A stochastic algorithm for the identification of solution spaces in high-dimensional design spaces

The volume of an axis-parallel hyperbox in a high-dimensional design space is to be maximized under the constraint that the objective values of all enclosed designs are below a given threshold. The hyperbox corresponds to a Cartesian product of intervals for each input parameter. These intervals are used to assess robustness or to identify relevant parameters for the improvement of an insufficient design.
A related algorithm which is applicable to any non-linear, high-dimensional and noisy problem with uncertain input parameters is presented and analyzed. Analytical solutions for high-dimensional benchmark problems are derived. The numerical solutions of the algorithm are compared with the analytical solutions to investigate the efficiency of the algorithm. The convergence behavior of the algorithm is studied. The speed of convergence decreases when the number of dimensions increases. An analytical model describing this phenomenon is derived. Relevant mechanisms are identified that explain how the number of dimensions affects the performance. The optimal number of sample points per iteration is determined depending on the preference for fast convergence or a large volume. The applicability of the method to a high-dimensional and non-linear engineering problem from vehicle crash analysis is demonstrated. Moreover, we consider a problem from a forming process and a problem from the rear passenger safety.
Finally, the method is extended to minimize the effort to turn a bad into a good design. We maximize the size of the hyperbox under the additional constraint that all parameter values of the bad design are within the resulting hyperbox except for a few parameter values. These parameters are called key parameters because they have to be changed to lie within their desired intervals in order to turn the bad into a good design. The size of the intervals represents the tolerance to variability caused, for example, by uncertainty. Two-dimensional examples are presented to demonstrate the applicability of the extended algorithm. Then, for a high-dimensional, non-linear and noisy vehicle crash design problem, the key parameters are identified. From this, a practical engineering solution is derived which would have been difficult to find by alternative methods.