Сравнение интерполяционного и мозаично-скелетонного методов для решения интегральных уравнений со сверточным ядром

Журнал вычислительной математики и математической физики. 2025. Т. 65. № 6. С. 861–874.

Валиахметов Б. И., Гладков А. О., Самохин А. Б., Тыртышников Е. Е.

In the article, interpolation-based and mosaic-skeleton approaches for solving the problem of the potential flow of a plate are compared. They compress a dense matrix of a linear system, arising in the collocation method on the non-uniform grid. The first method relies on Fast Fourier Transform and linear interpolation from an additional uniform grid. The second uses block low-rank approximation of a matrix. Both techniques demonstrate effectiveness in time and memory but distinguish different structures in a matrix, which affects the solution of a linear system. In our implementation, the mosaic-skeleton method solves the problem quicker than its competitor. Still, it consumes more memory, and its duration grows more significantly with an increase in the size of a system.

Keywords: интегральные уравнения, Fast Fourier Transform, быстрое преобразование Фурье, integral equations, неравномерная сетка по пространству, non-uniform grid, collocation method, mosaic-skeleton method, метод коллокаций, мозаично-скелетонный метод