An autonomous uncertainty quantification method for the digital age: Transonic flow simulations using multivariate Padé approximations

There is a need to automate stochastic uncertainty quantification codes in the digital age. Problems in turbomachinery Computational Fluid Dynamics (CFD) are characterised by non-linear and discontinuous responses and long run times. Ensuring the reliability of Uncertainty Quantification (UQ) codes in such conditions, in an autonomous way, is a challenging problem. Human involvement has always been required. In this work, we therefore suggest a new approach that combines three state-of-the-art methods: multivariate Padé approximations, Optimal Quadrature Subsampling and Statistical Learning. Its main component is the generalised least squares multivariate Padé-Legendre (PL) approximation. PL approximations are globally fitted rational functions that can accurately describe discontinuous non-linear behaviour. They need fewer model evaluations than local or adaptive methods and do not cause the Gibbs phenomenon, like continuous Polynomial Chaos methods. We describe a series of modifications of the Padé algorithm that allow us to apply it to arbitrary input points instead of optimal quadrature locations. This property is particularly useful for industrial applications, where a database of CFD runs is already available, but not in optimal parameter locations. One drawback of the PL approximation is that it is non-trivial to ensure reliability for multiple input parameters. We therefore suggest a new method to improve stability in this work: Optimal Quadrature Subsampling. Our argument is that least squares errors, caused by an ill-conditioned design matrix, are the main source of error. Finally, we use statistical learning techniques to automatically guarantee smoothness and convergence. The resulting method is shown to efficiently and correctly fit thousands of partly discontinuous response surfaces for an industrial film cooling and shock interaction problem automatically and using only 9 CFD simulations. It can be applied to any other UQ problem that is characterised by a limited amount of data and the presence of discontinuities.