We propose a joint optimization algorithm that combines the optimal shape parameter–Gaussian radial basis function (G-RBF) surrogate model with global and local optimization techniques to improve accuracy and reduce costs. We analyze factors that affect the accuracy of the G-RBF surrogate model and use the particle swarm optimization (PSO) algorithm to determine the optimal shape parameter and control the number and spacing of the sampling points for a high-precision surrogate model. Global optimization refines the surrogate model, serving as the initial value for local optimization to further refine the problem. Our experiments show that this method significantly reduces computation costs. We optimize the section size of cantilever beams for different materials, obtaining the optimal section size and mass for each. We find that hard aluminum alloy is the optimal choice, meeting yield strength and deflection requirements through finite element analysis verification. Our work highlights the effectiveness of the joint optimization algorithm based on the surrogate model, providing valuable tools and insights into optimizing various structures.

The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations.

In multiquadratic radial basis function (MQ-RBF) interpolation, shape parameters have a direct effect on the interpolation accuracy. The paper presents an MQ-RBF interpolation technique with optimized shape parameters for estimating the parameters of sine wave signals. At first, we assessed the impact of basic sinusoidal parameters on the MQ-RBF interpolation outcomes through numerical experiments. The results indicated that the angular frequency of a sine wave is a crucial determinant of the corresponding MQ-RBF interpolation shape parameters. A linear regression method was then used to establish the optimal parameter selection formula for a single-frequency sine wave, based on a large volume of experimental data. For multi-frequency sinusoidal signals, appropriate interpolation shape parameters were selected using the random walk algorithm to create datasets. These datasets were subsequently used to train several regression models, which were then evaluated and compared. Based on its operational cost and prediction accuracy, the random forest algorithm was chosen to establish the shape parameter selection model for multi-frequency sinusoidal signals. The inclusion of the Bayesian optimizer resulted in a highly accurate model. The establishment of this model enabled the adaptive selection of the corresponding shape parameters for any sine wave signal, providing a convenient means of selecting MQ-RBF interpolation shape parameters. Furthermore, the paper proposes an MQ-RBF interpolation subdivision least squares method that significantly improves the estimation accuracy of sine wave parameters. The practicality of the method was validated by successfully applying it in the calibration of the clock delay mismatch of a time-interleaved analog-to-digital converter system.

It is known that random noises have a significant impact on differential systems. Recently, the influences of random noises for impulsive systems have been started. Nevertheless, the existing references on this issue ignore the significant phenomena of nonlinear impulses and time-varying delays. Therefore, we see the necessity to study the influences of random noises for impulsive systems with the above two factors. Stimulated by the above, a polynomial random noise is introduced to suppress the potential explosive behavior of the nonlinear impulsive differential system with time-varying delay. Fortunately, the stochastically controlled impulsive delay differential system admits a unique global solution, is bounded, and grows at most in the polynomial form.