Multi-objective optimization using hyper-dual numbers
Abstract/Contents
- Abstract
- High-fidelity analysis tools can provide accurate predictions of the performance of a design. However, these tools often have a higher computational cost when compared to lower-fidelity tools. This can greatly increase the computational expense of tasks that require many repeated calls to the analysis tool, such as numerical optimization methods or uncertainty quantification techniques. There are various strategies that can be employed to reduce the computational cost. One strategy is to make use of additional information, such as the gradient (first-derivative information) or Hessian (second-derivative information). Gradient-based optimization methods can converge faster and require fewer function evaluations than non-gradient based methods. Optimization methods that use the Hessian, or an approximation to the Hessian, can further improve the rate of convergence and may require even fewer function evaluations. Response surfaces created using a few evaluations of the high-fidelity tool can also be used as surrogates for the high-fidelity analysis to reduce the computational cost. More accurate response surfaces can be created, using fewer sample locations, by incorporating first-derivative information. Including second-derivative information may provide a further benefit, provided it can be computed accurately and efficiently. Generalized complex numbers can be used to compute effectively exact first derivatives, by reducing the truncation error to below machine precision. Higher-dimensional extensions of generalized complex numbers can produce effectively exact second (or higher) derivatives, provided they have the property that multiplication is commutative. This dissertation develops a new number of this type, called hyper-dual numbers, in order to produce exact second derivatives. The mathematics of hyper-dual numbers enable first- and second-derivative calculations that are free from both subtractive cancellation error and truncation error, and therefore exact. This dissertation develops the mathematical properties of hyper-dual numbers and discusses their implementation in several programming languages. The implementations allow hyper-dual numbers to be easily applied to codes of arbitrary complexity. This is demonstrated on a 3D, parallel, unstructured computational fluid dynamics code, and the resulting derivative calculations are validated against exact results. Hyper-dual numbers are then applied to a multi-objective optimization method and two response surface creation methods.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2013 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Fike, Jeffrey Alan |
|---|---|
| Associated with | Stanford University, Department of Aeronautics and Astronautics. |
| Primary advisor | Alonso, Juan José, 1968- |
| Thesis advisor | Alonso, Juan José, 1968- |
| Thesis advisor | Jameson, Antony, 1934- |
| Thesis advisor | Senesky, Debbie |
| Advisor | Jameson, Antony, 1934- |
| Advisor | Senesky, Debbie |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Jeffrey Alan Fike. |
|---|---|
| Note | Submitted to the Department of Aeronautics and Astronautics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2013. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2013 by Jeffrey Alan Fike
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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