On the Visualization of the Ensemble of Parametric Numerical Solutions Using Tensor Decomposition
Abstract:
An approximation of the tensor obtained at a discretization of the multidimensional function on a uniform grid is addressed from the viewpoint of the storage and treating the result of the parametric computations in the CFD problems. The tensor decompositions are considered for this purpose. The new algorithm of the calculation of the canonical decomposition using the gradient descent for an approximately decomposable goal functional. This algorithm applies an ensemble of points on the hyperplane orthogonal to the computed core of the canonical decomposition (”umbrella”) that enables its flexible application for the approximation of the tensors with a priori unknown rank. This algorithm is naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for model six-dimensional functions and for the ensemble of the numerical solutions for two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. Mach number and the the flow deflection angles are considered as the flow parameters.
