Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Abstract

Identification of the Volterra system is an ill-posed problem. We propose a regularization method for solving this ill-posed problem via a multiscale collocation method with multiple regularization parameters corresponding to the multiple scales. Many highly nonlinear problems such as flight data analysis demand identifying the system of a high order. This task requires huge computational costs due to processing a dense matrix of a large order. To overcome this difficulty a compression strategy is introduced to approximate the full matrix resulted in collocation of the Volterra kernel by an appropriate sparse matrix. A numerical quadrature strategy is designed to efficiently compute the entries of the compressed matrix. Finally, numerical results of three simulation experiments are presented to demonstrate the accuracy and efficiency of the proposed method. Communicated by Juan Manuel Peña. 422 M. Brenner et al.

Sparse Deep Neural Network for Nonlinear Partial Differential Equations

“…It can be seen from Fig. 4 that the function reconstructed by the standard DNN model (3.1) has large errors in the interval [3,5].…”

“…For neurons in different layers, corresponding to different transformation scales, the corresponding features have different levels of importance. Imposing different regularization parameters for different scales was proved to be an effective way to deal with multi-scale regularization problems [3,6,32]. Inspired by multiscale analysis, we propose a sparse regularization network model by applying different sparse regularization penalties to the neuron connections in different layers.…”

Unsteady aerodynamics modeling using Volterra series expansed by basis function

“…It should be noted that this identification problem is ill-posed in that the objective is to determine the structure of the system from input and output measurements [17]. Therefore, to obtain stable kernel estimation, a regularization method must be used to solve the least square problem.…”

Nonlinear updating method: a review

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