Hyperelastic Elastography in a Large-Scale Bayesian Inversion Setting

We consider the problem of recovering the material parameters of a hyperelastic material [1] in the Bayesian inversion setting. In the Bayesian setting we can extract the statistics associated with various sources of uncertainty, including noise, experimental deficiencies and incomplete observations of the domain. This will allow medical practitioners to make superior diagnosis decisions when presented with a quantitative measure of uncertainty in the recovered parameters.

On the assumption of a Gaussian additive noise model, a Gaussian prior and a linear forward model, the posterior distribution of the material parameters given the observations will also be Gaussian. To ensure that the assumption of a linear forward model is valid, and that the posterior is approximated sufficiently well by a Gaussian distribution, we place a limit on the strain regime in which our current methodology applies. We are developing MCMC methods for exploring the non-Gaussian statistics of the posterior distribution.

In the linear case, the covariance matrix of the posterior distribution is then characterised by the inverse of the Hessian of the objective functional evaluated at its minimiser. To extract statistical information from the large and dense Hessian we perform a low-rank approximation of the Hessian [2]. The eigenvectors associated with the lowest eigenvalues are the directions in parameter space that are least constrained by the observations.

We implement this work within the dolfin-adjoint [3] software package. We derive the MPI-parallel finite element discretisation of the forward, adjoint (1st and 2nd order), and tangent linear models using the high-level differentiation tools available within the FEniCS project.

We show results demonstrating the effects of partial observations and poor experimental design on the reliability of the recovered parameters.

[1] N. H. Gokhale, P. E. Barbone, and A. A. Oberai, “Solution of the nonlinear elasticity imaging inverse problem: the compressible case,” Inverse Problems, 10.1088/0266-5611/24/4/045010
[2] H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders, and O. Ghattas, “Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations,” SIAM J. Sci. Comput., 10.1137/090780717
[3] P. Farrell, D. Ham, S. Funke, and M. Rognes, “Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs,” SIAM J. Sci. Comput., 10.1137/120873558