Abstract

Many modern datasets arise from scattered and non-uniform samples in high-dimensional spaces, where traditional grid-based methods lose efficiency and accuracy. Samplets provide a principled mathematical framework for representing such data across multiple scales, enabling both theoretical insight and computational efficiency. Inspired by wavelets but tailored for scattered data, samplets act as localized building blocks that capture information at multiple resolutions. They make it possible to compress and manipulate large numerical systems with almost linear computational cost, turning previously intractable problems into manageable ones.
This multiscale approach shines in applications such as kernel interpolation with Matèrn functions, where data are reconstructed by successively refining details across different length scales. Representing each step in samplet coordinates produces sparse, well-conditioned systems that can be solved stably and efficiently.
Samplets are also very powerful in data analysis. By observing how samplet coefficients decay within the multiresolution levels, one can measure the smoothness and detect sharp transitions within irregular signals in near-linear time. In essence, samplets bridge structure and scalability, providing a multiresolution language for understanding and processing complex, scattered data.

Bio

Sara Avesani is a PhD student in computational science at Università della Svizzera Italiana (USI), where she has been enrolled since 2023. Her research focuses on multiscale approximation methods for scattered data, with particular emphasis on meshfree, adaptive multiresolution algorithms for detecting local smoothness classes and for the numerical solution of partial differential equations. She actively contributes to the C++ implementation of these methods within the FMCA (Fast Multiresolution Covariance Analysis) library, developing efficient and user-friendly components that combine high performance with flexible, application-oriented interfaces. Her contributions bridge theoretical advances in kernel methods and reproducing kernel Hilbert spaces with practical, scalable implementations for large-scale scientific computing.