Efficient truncated randomized SVD for mesh-free kernel methodsImage 1
Year: 2024
Type: article
Abstract: This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the \textit{scale} or \textit{shape} parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference.
Institutions: National Taiwan University, University of Turin
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,310
Funders: National Science and Technology Council, National Research Centre, Istituto Nazionale di Alta Matematica "Francesco Severi", Artificial Intelligence Research Center, European Commission, Gruppo Nazionale per il Calcolo Scientifico, NextGenerationEU, International Council of Shopping Centers
OpenAlex ID: W4393965264