Constrained Backward Mapping Algorithm for Kernel POD in CFD Problems

In modelling multi-dimensional correlated responses in Computational Fluid Dynamics, dimensionality reduction significantly reduces modelling complexity. For nonlinear problems, the kernel Proper Orthogonal Decomposition (kernel POD) outperforms linear POD in terms of reduced-dimensional space and numerical stability. When applying the kernel POD method to solve parameterised problems, it is necessary to map the reduced-dimensional variables back to the original multi-dimensional response. Building upon the framework of training samples weighting, we develop two enhanced approaches to solve for the weights: a modified constrained optimisation method and a dimensionality-reduced space projection method with additional constraints, ensuring the dual constraints of non-negative weights and summation to unity. Through three numerical examples, it is demonstrated that the proposed backward mapping algorithms achieve more accurate reconstruction of multi-dimensional response compared with existing methods. This improvement further validates the overall superiority of the kernel POD method.