"DEVELOPMENT OF ADAPTIVE PARALLEL PROCEDURES FOR CONTROLLING THE MODELING STEP USING SURROGATE ANALYSIS"

The work is devoted to the development of adaptive procedures for controlling the integration step in the numerical implementation of dynamic models described by ordinary differential equations or systems of equations. For stiff cases, efficient parallel procedures for adaptive step control were introduced into the numerical implementations of difference compositions of block methods. The size of a new step was determined by the behavior of the process under study, the order of accuracy of the difference compositions involved, and the requirements for the criteria of the objective function. Single-step block methods with different numbers of calculation points in a block and with corresponding orders of accuracy were used as step control schemes. If there were enough computing processors, then a separate processor was assigned to each computational point in the block. The discrepancy norm was estimated for making a decision on accepting the current integration step and determining the length of the next integration step in the coinciding nodes of the grid function. During the calculation, intervals of stability, reduction and increase of the integration step were automatically diagnosed. If the local error in the block on the marching schemes was not achieved immediately, additional clarifying iterations were performed. The step was abandoned only if such iterations were unsuccessful. To verify the parameters, an approach was used in the work focused on the use of surrogate models built on the basis of Legendre orthogonal polynomials. The studies were carried out on equations (systems of equations) whose degree of rigidity was easily varied. The estimated indicators of the objective function were the time to obtain a solution, the normalized calculation error, the ratio of the number of effective steps to the total number of completed ones. The data-driven machine learning methods used in the work made it possible to estimate a variety of parameters for control that deliver the extremum of the multi-criteria objective function or allow the formation of a Pareto front. Testing of the proposed algorithms on stiff Cauchy problems with the possibility of varying the rigidity parameter was carried out as a comparison of the results of applying classical approaches with the proposed ones.