Dishi Liu

Dishi Liu, Alexander Litvinenko, Claudia Schillings et al.

Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods since it is computationally expensive, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of Kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost.

Sergey Dolgov, Alexander Litvinenko, Dishi Liu

Combination of low-tensor rank techniques and the Fast Fourier transform (FFT) based methods had turned out to be prominent in accelerating various statistical operations such as Kriging, computing conditional covariance, geostatistical optimal design, and others. However, the approximation of a full tensor by its low-rank format can be computationally formidable. In this work, we incorporate the robust Tensor Train (TT) approximation of covariance matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is shown that here the computational complexity of Kriging is reduced to $\mathcal{O}(d r^3 n)$, where $n$ is the mode size of the estimation grid, $d$ is the number of variables (the dimension), and $r$ is the rank of the TT approximation of the covariance matrix. For many popular covariance functions the TT rank $r$ remains stable for increasing $n$ and $d$. The advantages of this approach against those using plain FFT are demonstrated in synthetic and real data examples.