We present a statistical method for recovering the material parameters of a heterogeneous hyperelastic body. Under the Bayesian methodology for statistical inverse problems, the posterior distribution encodes the probability of the material parameters given the available displacement observations and can be calculated by combining prior knowledge with a finite element model of the likelihood.

In this study we concentrate on a case study where the observations of the body are limited to the displacements on the surface of the domain. In this type of problem the Bayesian framework (in comparison with a classical PDE-constrained optimisation framework) can give not only a point estimate of the parameters but also quantify uncertainty on the parameter space induced by the limited observations and noisy measuring devices.

There are significant computational and mathematical challenges when solving a Bayesian inference problem in the case that the parameter is a field (i.e. exists infinite-dimensional Banach space) and evaluating the likelihood involves the solution of a large-scale system of non-linear PDEs. To overcome these problems we use dolfin-adjoint to automatically derive adjoint and higher-order adjoint systems for efficient evaluation of gradients and Hessians, develop scalable maximum aposteriori estimates, and use efficient low-rank update methods to approximate posterior covariance matrices.