Low-rank approximation of least squares fitting with bivariate tensor-product splines
Dominik Mokriš  1@  
1 : MTU Aero Engines AG

MTU Aero Engines is Germany's leading engine manufacturer and an established global player in the
industry. We engage in the design, development, manufacture and maintenance of aircraft engines in all thrust
and power categories as well as of stationary gas turbines. These activities are supported by a broad spectrum of
CAE (Computer-Aided Engineering) tools and processes. In particular, underlying the aerodynamic simulations
is the in-house geometry generation software.
The last years have seen growing precision and availability of 3D scanning tools. Hand-in-hand with it grows
also the interest of our engineers to use the scanned data as an input of the geometry generator and to convert
them into high-quality NURBS surfaces suitable for further use. While considerable effort has been spent on
improving this process (see, e.g., [1, 2, 3, 4]), high expectations of our users mean that there is always room for
further improvement.
In this talk, I will show an early-stage work from this area. Assuming that the input data are a two-
dimensional grid of scalars, one can write them in a matrix form. Instead of approximating them directly with
a bivariate tensor-product function, a low-rank approximation of this matrix can be constructed in the form
Z \approx \sum t_r u_r \otimes v_r .
For each ur and vr a univariate least-squares approximation can be computed, thus obtaining vectors d_r and
e_r , respectively, of control points. These control points can be re-assembled into a matrix form
C = \sum t_r d_r \otimes e_r .
Taking C as control points of a bivariate spline function yields a good approximation of the original data Z.
This method is a generalization of an existing approach presented in [5]. However, there are two new
contributions. First, using least-squares approximation instead of interpolation leaves more freedom in the
choice of the data parametrization and fitting basis. Second, we prove that when s is equal to the rank of the
data matrix Z, then the elements of C are in fact the control points of the bivariate least-squares approximation
of Z. Additionally, this approach is generalized to weighted least-squares with separable weights, which is of
advantage, e.g., when approximating functions in L2 -sense using quadrature rules.

Personnes connectées : 2 Vie privée