Fast Formation of Matrices for Least-Squares Fitting by Tensor-Product Spline Surfaces

Abstract

Least-squares fitting by tensor-product spline surfaces is a classical method for approximating unstructured data, which is widely used in industry. However, assembling the system of equations via the straightforward approach can be quite time–consuming. In this paper, we propose to accelerate this process by employing the technique of sum factorization, which is frequently used in the context of isogeometric analysis. Our approach consists of two steps. First, we introduce a regular grid onto which the parameters of the data are projected. Consequently, the expressions of the matrix entries take a form that admits the use of sum factorization, which is then employed in the second step. We provide a detailed complexity analysis and quantify the expected relative assembly costs. Several examples, including an example involving industrial data, demonstrate how the choice of the grid influences speed and precision, and confirm the expected time savings of the proposed method. • We use sum factorization to accelerate formation of least squares matrices. • We provide detailed complexity analysis. • Despite data projection to a regular grid, the error is comparable to the baseline. • Several examples (including an industrial geometry) confirm the findings.