# Sensitivity technologies for large scale simulation.

## Abstract

Sensitivity analysis is critically important to numerous analysis algorithms, including large scale optimization, uncertainty quantification,reduced order modeling, and error estimation. Our research focused on developing tools, algorithms and standard interfaces to facilitate the implementation of sensitivity type analysis into existing code and equally important, the work was focused on ways to increase the visibility of sensitivity analysis. We attempt to accomplish the first objective through the development of hybrid automatic differentiation tools, standard linear algebra interfaces for numerical algorithms, time domain decomposition algorithms and two level Newton methods. We attempt to accomplish the second goal by presenting the results of several case studies in which direct sensitivities and adjoint methods have been effectively applied, in addition to an investigation of h-p adaptivity using adjoint based a posteriori error estimation. A mathematical overview is provided of direct sensitivities and adjoint methods for both steady state and transient simulations. Two case studies are presented to demonstrate the utility of these methods. A direct sensitivity method is implemented to solve a source inversion problem for steady state internal flows subject to convection diffusion. Real time performance is achieved using novel decomposition into offline and online calculations. Adjoint methods are used to reconstruct initialmore »

- Authors:

- Rice University, Houston, TX
- Brown University, Providence, RI
- Carnegie Mellon University, Pittsburgh, PA
- University of UppSala, Sweden

- Publication Date:

- Research Org.:
- Sandia National Laboratories

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 921606

- Report Number(s):
- SAND2004-6574

TRN: US0802152

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGEBRA; ALGORITHMS; FLUID FLOW; IMPLEMENTATION; MULTIPLE PRODUCTION; NEWTON METHOD; OPTIMIZATION; SENSITIVITY; SENSITIVITY ANALYSIS; SIMULATION; SPATIAL RESOLUTION; Sensitivity analysis.; Uncertainty-Analysis.; Error analysis (Mathematics); Algorithms.

### Citation Formats

```
Collis, Samuel Scott, Bartlett, Roscoe Ainsworth, Smith, Thomas Michael, Heinkenschloss, Matthias, Wilcox, Lucas C, Hill, Judith C, Ghattas, Omar, Berggren, Martin Olof, Akcelik, Volkan, Ober, Curtis Curry, van Bloemen Waanders, Bart Gustaaf, and Keiter, Eric Richard.
```*Sensitivity technologies for large scale simulation.*. United States: N. p., 2005.
Web. doi:10.2172/921606.

```
Collis, Samuel Scott, Bartlett, Roscoe Ainsworth, Smith, Thomas Michael, Heinkenschloss, Matthias, Wilcox, Lucas C, Hill, Judith C, Ghattas, Omar, Berggren, Martin Olof, Akcelik, Volkan, Ober, Curtis Curry, van Bloemen Waanders, Bart Gustaaf, & Keiter, Eric Richard.
```*Sensitivity technologies for large scale simulation.*. United States. doi:10.2172/921606.

```
Collis, Samuel Scott, Bartlett, Roscoe Ainsworth, Smith, Thomas Michael, Heinkenschloss, Matthias, Wilcox, Lucas C, Hill, Judith C, Ghattas, Omar, Berggren, Martin Olof, Akcelik, Volkan, Ober, Curtis Curry, van Bloemen Waanders, Bart Gustaaf, and Keiter, Eric Richard. Sat .
"Sensitivity technologies for large scale simulation.". United States. doi:10.2172/921606. https://www.osti.gov/servlets/purl/921606.
```

```
@article{osti_921606,
```

title = {Sensitivity technologies for large scale simulation.},

author = {Collis, Samuel Scott and Bartlett, Roscoe Ainsworth and Smith, Thomas Michael and Heinkenschloss, Matthias and Wilcox, Lucas C and Hill, Judith C and Ghattas, Omar and Berggren, Martin Olof and Akcelik, Volkan and Ober, Curtis Curry and van Bloemen Waanders, Bart Gustaaf and Keiter, Eric Richard},

abstractNote = {Sensitivity analysis is critically important to numerous analysis algorithms, including large scale optimization, uncertainty quantification,reduced order modeling, and error estimation. Our research focused on developing tools, algorithms and standard interfaces to facilitate the implementation of sensitivity type analysis into existing code and equally important, the work was focused on ways to increase the visibility of sensitivity analysis. We attempt to accomplish the first objective through the development of hybrid automatic differentiation tools, standard linear algebra interfaces for numerical algorithms, time domain decomposition algorithms and two level Newton methods. We attempt to accomplish the second goal by presenting the results of several case studies in which direct sensitivities and adjoint methods have been effectively applied, in addition to an investigation of h-p adaptivity using adjoint based a posteriori error estimation. A mathematical overview is provided of direct sensitivities and adjoint methods for both steady state and transient simulations. Two case studies are presented to demonstrate the utility of these methods. A direct sensitivity method is implemented to solve a source inversion problem for steady state internal flows subject to convection diffusion. Real time performance is achieved using novel decomposition into offline and online calculations. Adjoint methods are used to reconstruct initial conditions of a contamination event in an external flow. We demonstrate an adjoint based transient solution. In addition, we investigated time domain decomposition algorithms in an attempt to improve the efficiency of transient simulations. Because derivative calculations are at the root of sensitivity calculations, we have developed hybrid automatic differentiation methods and implemented this approach for shape optimization for gas dynamics using the Euler equations. The hybrid automatic differentiation method was applied to a first order approximation of the Euler equations and used as a preconditioner. In comparison to other methods, the AD preconditioner showed better convergence behavior. Our ultimate target is to perform shape optimization and hp adaptivity using adjoint formulations in the Premo compressible fluid flow simulator. A mathematical formulation for mixed-level simulation algorithms has been developed where different physics interact at potentially different spatial resolutions in a single domain. To minimize the implementation effort, explicit solution methods can be considered, however, implicit methods are preferred if computational efficiency is of high priority. We present the use of a partial elimination nonlinear solver technique to solve these mixed level problems and show how these formulation are closely coupled to intrusive optimization approaches and sensitivity analyses. Production codes are typically not designed for sensitivity analysis or large scale optimization. The implementation of our optimization libraries into multiple production simulation codes in which each code has their own linear algebra interface becomes an intractable problem. In an attempt to streamline this task, we have developed a standard interface between the numerical algorithm (such as optimization) and the underlying linear algebra. These interfaces (TSFCore and TSFCoreNonlin) have been adopted by the Trilinos framework and the goal is to promote the use of these interfaces especially with new developments. Finally, an adjoint based a posteriori error estimator has been developed for discontinuous Galerkin discretization of Poisson's equation. The goal is to investigate other ways to leverage the adjoint calculations and we show how the convergence of the forward problem can be improved by adapting the grid using adjoint-based error estimates. Error estimation is usually conducted with continuous adjoints but if discrete adjoints are available it may be possible to reuse the discrete version for error estimation. We investigate the advantages and disadvantages of continuous and discrete adjoints through a simple example.},

doi = {10.2172/921606},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2005},

month = {1}

}