Chapter 5: Trivariate Tensor-Product Splines
Larry L. Schumaker · Society for Industrial and Applied Mathematics eBooks · 2024
Tools for working with bivariate tensor-product splines were introduced in Chapter 2 of [[Schum15]] along with a discussion of their use in solving various interpolation and fitting problems for functions defined on a rectangle in 2D. For convenience, we review the basics here since we will be using bivariate tensor-product splines to solve elliptic boundary-value problems in 2D in Chapters 9 and 10. In the remainder of this chapter we introduce analogous tools for working with tensor-product splines defined on rectangular boxes in 3D. We then show how to use them for several applications, including quasi-interpolation, not-a-knot interpolation, least-squares, and penalized least-squares fitting. The use of trivariate tensor-product splines to solve boundary-value problems in 3D is discussed in Chapter 15.