Optimization and design space exploration
What is optimization?
In plain english, optimization is the act of obtaining the best result under given circumstances. This applies to any field (finance, health, construction, operations, manufacturing, transportation, engineering design, sales, public services, mail, and so on). The ultimate goal is either to minimize, maximize or zeroed a quantity of interest (QoI).
To find the optimal value we can choose between gradient-based methods and derivative-free methods:
Gradient-based methods look for improvement based on derivative information.
Derivative-free methods look for the optimal value using sampling with bias/rules toward improvement, or they do broad exploration with selective exploitation (e.g. genetic algorithms).
In general, gradient-based methods will converge to local extremes, while derivative-free methods can find local and global extremes in uni-modal and multi-modal problems.
When conducting optimization studies, we can also rely on parametrical studies and design of experiments, this is part of design space exploration (DSE), and based on the information obtained from DSE we can conduct surrogate-based optimization (SBO).
 |
 |
| Analytical model | Surrogate model |
When we do SBO, we use a surrogate model (also known as meta-model, curve fitting or response surface), to approximate an original high fidelity model (e.g., expensive CFD simulations or costly experiments). The surrogate acts as data fit to the observations so that new results can be predicted without recurring to expensive experiments (numerical or physical).
 |
| SBO workflow |
Once the surrogate is built, we can use any kind of optimization or calibration method. Evaluating the QoI at the surrogate level is inexpensive (working at the surrogate level is order of magnitude faster than using high fidelity models).
 |
| Gradient-based optimization at the surrogate level |
Optimization can be single-objective or multi-objective. In multi-objective optimization (MOO) we are interested in optimizing more than one QoI simultaneously. The final goal in MOO is to find a representative set of optimal solutions (Pareto front), quantify the trade-offs, and finding a single or set of solutions that satisfy the subjective preferences of a human decision maker.
 |
| Pareto front using a derivative-free method (MOGA) and SBO |
MOO is expensive, it often requires a lot of function evaluations. Hence, SBO is an economical and attractive way to do MOO and DSE. When conducting SBO and DSE studies, the observations (or experiments) can be used for data mining and data analytics. This information can also be used for initial screening and to provide information on the sensitivities of the data. The observations can be used to ask and answer questions about the data. This is data analytics in action.
 |
|
| Scatter plot matrix obtained from a DACE experiment (space exploration) |
 |
| Correlation matrix obtained from a DACE experiment (space exploration) |
At Wolf Dynamics, we have extensive experience in conducting design optimization studies (DO), design space exploration studies (DSE), and data analytics (DA). We have successfully used our expertise in numerical optimization to conduct shape optimization studies, parameter estimation, reverse engineering and DSE studies in different industrial applications.