Сравнение целевых функций в задаче Прони для аппроксимации данных измерений

Abstract

In the article, we compare two objective functions in the Prony’s problem of approximation of measurement data by solutions of a linear differential equation of a given order with constant coefficients. The target functions differ in the type of dependence of the gradient on the coefficients of the equation (linear or with complex nonlinearity) and are 1) the norm of the residual of the equation (linear least squares method) or 2) the norm of the approximation error according to A. Householder (variational identification method). In the latter case, the coefficients of the differential equation and the initial conditions of the solution are jointly optimized. For the considered objective functions, the local stability constants of the solution to the Prony’s problem are calculated using local expansions of the dependencies of the optimal coefficients of the equation as implicit functions of the data with the condition that the gradient of the objective function is identically equal to zero. On this basis, a method is proposed for determining the permissible error in the data to ensure a given level of deviation of the solution from the true value. We use the example of K. Lanczos of calculating the exponents given observations of the sum of three exponents with rounding errors to confirm a significant advantage (in terms of the allowable errors in the data) of using the variational objective function. The adequacy of the used local stability indices for considerable perturbations is verified by numerical experiment.