Multiparameter Shape Optimization

Abstract

Optimization tools in engineering design often require a high computational cost. This cost originates from two main sources: First, the evaluation of the objective function involved in such problems is in general very expensive. Then, depending on the method employed and on the dimension of the design vector, the optimization procedure requires a high number of evaluations of the objective function to reach the final solution. Many authors proposed hierarchical techniques to make the optimization algorithm cheaper. Among these techniques, we can cite the use of a simplified model of the physical problem (for exemple, the use of Euler equations instead of the Navier-Stokes ones), the use of a metamodel instead of the exact model, or the use of a hierarchical parameterization instead of a single level one. This means that the optimization is carried out, at some steps, on a coarse level where not all the design parameters are considered. This idea is inspired from the multigrid theory used to solve problems with differential equations. In the present study, we propose a more efficient and more general method that can accelerate the convergence of the optimization algorithm and can be employed for any kind of problem. This method combines the multigrid concept with the spectral decomposition of the Hessian matrix of the cost function. Indeed, the smallest eigenvalues of the Hessian matrix correspond to directions where the convergence of the optimization algorithm is very slow, while the highest eigenvalues correspond to directions where the convergence is fast when descent optimization algorithms are used. Thus, instead of iterating on the entire design space, our algorithm serach for the solution in a selected subspace in order to accelerate the resolution in the directions of low convergence rate. Then it pursues the search on the entire design space. This can be done by several strategies analogous to those of the multigrid methods.