  Tensor B-spline methods are a high-performance alternative to solve partial
differential equations (PDEs). This paper gives an overview on the principles
of Tensor B-spline methodology, shows their use and analyzes their performance
in application examples, and discusses its merits. Tensors preserve the
dimensional structure of a discretized PDE, which makes it possible to develop
highly efficient computational solvers. B-splines provide high-quality
approximations, lead to a sparse structure of the system operator represented
by shift-invariant separable kernels in the domain, and are mesh-free by
construction. Further, high-order bases can easily be constructed from
B-splines. In order to demonstrate the advantageous numerical performance of
tensor B-spline methods, we studied the solution of a large-scale heat-equation
problem (consisting of roughly 0.8 billion nodes!) on a heterogeneous
workstation consisting of multi-core CPU and GPUs. Our experimental results
nicely confirm the excellent numerical approximation properties of tensor
B-splines, and their unique combination of high computational efficiency and
low memory consumption, thereby showing huge improvements over standard
finite-element methods (FEM).
