# Multi-fidelity regression for computational simulations

In the blog post “Uncertainty quantification for computational simulations” a polynomial approximation is constructed on the input-output data of Computational Fluid Dynamics (CFD) simulations i.e. y=f(s_1,\dots,s_D) where s_1 to s_D could be independent parameters such as angle of attack, Mach number and/or geometric parameters.

The resulting polynomial approximation can be used for uncertainty quantification (UQ), sensitivity analysis or dimension reduction (via active subspaces). However, in many applications the usefulness of the computed moments (for UQ) or the dimension reducing subspaces is limited by the accuracy of the polynomial approximation. This can be an issue for many industrial flows (such as those in aerospace) which are fundamentally turbulent in nature. The RANS turbulence models used can introduce significant errors in predictions, adversely influencing the usefulness of the resulting polynomial approximation.

Large Eddy Simulation of boundary layer transition over a gas turbine compressor blade.
Source: Ashley Scillitoe

High fidelity turbulence resolving CFD simulations such as Large Eddy Simulation (LES) offer a more accurate alternative, however they are too computationally expensive to be pratical to run for all the points in the Design of Experiment (DoE).

A realistic compromise is multi-fidelity regression. whereby a small number of high-fidelity (HF) simulations are used to augment the cheap to run low-fidelity (LF) simulations. One options [1] is to construct an additional correction polynomial, which accounts for the error between the LES (HF) and RANS (LF). The resulting multi-fidelity (MF) polynomial can then be used as normal for UQ, sensitivity analysis and dimension reduction.

Left: Correction points from error between LF and HF. Right: MF polynomial constructed from correction and LF polynomials.
Source: [1].

Something like the above shouldn’t be too challenging to implement in Effective Quadratures, and would offer real value when combined with the existing machinery. Cheaper simulations could be used for testing e.g. inviscid CFD for low-fidelity, and RANS for high-fidelity.

References:
[1] Palar, P. S., Tsuchiya, T., & Parks, G. T. (2016). Multi-fidelity non-intrusive polynomial chaos based on regression. Computer Methods in Applied Mechanics and Engineering , 305 , 579–606. https://doi.org/10.1016/j.cma.2016.03.022.