A nonintrusive reduced order modelling approach using Proper Orthogonal Decomposition and locally adaptive sparse grids

Abstract

Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems.To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.