Explicit Design Space Decomposition (EDSD) was originally designed to accurately and efficiently identify regions of the design space where specific system behaviors occur. The idea is to approximate iso-contour of decision functions (e.g., constraint for optimization or limit state for reliability assessment) as detailled in Basudhar and Missoum (2008). EDSD aims at solving problems with the following features:

• Computationnaly expensive black-box model
• Discontinuous or binary responses

As most metamodels cannot handle discontinuities, a classification approach, Support Vector Machine (SVM), was selected as a surrogate for the boundary of the feasible or failure domains. In order to reduce the computational cost associated with the construction of the SVM boundary an adaptive sampling scheme was developed (Basudhar and Missoum, 2010). In its most basic form, the sampling is based on the search for a sample located as far away as possible from exisiting samples (i.e., in sparse regions) as well as on the SVM (that is where the probability of misclassification is the largest). This can be written using the following global and non-smooth optimization problem:

\begin{align*} \underset{\mathbf{x}}{\max}&&& \underset{i=1,\ldots,N_s}{\min}\left|\left|\mathbf{x}-\mathbf{x}^{(i)}\right|\right|\\ \text{s.t.}&&& s(\mathbf{x})=0\\ &&& l_i\leq x_i\leq u_i&&& i=1,\ldots,N_{v} \end{align*}

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Note that the "max-min" problem can be made differentiable using the ``p-norm" (Lacaze and Missoum, 2013).

In order to improve the convergence properties of the algorithm, a secondary sample, whose purpose is to remove the "locking of the SVM", is also used. Details can be found in Basudhar and Missoum (2010).

EDSD has formed the basis of several new methods:

It has also been applied to several aerospace, mechanical, and biomedical problems.

## References:

Limit state function identification using Support Vector Machines for discontinuous responses and disjoint failure domains,Probabilistic Engineering Mechanics, vol. 23, 2008, pp. 1 - 11.

Adaptive explicit decision functions for probabilistic design and optimization using support vector machinesComputers & Structures, vol. 86, 2008, pp. 1904 - 1917.

An improved adaptive sampling scheme for the construction of explicit boundariesStructural and Multidisciplinary Optimization, vol. 42, 2010, pp. 517-529.

A generalized “max-min” sample for surrogate update,Structural and Multidisciplinary Optimization, 2013, pp. 1-5.

Constrained efficient global optimization with support vector machinesStructural and Multidisciplinary Optimization, vol. 46, 2012, pp. 201-221.

A sampling-based approach for probabilistic design with random fieldsComputer Methods in Applied Mechanics and Engineering, vol. 198, 2009, pp. 3647 - 3655.

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