Algorithms for Function Approximation with Redundancy on Complex Geometries

The discretization of partial differential equations on domains with a complicated geometry  requires a difficult meshing phase. This is often a laborious, manual process that typically starts from a CAD description of the geometry. It is not a secret that this process may take longer than the actual solver. A major current research interest is iso-geometric analysis, in which the basis functions of the CAD model are reused in the discretization of the governing PDE. This way, the meshing step can be avoided.

In this project we intend to investigate a novel alternative, in which we embed the complicated domain in a larger bounding box. A mesh is not created: one can use a regular grid on the bounding box, and simply restrict that grid to the domain at hand. The major difficulty in this approach is that any basis for a function space on the bounding box constitutes a so-called frame on the embedded domain. Frames are redundant sets. In this setting it is clear that the frame is redundant, or overcomplete, because any function on the domain can be extended in many different ways to a function on the bounding box. Thus, one function on the domain can have many different representations in the frame. This redundancy leads to severe ill-conditioning. Fortunately, recent research in the division shows that this particular kind of ill-conditioning is harmless, and in fact benign. It can actually be exploited to improve numerical stability. The problem of approximating functions using a frame is well understood for Fourier-based approximations and efficient algorithms are becoming available at least for a few domains. It can be expected that similar, or better, algorithms can be devised using wavelets.