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Global Optimization Based on Sparse Grid Surrogate Models for Black-box Expensive Functions
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery, Sep 2014, Volume 2014, p.1 - 11
Abstract
In the context of oil industry, many problems consist in a minimization process of a computationally expensive function with bound constraints. The values of the objective function are in general the output of a complex simulator for which we don’t have any explicit expression. And the absence of any information on the function gradient narrows the resolution field to algorithms using no first or second order derivatives.
There exists many different approaches in derivative free optimization, among which the most popular are space partitioning methods like DIRECT, direct search methods like Nelder Mead or MADS but also evolutionary algorithms like genetic algorithms, evolution strategies or other similar methods. The main drawback for all these methods is that they can suffer from a poor convergence rate and a high computational cost, especially for high dimensional cases. However, they can succeed in finding a global optimum where all other classical methods fail.
In this work we consider surrogate based optimization which has already been widely used for many years. A surrogate model is a framework used to minimize a function by sequentially building and minimizing a simpler model (surrogate) of the original function. In this work we build a new surrogate model by using the sparse grid interpolation method. Basically, the sparse grid approach is a grid error-controlled hierarchical approximation method which neglects the basis functions with the smallest supports. This approach was introduced in 1963 by Smolyak in order to evaluate integrals in high dimensions. It was then applied for PDE approximations but also for optimization and more recently for sensitivity analysis.
Compared to the first optimization algorithm based on sparse grids proposed by Klimke et al, a local refinement step is constructed here in order to explore the more promising regions. Moreover, no optimization steps are performed over the objective function, which reduces significantly the number of function evaluations employed.
In this talk we present our GOSGrid optimization algorithm based on sparse grids. We also compare it with other derivative free global algorithms. The comparison is based on a parameter estimation problem in reservoir engineering.