Improved forms of iterative methods for systems of linear algebraic equations

Рассматриваются особенности численной реализации решения трехмерной обратной задачи обращения полных тензорных магнитно-градиентных данных, которая моделируется системой двух трехмерных интегральных уравнений Фредгольма 1-го рода. Для решения этой некорректно поставленной задачи применяется алгоритм, основанный на минимизации функционала А.Н. Тихонова. В качестве метода минимизации используется метод сопряженных градиентов. Выбор параметра регуляризации осуществляется в соответствии с версией обобщенного принципа невязки, в которой учитываются ошибки округления, существенные при решении задач большой размерности. Features of numerical solution of the three-dimensional ill-posed problem devoted to the inversion of full tensor magnetic gradient data are considered. This problem is simulated by a system of two three-dimensional Fredholm integral equations of the first kind. The Tikhonov regularization is applied to solve this ill-posed problem. The conjugate gradient method is used as a minimization method. The choice of the regularization parameter is realized according to the generalized residual principle with consideration of round-off errors capable of affecting the final result of calculations significantly.

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Accounting for Round-Off Errors When Using Gradient Minimization Methods

Lukyanenko

Shinkarev

Yagola

2022

Algorithms

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This paper discusses a method for taking into account rounding errors when constructing a stopping criterion for the iterative process in gradient minimization methods. The main aim of this work was to develop methods for improving the quality of the solutions for real applied minimization problems; these solutions require significant amounts of calculations, and as a result, they can be sensitive to the accumulation of rounding errors. However, this paper demonstrates that the developed approach can also be useful in solving computationally small problems. The main ideas of this work are demonstrated using one of the possible implementations of the conjugate gradient method for solving an overdetermined system of linear algebraic equations with a dense matrix.

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Improved forms of iterative methods for systems of linear algebraic equations

Cited by 3 publications

References 2 publications

Comparing robust forms of iterative methods of conjugate directions

Comparing robust forms of iterative methods of conjugate directions

Regularized inversion of full tensor magnetic gradient data

Accounting for Round-Off Errors When Using Gradient Minimization Methods

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