A surrogate model based nested optimization framework for inverse problem considering interval uncertainty

The inverse problem is a kind of engineering problem that estimates the input through the given output. In this paper, the given output described as interval uncertainty parameter is concerned, and the interval is formed by the interval midpoint and the interval radius. The two-step framework that estimates the midpoint and the radius separately is used. A novel nested optimization framework is proposed to estimate the input interval radius with more inputs than outputs. The nested framework has two loops: (i) the inner loop quantifies the lower and upper bounds of the output with given interval radiuses of inputs from the outer loop by two optimizations, and the results will be feedback to the outer loop as constraint values of outer loop; (ii) the outer loop maximizes the input interval radiuses to reduce the cost while meeting the constraints transformed from the given interval. The nested framework induces a high number of forward model computations in the loops and may lead to an unacceptable computational burden for most engineering applications. Therefore, the surrogate model is suggested. The Radial Basis Functions (RBF) surrogate model is used to relieve the computational burden. The effectiveness and the accuracy of the framework are verified through a mathematical example, a cantilever tube example and an airfoil example.