### Abstract

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 language | English (US) |
---|---|

Journal | Journal of Applied Statistics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Applied Statistics*. https://doi.org/10.1080/02664763.2019.1686131

**Modeling material stress using integrated Gaussian Markov random fields.** / Marcy, Peter W.; Vander Wiel, Scott A.; Storlie, Curtis B.; Livescu, Veronica; Bronkhorst, Curt A.

Research output: Contribution to journal › Article

*Journal of Applied Statistics*. https://doi.org/10.1080/02664763.2019.1686131

}

TY - JOUR

T1 - Modeling material stress using integrated Gaussian Markov random fields

AU - Marcy, Peter W.

AU - Vander Wiel, Scott A.

AU - Storlie, Curtis B.

AU - Livescu, Veronica

AU - Bronkhorst, Curt A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Bayesian analysis

KW - blind deconvolution

KW - Gaussian Markov random field

KW - large-scale inverse problem

KW - materials science

KW - process convolution

KW - robust regression

UR - http://www.scopus.com/inward/record.url?scp=85075069159&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075069159&partnerID=8YFLogxK

U2 - 10.1080/02664763.2019.1686131

DO - 10.1080/02664763.2019.1686131

M3 - Article

AN - SCOPUS:85075069159

JO - Journal of Applied Statistics

JF - Journal of Applied Statistics

SN - 0266-4763

ER -