Parallel Solving of Fredholm Integral Equations of the First Kind by Tikhonov Regularization Method Using OpenMP Technology

  • Lesia MochuradEmail author
  • Khrystyna Shakhovska
  • Sergio Montenegro
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1080)


The paper considers the software approach to solving the Fredholm integral equation of the first kind by the Tikhonov regularization method, which uses the properties of modern architectures of computing systems as multi-core and the standard of parallel programming of OpenMP. A numerical algorithm for solving a Fredholm integral equation of the first kind is proposed such as the Tikhonov regularization method. It is based on the method of collocation under conditions of piecewise constant approximation of the desired function. When choosing even this most economical method of approximation, we obtain systems of linear algebraic equations of large dimensions with densely filled matrices to achieve the necessary accuracy. The variation of the error is analyzed, depending on the choice of the regularization parameter, which is selected experimentally. In order to optimize the computational process, the procedure for solving the integral equation is compared with the use of the OpenMP parallel programming technique. Improved acceleration and performance. The results, which indicate the possibility of further optimization of the computational process due to the variation in the number of parallel streams and computer processor cores, are obtained. A series of numerical experiments that confirm the effectiveness and feasibility of the proposed approach to numerical solution of Fredholm integral equations of the first kind.


Incorrectness problem Poorly conditioned matrix Regularizing operator Multicore OpenMP software standard 


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Lviv Polytechnic National UniversityLvivUkraine
  2. 2.Julius-Maximilians-University WürzburgWürzburgGermany

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