  The equations of a physical constitutive model for material stress within
tantalum grains were solved numerically using a tetrahedrally meshed volume.
The resulting output included a scalar vonMises stress for each of the more
than 94,000 tetrahedra within the finite element discretization. In this paper,
we define an intricate statistical model for the spatial field of vonMises
stress which uses the given grain geometry in a fundamental way. Our model
relates the three-dimensional field to integrals of latent stochastic processes
defined on the vertices of the one- and two-dimensional grain boundaries. An
intuitive neighborhood structure of said boundary nodes suggested the use of a
latent Gaussian Markov random field (GMRF). However, despite the potential for
computational gains afforded by GMRFs, the integral nature of our model and the
sheer number of data points pose substantial challenges for a full Bayesian
analysis. To overcome these problems and encourage efficient exploration of the
posterior distribution, a number of techniques are now combined: parallel
computing, sparse matrix methods, and a modification of a block update strategy
within the sampling routine. In addition, we use an auxiliary variables
approach to accommodate the presence of outliers in the data.
