The solution of infinitely ill-conditioned weakly-singular problems

Error bounds for extremely ill-conditioned problems

We discuss methods to compute error bounds for extremely ill-conditioned problems. As a model problem we treat matrix inversion. We demonstrate that additive corrections to improve an approximate inverse are useful for ill-conditioned problems, but hardly usable for extremely ill-conditioned problems. Here multiplicative corrections can be used, including the possibility to compute guaranteed e...

Regularization of Large-scale Ill-conditioned Least Squares Problems Regularization of Large{scale Ill{conditioned Least Squares Problems

Ill{conditioned problems arise in important areas like geophysics, medical imaging and signal processing. The fact that the ill{cond-itioning is an intrinsic feature of these problems makes it necessary to develop special numerical methods to treat them. Regularization methods belong to this class. The lack of robust regularization methods for large{scale ill{cond-itioned problems motivated thi...

Numerical solution of a type of weakly singular nonlinear Volterra integral equation by Tau Method

‎In this paper‎, ‎a matrix based method is considered for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $s^{beta}(t-s)^{-alpha}G(y(s))$ based on the Tau method‎. ‎In this method‎, ‎a transformation of the independent variable is first introduced in order to obtain a new equation with smoother solution‎. ‎Error analysis of this method is also ...

Information Complexity-Based Regularization Parameter Selection for Solution of Ill-Conditioned Inverse Problems

We propose an information complexity-based regularization parameter selection method for solution of ill-conditioned inverse problems. The regularization parameter is selected to be the minimizer of the Kullback-Leibler (KL) distance between the unknown data-generating distribution and the fitted distribution. The KL distance is approximated by an information complexity (ICOMP) criterion develo...

A Method to Estimate the Solution of a Weakly Singular Non-linear Integro-differential Equations by Applying the Homotopy Methods