What the study found
The paper studies bivariate polynomial approximation of a real-valued function from averages over triangles in a two-dimensional domain. It reports that histopolation, with an optional regression step, can be implemented with Padua points, discrete Leja sequences, and approximate Fekete nodes, and that numerical experiments demonstrate effectiveness in function approximation.
Why the authors say this matters
The authors say the method is relevant because it uses only triangle averages to reconstruct a function and because selecting a subset of triangles can make the histopolation problem solvable and well-conditioned. They also suggest that using the remaining triangles for regression can improve approximation accuracy.
What the researchers tested
The researchers tested bivariate polynomial histopolation on a triangulation of a domain in u211d^2, where only averages of the target function over triangles were available. They introduced histopolation and combined histopolation-regression methods using Padua points, discrete Leja sequences, and approximate Fekete nodes.
What worked and what didn't
The abstract says the proposed algorithms were implemented and evaluated through numerical experiments, and these experiments showed effectiveness in function approximation. It also states that choosing a subset of triangles is necessary for solvability and good conditioning, while the remaining triangles may be used to improve accuracy through regression.
What to keep in mind
The abstract does not provide detailed numerical results, specific error measures, or comparisons among the point sets. It also does not state limitations beyond the need to select an appropriate subset of triangles for histopolation.
Key points
- The study focuses on reconstructing a real-valued function from triangle averages over a triangulated domain in two dimensions.
- It introduces histopolation and combined histopolation-regression methods.
- The methods use Padua points, discrete Leja sequences, and approximate Fekete nodes.
- The abstract says selecting a subset of triangles is important for solvability and good conditioning.
- Numerical experiments are reported to show effectiveness in function approximation.
Disclosure
- Research title:
- Polynomial histopolation and regression are evaluated on triangulations
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