TY - JOUR
T1 - A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation
AU - Chen , Yannan
AU - Fu , Kaidong
AU - Li , Can
AU - Ye , Qi
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 590
EP - 614
PY - 2024
DA - 2024/08
SN - 5
DO - http://doi.org/10.4208/csiam-am.SO-2023-0045
UR - https://global-sci.org/intro/article_detail/csiam-am/23309.html
KW - Scattered data interpolation, generalized Mercel kernel, structural tensor, multilinear system, optimization, Łojasiewicz inequality.
AB - <p style="text-align: justify;">Scattered data interpolation aims to reconstruct a continuous (smooth) function that approximates the underlying function by fitting (meshless) data points. There
are extensive applications of scattered data interpolation in computer graphics, fluid
dynamics, inverse kinematics, machine learning, etc. In this paper, we consider a novel
generalized Mercel kernel in the reproducing kernel Banach space for scattered data
interpolation. The system of interpolation equations is formulated as a multilinear system with a structural tensor, which is an absolutely and uniformly convergent infinite
series of symmetric rank-one tensors. Then we design a fast numerical method for
computing the product of the structural tensor and any vector in arbitrary precision.
Whereafter, a scalable optimization approach equipped with limited-memory BFGS
and Wolfe line-search techniques is customized for solving these multilinear systems.
Using the Łojasiewicz inequality, we prove that the proposed scalable optimization
approach is a globally convergent algorithm and possesses a linear or sublinear convergence rate. Numerical experiments illustrate that the proposed scalable optimization approach can improve the accuracy of interpolation fitting and computational
efficiency.</p>