# Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition

## Abstract

Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient analysis of variance-based stochastic circuit/microelectromechanical systems simulator to efficiently extract the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is efficiently simulated by our simulator at the cost of only 10min in MATLAB on a regular personal computer.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1274892

- Report Number(s):
- PNNL-SA-109438

Journal ID: ISSN 0278-0070; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: IEEE Transations on Computer-Aided Design of Integrated Circuits and Systems; Journal Volume: 34; Journal Issue: 1

- Country of Publication:
- United States

- Language:
- English

- Subject:
- ANOVA; gPC; UQ; MEMS; tensor-train

### Citation Formats

```
Zhang, Zheng, Yang, Xiu, Oseledets, Ivan V., Karniadakis, George E., and Daniel, Luca.
```*Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition*. United States: N. p., 2015.
Web. doi:10.1109/TCAD.2014.2369505.

```
Zhang, Zheng, Yang, Xiu, Oseledets, Ivan V., Karniadakis, George E., & Daniel, Luca.
```*Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition*. United States. doi:10.1109/TCAD.2014.2369505.

```
Zhang, Zheng, Yang, Xiu, Oseledets, Ivan V., Karniadakis, George E., and Daniel, Luca. Thu .
"Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition". United States.
doi:10.1109/TCAD.2014.2369505.
```

```
@article{osti_1274892,
```

title = {Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition},

author = {Zhang, Zheng and Yang, Xiu and Oseledets, Ivan V. and Karniadakis, George E. and Daniel, Luca},

abstractNote = {Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient analysis of variance-based stochastic circuit/microelectromechanical systems simulator to efficiently extract the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is efficiently simulated by our simulator at the cost of only 10min in MATLAB on a regular personal computer.},

doi = {10.1109/TCAD.2014.2369505},

journal = {IEEE Transations on Computer-Aided Design of Integrated Circuits and Systems},

number = 1,

volume = 34,

place = {United States},

year = {Thu Jan 01 00:00:00 EST 2015},

month = {Thu Jan 01 00:00:00 EST 2015}

}