Graff, Lavinia. A stochastic algorithm for the identification of solution spaces in highdimensional design spaces. 2013, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_10636
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Abstract
The volume of an axisparallel hyperbox in a highdimensional 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 nonlinear, highdimensional and noisy problem with uncertain input parameters is presented and analyzed. Analytical solutions for highdimensional 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 highdimensional and nonlinear 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. Twodimensional examples are presented to demonstrate the applicability of the extended algorithm. Then, for a highdimensional, nonlinear 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.
A related algorithm which is applicable to any nonlinear, highdimensional and noisy problem with uncertain input parameters is presented and analyzed. Analytical solutions for highdimensional 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 highdimensional and nonlinear 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. Twodimensional examples are presented to demonstrate the applicability of the extended algorithm. Then, for a highdimensional, nonlinear 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.
Advisors:  Harbrecht, Helmut 

Committee Members:  Krause, Rolf 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht) 
UniBasel Contributors:  Graff, Lavinia and Harbrecht, Helmut 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  10636 
Thesis status:  Complete 
Number of Pages:  166 p. 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  22 Apr 2018 04:31 
Deposited On:  06 Jan 2014 12:55 
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