Quasi-smoothing Method For Solving Ill-posed Problems And Its Application

Based on Phillips smoothing method, this paper proposed the quasi-smoothing method by establishing homotopy relations between the first kind of operator equations and the second kind of operator equations, and the selection of smoothing parameter is studied. Numerical results show that the method is feasible to solve the problem which is oscillation slightly intense,but is not as effective as smoothing method on the problem without oscillation. So we proposed modified quasi-smoothing method. It's difficult to known the oscillation of the true solution actually, which make it difficult to decide which method. So we proposed the modified quasi-smoothing method, which integrate the effects of the two method by controlling the weight factor. In this way, smoothing method which can effective simulate the slight oscillation play a major role if the weight factor is small and contrarily quasi-smoothing method play a major role.Although quasi-smoothing method and modified quasi-smoothing method can relatively well simulate the true solution which is oscillation slightly dramatic, to more dramatic oscillation situation, neither of them is effective. Therefore, piecewise interval method is proposed to solve the problem which is oscillation intense. That is combine the solutions on each interval computing on smoothing method and modified quasi-smoothing method. Numerical results show that piecewise interval method using modified quasi-smoothing method on each interval is very effective,. It is noteworthy that it is not good to divide the interval into too many sub-intervals, otherwise it will affect the speed.Solve the integral equation of the first kind using modified quasi-smoothing method and cubic spline function method. First of all, discrete the integral equation using cubic spline function method to ill linear equations. Then the modified quasi-smoothing method is applied to solve the ill-posed problem. Simulation results show that the modified quasi-smoothing method is effective.