A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation (CSIAM Transactions on Applied Mathematics, 2024)
Scattered data interpolation aims to reconstruct a continuous (smooth) function that approximates the underlying function by fitting (meshless) data points. Thereare extensive applications of scattered data interpolation in computer graphics, fluiddynamics, inverse kinematics, machine learning, etc. In this paper, we consider a novelgeneralized Mercel kernel in the reproducing kernel Banach space for scattered datainterpolation. The system of interpolation equations is formulated as a multilinear system with a structural tensor, which is an absolutely and uniformly convergent infiniteseries of symmetric rank-one tensors. Then we design a fast numerical method forcomputing the product of the structural tensor and any vector in arbitrary precision.Whereafter, a scalable optimization approach equipped with limited-memory BFGSand Wolfe line-search techniques is customized for solving these multilinear systems.Using the Łojasiewicz inequality, we prove that the proposed scalable optimizationapproach 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 computationalefficiency.