Optimal Choice of the Shape Parameter for the Radial Basis Functions Method in One-Dimensional Parabolic Inverse Problems

Inverse problems have numerous important applications in science, engineering, medicine, and other disciplines. In this study, we present a numerical solution for a one-dimensional parabolic inverse problem with energy overspecification at a fixed spatial point, using the radial basis function (RBF) method. The collocation matrix arising in RBF-based approaches is typically highly ill-conditioned, and the method’s performance is strongly influenced by the choice of the radial basis function and its shape parameter. Unlike previous studies that focused primarily on Gaussian radial basis functions, this work investigates and compares the performance of three RBF types—Gaussian (GRBF), Multiquadrics (MQRBF), and Inverse Multiquadrics (IMQRBF). By transforming the inverse problem into an equivalent direct problem, we apply the RBF collocation method in both space and time. Numerical experiments on two test problems with known analytical solutions are conducted to evaluate the approximation error, optimal shape parameters, and matrix conditioning. Results indicate that both MQRBF and IMQRBF generally provide better accuracy than GRBF. Furthermore, IMQRBF enhances numerical stability due to its lower condition number, making it a more robust choice for solving ill-posed inverse problems where both stability and accuracy are critical.