ABSTRACT
This chapter examines the use of the conjugate gradient method in conjunction with wavelet expansions for the solution of first kind integral equations. Fredholm integral equations of the first kind with non-degenerate kernels represent some of the most frequent examples of ill-posed problems. The chapter utilizes the conjugate gradient method to solve the discretized system, and the number of iterations can be considered to be the regularization parameter. It explores the use of three types of scaling functions: Daubechies orthonormal, quadratic B-spline, and truncated, shifted Shannon; three types of image functions: twin peaks, three triangles, and parabola; and, in addition, two kernels: sinc and stove pipe. Considering the parabolic object function, and either type of kernel, both the quadratic B-spline wavelets and the Shannon wavelet expansions give better results than the Daubechies wavelets.
