Last updated: September 26

ANR Project REWARD

Reverse Weak Formulation for parameter identification

Application to medical elastography

ANR-22-CE40-0005; funding: €250k.

Project overview

The main goal of the project is to deepen and broaden the analysis of a recently introduced mathematical method called the Reverse Weak Formulation, which aims to recover the elastic properties of materials from internal displacement data. In abstract terms, consider a variational problem in the canonical form

\[ a(u,v) = L(v), \qquad \forall v \in H, \]

where \(a\) and \(L\) are, respectively, a bilinear form and a linear form on a Hilbert space \(H\). The Reverse Weak Formulation consists in identifying the model \(a\) from a set of solutions \(\{u_i\}_{i\in I}\) corresponding to known source terms \(\{L_i\}_{i\in I}\):

\[ \text{Find}\quad a\quad\text{such that}\qquad a(u_i,v) = L_i(v),\quad \forall v\in V\subset H,\quad \forall i\in I. \tag{RWF} \]

This formulation defines a general class of inverse problems that is particularly well suited to parameter identification in linear systems of partial differential equations, such as linear elasticity. Under a priori assumptions on the model \(a\), we aim to prove stability estimates for both the theoretical and numerical aspects of this inverse problem. We also aim to extend the method to novel random and multiscale models and to test it on various types of experimental data obtained using state-of-the-art imaging methods. Furthermore, we will extend the method to viscoelastic models and adapt it to other inverse problems involving internal data arising from hybrid imaging methods, such as Optical Coherence Elastography.

Consortium

Lyon hub
Paris hub
Partners
Involved students
  • Eliott Kacedan — ICJ, Centrale Lyon (funded by this ANR project)
  • Paul Kaassis — ICJ, Centrale Lyon

Structure

The project is divided into the main tasks briefly presented below.

  1. Stability estimates for anisotropic elastic parameter recovery
  2. Discretization and discrete stability estimates for the general RWF
  3. Parameter recovery from partial internal data, with a focus on linear and planar data acquisition
  4. Uncertainty analysis and data processing for experimental and in vivo data
  5. Random and multiscale elastic models for biological tissues: recovering microscale properties from macroscale measurements
  6. Analysis of the shear-wave elastography problem
  7. Extension of the RWF approach to viscoelastic media
  8. Application of the RWF approach to dynamic optical coherence tomography data

E. Kacedan’s thesis focuses mainly on Tasks 1, 3, and 6, while P. Kaassis’s thesis focuses on Task 5.

See the project’s HAL page.