# A probabilistic graphical model based stochastic input model construction

## Abstract

Model reduction techniques have been widely used in modeling of high-dimensional stochastic input in uncertainty quantification tasks. However, the probabilistic modeling of random variables projected into reduced-order spaces presents a number of computational challenges. Due to the curse of dimensionality, the underlying dependence relationships between these random variables are difficult to capture. In this work, a probabilistic graphical model based approach is employed to learn the dependence by running a number of conditional independence tests using observation data. Thus a probabilistic model of the joint PDF is obtained and the PDF is factorized into a set of conditional distributions based on the dependence structure of the variables. The estimation of the joint PDF from data is then transformed to estimating conditional distributions under reduced dimensions. To improve the computational efficiency, a polynomial chaos expansion is further applied to represent the random field in terms of a set of standard random variables. This technique is combined with both linear and nonlinear model reduction methods. Numerical examples are presented to demonstrate the accuracy and efficiency of the probabilistic graphical model based stochastic input models. - Highlights: • Data-driven stochastic input models without the assumption of independence of the reduced random variables. •more »

- Authors:

- Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, 101 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 22314901

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 272; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; ALGORITHMS; CHAOS THEORY; COMPUTERIZED SIMULATION; KERNELS; LIMITING VALUES; MATHEMATICAL MODELS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; POLYNOMIALS; PROBABILISTIC ESTIMATION; RANDOMNESS; STOCHASTIC PROCESSES

### Citation Formats

```
Wan, Jiang, Zabaras, Nicholas, E-mail: nzabaras@gmail.com, and Center for Applied Mathematics, 657 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801.
```*A probabilistic graphical model based stochastic input model construction*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.05.002.

```
Wan, Jiang, Zabaras, Nicholas, E-mail: nzabaras@gmail.com, & Center for Applied Mathematics, 657 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801.
```*A probabilistic graphical model based stochastic input model construction*. United States. doi:10.1016/J.JCP.2014.05.002.

```
Wan, Jiang, Zabaras, Nicholas, E-mail: nzabaras@gmail.com, and Center for Applied Mathematics, 657 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801. Mon .
"A probabilistic graphical model based stochastic input model construction". United States.
doi:10.1016/J.JCP.2014.05.002.
```

```
@article{osti_22314901,
```

title = {A probabilistic graphical model based stochastic input model construction},

author = {Wan, Jiang and Zabaras, Nicholas, E-mail: nzabaras@gmail.com and Center for Applied Mathematics, 657 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801},

abstractNote = {Model reduction techniques have been widely used in modeling of high-dimensional stochastic input in uncertainty quantification tasks. However, the probabilistic modeling of random variables projected into reduced-order spaces presents a number of computational challenges. Due to the curse of dimensionality, the underlying dependence relationships between these random variables are difficult to capture. In this work, a probabilistic graphical model based approach is employed to learn the dependence by running a number of conditional independence tests using observation data. Thus a probabilistic model of the joint PDF is obtained and the PDF is factorized into a set of conditional distributions based on the dependence structure of the variables. The estimation of the joint PDF from data is then transformed to estimating conditional distributions under reduced dimensions. To improve the computational efficiency, a polynomial chaos expansion is further applied to represent the random field in terms of a set of standard random variables. This technique is combined with both linear and nonlinear model reduction methods. Numerical examples are presented to demonstrate the accuracy and efficiency of the probabilistic graphical model based stochastic input models. - Highlights: • Data-driven stochastic input models without the assumption of independence of the reduced random variables. • The problem is transformed to a Bayesian network structure learning problem. • Examples are given in flows in random media.},

doi = {10.1016/J.JCP.2014.05.002},

journal = {Journal of Computational Physics},

number = ,

volume = 272,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}