Vector-Matrix Method of Numerical Implementation of the Polynomial Integral Volterra Operators

Abstract

The article deals with the quadrature method for the numerical implementation of polynomial integral operators. With the computer implementation of Volterra-type integral models, the typical problem is the accumulation of calculations at each step of the computational process. For its acceleration it is suggested to apply the vector-matrix approach. The suggested approach is based on quadrature methods: rectangles, trapezoids, and Simpson's. For homogeneous polynomial integral Volterra operators of the first-, second-and third-degree, respectively, the objects in the form of vectors, matrices, and threedimensional structures containing the coefficients of the corresponding quadrature formulas have been constructed. The suggested vectormatrix approach involves the reduction of computational operations to the elementary multiplication of elements of the corresponding structures and allows efficient use of parallel algorithms, which significantly accelerates the execution of computational tasks for the implementation of integral operators. In the research work the complexity of implementation is estimated depending on the number of possible parallel flows. The estimation of the suggested approximations of integral representations is researched by model examples, in which there are models in the form of second-and third-degree polynomial integrals of Volterra. The results of computational experiments showed that among the considered quadrature methods, the trapezoidal method is optimal in terms of «precision-complexity of implementation». The accuracy of the numerical implementation of integral models depends on the chosen method, the simulation step, the type of kernel, and does not depend on the dimensionality of the operator. The vector-matrix approach allows building of efficient algorithms for the numerical implementation of integral models and greatly simplifies their software implementation, as it allows easy scaling to a multidimensional case. Such representation allows to use advantages of matrix-oriented packages of applications (Matlab, Octave, Scilab), the peculiarity of which is the high speed of execution of matrix operations.

Vector-Matrix Implementation of the Integrated Method of Recovery of Input Signals of Nonlinear Dynamic Systems

“…1 and fig. 2 a structural representation of the initial data of these operations for different homogeneous operators [5] is given. The program analogy of such a representation for the implementation of the operator of the first degree has the form: sum(A.*K(1:j).…”

“…This approach greatly simplifies the software implementation of multidimensional operators, as it allows easy generalization to the multidimensional case [5]. The suggested approach is studied in model experiments.…”

“…It can be seen that expression (3) is determined by the complexity of the description and the significant number of computational actions. To overcome these difficulties, it is advisable to use a vector-matrix approach with the reduction of all operations in (3) to element-by-element addition and multiplication [5].…”

“…The inverse problems of the dynamic usually reduced to solving Volterra integral equations of the first kind [2,3,6]. In the case of nonlinear dynamical systems, the problem is to determine the input effect () t  based on a given response () yt and a mathematical model in the form of a polynomial Volterra integral equation of the first kind, in particular the second degree [1,7,8,9]: The classic approach to solving this problem is the use of quadrature methods [2,5,8,9]. Since the problem is incorrect and experimental data are usually given with inaccuracy, it is necessary to use regularization methods.…”

“…It should be noted that the numerical implementation of this approach is accompanied by a significant number of computational procedures [2,5].…”