Statistical Parameter Identification

Core Team Members: Jack S. Hale, Team Legato, University of Luxembourg.
Collaborators: Patrick Farrell, Oxford University.

« Supported by the Fonds National de la Recherche, Luxembourg (#6693582) »

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In this project we are developing computational methods to understand the uncertainty in the recovered material parameters from limited and noisy displacement observations of a hyperelastic material body. This has important applications in developing patient specific models for surgical simulation when direct observations of material properties cannot be made. The addition of statistical information allows the user understand limitations in experimental methodologies, equipment and poor model selection.

Using any suitable medical imaging method we calculate a displacement field of the body and then perform an adjoint based PDE-constrained optimisation problem to recover the material parameters.

Using Bayes’ theorem, we recover the statistical information about the posterior distribution (the probability that we have the material properties given the limited and noisy observations) by forming a low-rank decomposition of the Hessian matrix of the minimisation functional evaluated at the maximum aposteriori point.

Trailing Eigenvector of Hessian evaluated at the maximum aposteriori point.

Trailing Eigenvector of Hessian evaluated at the maximum aposteriori point. The truth value is a circular inclusion and limited and noisy observations are available only on the boundary of the domain. This eigenvector corresponds to the parameters that are least constrained by the given observation data, e.g. across the interface and inside the inclusion.