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On the Use of the Conjugate Gradient Method for the Numerical Solution of First-Kind Integral Equations in Two Variables

In: Integral Methods in Science and Engineering

Author

Listed:
  • Barbara Bertram
  • Haiyan Cheng

Abstract

In this chapter we study the use of the conjugate gradient method for solving Fredholm integral equations of the first kind of the form 8.1 $$\int_{0}^{1} {\int_{0}^{1} {K(x - s)K(y - t)f(s,t)dsdt = g(x,y),} }$$ where the limits of integration are taken to be 0 and 1 without loss of generality. Such equations occur often in imaging problems, and when the kernels are nondegenerate, are marked by being ill-posed [1]. These problems yield discretizations that are linear systems and must be handled with great care because the coefficient matrices are quite ill-conditioned. To cope with this ill conditioning, we choose the method of conjugate gradients (see [2] and [3]), which has several advantages. The dependence upon the condition number is milder and in addition we may use the number of iterations as a regularizing parameter.

Suggested Citation

  • Barbara Bertram & Haiyan Cheng, 2002. "On the Use of the Conjugate Gradient Method for the Numerical Solution of First-Kind Integral Equations in Two Variables," Springer Books, in: Christian Constanda & Peter Schiavone & Andrew Mioduchowski (ed.), Integral Methods in Science and Engineering, chapter 8, pages 51-56, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0111-3_8
    DOI: 10.1007/978-1-4612-0111-3_8
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