Explicit quasi-optimal point sets for total-degree polynomial interpolation on the square.
Research
Approximation theory as a bridge between geometry, computation and data.
The programme ranges from foundational questions in interpolation to numerical tools for imaging, PDEs and data analysis.
01
Kernel methods
Radial basis functions and meshless approximation
Stable bases, point selection and adaptive kernels for scattered-data approximation and numerical computation.
- Near-optimal and greedy centre selection
- Stable orthonormal bases for RBF spaces
- Rational and rescaled kernel approximation
- Partition of unity and RBF-FD methods
- Conditionally positive definite kernels
02
Polynomial approximation
Multivariate interpolation, cubature and extremal points
Geometric and computational approaches to stable approximation in several variables.
- Padua points and Lissajous points
- Weakly admissible meshes
- Approximate Fekete and discrete Leja points
- Polynomial approximation from diffused data
- Lebesgue functions and stability
03
Discontinuities & imaging
Mapped bases for non-smooth data
Approximation methods that use geometry or prior structure to control oscillations and preserve edges.
- Fake nodes and mapped polynomial bases
- Variably scaled discontinuous kernels
- Gibbs and Runge phenomena
- Magnetic particle, CT and MR imaging
- Solar imaging and inverse problems
04
Data & learning
Topology, kernels and neural approximation
Kernel-based ideas applied to point clouds, classification, intrinsic dimension and machine-learning architectures.
- Persistent homology and persistence kernels
- Intrinsic dimension of point clouds
- RBF neural networks and RBF-KANs
- Max-min neural network operators
- Computational medicine and brain dynamics
Selected contributions
Ideas that have shaped the programme.
Numerical linear-algebra constructions for stable interpolation in native spaces.
Fake nodes and discontinuous kernels for functions with jumps, edges and steep gradients.
Building communities through UMI–TAA, GNCS–INdAM and international approximation initiatives.