Development of Gradient-Enhanced Kriging Approximations for Multidisciplinary Design Optimization

Doctoral Dissertation


Techniques for improving the accuracy of the global approximations used in various Multidisciplinary Design Optimization (MDO) procedures, while reducing the amount of design space information required to develop the approximations, were studied in this research. These improvements can be achieved by incorporating gradient or sensitivity information into the existing approximation techniques. An approach to develop response surface approximations based upon artificial neural networks trained using both state and sensitivity information is developed. Compared to previous approaches, this approach does not require weighting the residuals of the targets and gradients and is able to approximate gradient-consistent response surfaces with a relatively compact network architecture. Numerical simulation on selected problems shows that this approach possesses the capability to develop improved response surface approximations compared to the non-gradient neural network training approach.

One issue that this approach cannot address properly, however, is to determine the step size for the design variables in the Taylor Series expansion that is used to utilize sensitivity-based, approximate information. It is also a common challenge associated with a particular gradient-enhanced approach, Database Augmentation. This research develops another gradient-enhanced approach based on Kriging models to solve the problem by including the step size as one of model parameters. This approach can also characterize the uncertainty of approximations, which is another goal of this research. Based on Database Augmentation, the approach develops Kriging models by minimizing the Integrated Mean Squared Error (IMSE) criterion instead of the Maximum Likelihood Estimation (MLE) process often used.

Numerical simulation on selected, small-scale problems shows that this IMSE-based gradient-enhanced Kriging (IMSE-GEK) approach can improve approximation accuracy by 60~80% over the non-gradient Kriging approximation. An analytical approach to compute IMSE was developed to reduce the prohibitive computing cost associated with applying the IMSE-GEK approach to high-dimensional problems. Some additional implementation issues associated with the approach, such as the database augmenting scheme, the use of variable step sizes and the inclusion of nugget effects at added points, are also presented.


Attribute NameValues
  • etd-07012003-215221

Author Weiyu Liu
Advisor Dr. Stephen M. Batill
Contributor Dr. Stephen M. Batill, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Aerospace and Mechanical Engineering
Degree Name Doctor of Philosophy
Defense Date
  • 2003-05-12

Submission Date 2003-07-01
  • United States of America

  • Kriging

  • Neural Network

  • integrated mean squared error

  • optimization

  • gradient enhanced approximation

  • University of Notre Dame

  • English

Access Rights Open Access
Content License
  • All rights reserved


Please Note: You may encounter a delay before a download begins. Large or infrequently accessed files can take several minutes to retrieve from our archival storage system.