and Computational Methods

doi: 10.18698/2309-3684-2024-3-120139

Application of generalized decomposition of polynomial chaos (RPH) in problems of quantitative estimation of uncertainty is considered. A program code has been implemented to study the influence of the input data generation scheme on the quality of the model whose coefficients are calculated by the least squares method. Standard error and sliding control values were used as quality criteria. Along with the classical method of filling the space of the input features on the scheme of the Latin hypercube, two variants of modelling coherent-optimal sample are considered: using the Markov chain and with additional thinning on the D-criterion. While the Latin hypercube sample evenly distributes points across the whole space of random variables, coherent optimum methods aim to distribute samples more densely in areas with greater variance and more rarely in areas with small variance. This approach allows for a better integration of information about the real model, which leads to a reduction in the number of samples in the planning of the experiment and as a result save costly CPU time. The implemented methods were compared on the Ishigami model function and the farm design with random values of physical characteristics. As a result of comparative modeling, it is established that in case of small range of change of random parameters, when their gradients slowly change, the design of the Latin hypercube shows the lowest values of error and sliding control. At the same time, in the case of strong non-linearity, the application of coherent-optimal design leads to a more stable and efficient model, and additional thinning according to the criterion of D-optimality gives the best result and is the most sustainable. It has also been shown that both the planning algorithms of the experiment are unstable and incorrect if there are insufficient samples.

Облакова Т.В., Фам Куок Вьет. Об оптимальной конструкции моделирования разложения полиномиального хаоса в задачах количественной оценки неопределенности. Математическое моделирование и численные методы, 2024, № 3, с. 120–139.

doi: 10.18698/2309-3684-2023-4-93108

The application of the generalized expansion of polynomial chaos (PC) and models based on Kolmogorov-Gabor polynomials in regression problems is considered. When choosing PC expansion, the Wiener-Askey scheme was used, which sets the correspondence between the feature distribution law and the orthogonal polynomial basis. To calculate the expansion coefficients, non-intrusive methods were used: least squares, elastic network, as well as Ivakhnenko's inductive evolutionary algorithm. Kolmogorov-Gabor polynomials are used as a reference function of a polynomial neural network. Model errors and performance were calculated on a test set. Models were compared on a linear transport problem under uncertainty: the diffusion coefficient and drift were modeled by uniformly distributed random variables. It is shown that with a small interval of variation in the values of random variables, both models give good efficiency, but the PC model demonstrates a smaller spread of errors and is faster in time. For the de-cay equation with random coefficients distributed according to the Gaussian law, the influence of the correlation of these coefficients on the rate of convergence is studied. It is shown that with dependent coefficients, the best performance is observed in higher-order PC models. On the basis of comparative modeling, it has been established that the use of PC is unambiguously preferable in the following cases: a small dimension of the space of input features, a known law of distribution of input data, and correlated features. It is also shown that the use of PC with a large dimension of the space of input features is inefficient due to the rapid increase in the number of terms in the expansion, leading to a sharp increase in the time to process the task. In this case, the regression model based on the Kolmogorov-Gabor polynomials in combination with the GMDH turned out to be clearly preferable.

Облакова Т.В., Фам Куок Вьет. Сравнительное моделирование на основе многочленов Колмогорова-Габора в задачах полиномиального хаоса и регрессии. Математическое моделирование и численные методы, 2023, № 4, с. 93–108.