On Application of Canonical Decomposition for the Visualization of Results of Multiparameter Computations

Abstract

The approximation of the tensor appearing at a discretization of the multidimensional function is considered from the viewpoint of storing and treating of the results of parametric computations obtained in computational aerogasdynamics. The new algorithm for the computation of the canonical decomposition using gradient descent and approximately decomposable goal functional is described. This algorithm applies the random set of points on the hyperplane orthogonal to the computed core of the canonical decomposition (“umbrella”) that ensures its flexible application for an approximation of the tensors with a priori unknown rank and may be naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for the model six-dimensional functions and for an ensemble of the numerical solutions for the two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. The Mach number and angles of the flow deflection serve as the flow parameters. The results are provided for the dimensionality 3 (simple numerical solution) and 4 (the ensemble of the numerical solutions in dependence on the Mach number).