$ and $\rho(x)$ from the Neumann-to-Dirichlet map $\Lambda_{\lambda,\mu,\rho}$ on $d\ge 2$ dimensions. It first treats the density reconstruction with known Lamé parameters, proving Lipschitz stability via a monotonicity relation and the Runge approximation, and provides a constructive Lipschitz estimate for piecewise-constant densities. It then extends to simultaneous recovery of all coefficients under a priori bounds and finite-dimensional subspace constraints, employing monotonicity and localized potentials to obtain Lipschitz stability, including a constructive scheme. The results enable robust numerical reconstruction and provide theoretical guarantees for finite-dimensional parameter spaces, with potential impact on non-destructive tissue assessment and related elastic imaging problems. Overall, the work advances the understanding of stability in multiparameter inverse elasticity problems beyond scalar conductivity analogies by leveraging monotonicity and localized potentials in arbitrary dimensions.

Abstract

In this study, we address the inverse problem of recovering the Lamé parameters ($λ, μ$) and the density $ρ$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters ($λ, μ$) are know and we look for the inverse problem of recovering the density $ρ$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $λ$, $μ$ and $ρ$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $λ$, $μ$ and $ρ$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.

Stability analysis of an inverse coefficients problem in a system of partial differential equations

TL;DR

The paper tackles the inverse coefficients problem in linear elasticity by aiming to recover , and from the Neumann-to-Dirichlet map on dimensions. It first treats the density reconstruction with known Lamé parameters, proving Lipschitz stability via a monotonicity relation and the Runge approximation, and provides a constructive Lipschitz estimate for piecewise-constant densities. It then extends to simultaneous recovery of all coefficients under a priori bounds and finite-dimensional subspace constraints, employing monotonicity and localized potentials to obtain Lipschitz stability, including a constructive scheme. The results enable robust numerical reconstruction and provide theoretical guarantees for finite-dimensional parameter spaces, with potential impact on non-destructive tissue assessment and related elastic imaging problems. Overall, the work advances the understanding of stability in multiparameter inverse elasticity problems beyond scalar conductivity analogies by leveraging monotonicity and localized potentials in arbitrary dimensions.

Abstract

In this study, we address the inverse problem of recovering the Lamé parameters () and the density of a medium from the Neumann-to-Dirichlet map for any dimension . This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters () are know and we look for the inverse problem of recovering the density . In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering , and simultameousely. We establish Lipschitz stability estimate, provided that the parameters , and have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.
Paper Structure (13 sections, 16 theorems, 109 equations)

This paper contains 13 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Recovery of the density $\rho$
  4. Monotonicity, localized potentials and Lipschitz stability
  5. Monotonicity
  6. Runge approximation and localized potentials
  7. Lipschitz stability result
  8. Constructive Lipschitz stability estimate
  9. Simultaneous recovery of $\lambda, \mu$ and $\rho$
  10. Monotonicity and localized potentials
  11. Lipschitz stability
  12. Proof of Theorem \ref{['stability1']}
  13. Conclusion

Key Result

Lemma 1

Let $\rho_1, \rho_2 \in L^\infty_+(\Omega)$ and let $g\in L^2(\Gamma_{\textup{N}},\mathbb{R}^d)$ be an applied boundary load. The corresponding solutions of (direct) are denoted by $u_1:=u^g_{\rho_1},\ u_2:=u^{g}_{\rho_2}\in \mathcal{V}$. Then we have

Theorems & Definitions (31)

  • Lemma 1: Monotonicity relation
  • proof
  • Corollary 1: Monotonicity
  • Lemma 2: Runge approximation
  • proof
  • Theorem 1: Localized potentials
  • proof
  • Theorem 2: Lipschitz stability
  • Lemma 3
  • proof
  • ...and 21 more