Adaptive reduced-basis generation for reduced-order modeling for the solution of stochastic nondestructive evaluation problems

A novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented. More specifically, the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation, particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty. In order to achieve this computational efficiency, the approach combines reduced-basis reduced-order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters, provided the distributions are known. The reduced-order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique, to then create the reduced-order basis. The capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty. The first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing. Then, the second case study considered characterization of an unknown temperature-dependent thermal conductivity of a solid from simulated thermal testing. Overall, the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost.