## Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting

## Abstract

Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1184503

- Alternate Identifier(s):
- OSTI ID: 1250080

- Report Number(s):
- SAND-2014-19077J

Journal ID: ISSN 0045-7825; 540696

- Grant/Contract Number:
- AC04-94AL85000; AC04-94-AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 289; Journal Issue: C; Journal ID: ISSN 0045-7825

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; nonlinear model reduction; Gappy POD; temporal correlation; forecasting; initial guess

### Citation Formats

```
Carlberg, Kevin, Ray, Jaideep, and van Bloemen Waanders, Bart. Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. United States: N. p., 2015.
Web. doi:10.1016/j.cma.2015.02.013.
```

```
Carlberg, Kevin, Ray, Jaideep, & van Bloemen Waanders, Bart. Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. United States. doi:10.1016/j.cma.2015.02.013.
```

```
Carlberg, Kevin, Ray, Jaideep, and van Bloemen Waanders, Bart. Sat .
"Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting". United States. doi:10.1016/j.cma.2015.02.013. https://www.osti.gov/servlets/purl/1184503.
```

```
@article{osti_1184503,
```

title = {Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting},

author = {Carlberg, Kevin and Ray, Jaideep and van Bloemen Waanders, Bart},

abstractNote = {Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.},

doi = {10.1016/j.cma.2015.02.013},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = C,

volume = 289,

place = {United States},

year = {2015},

month = {2}

}

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