Modeling material stress using integrated Gaussian Markov random fields

Peter W. Marcy, Scott A. Vander Wiel, Curtis B. Storlie, Veronica Livescu, Curt A. Bronkhorst

Research output: Contribution to journalArticlepeer-review


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 the 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.

Original languageEnglish (US)
Pages (from-to)1616-1636
Number of pages21
JournalJournal of Applied Statistics
Issue number9
StatePublished - Jul 3 2020


  • Bayesian analysis
  • Gaussian Markov random field
  • blind deconvolution
  • large-scale inverse problem
  • materials science
  • process convolution
  • robust regression

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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