Solving ill-posed problems with mollification and an application in biometrics

Abstract: This is a thesis about how mollification can be used as a regularization method to reduce noise in ill-posed problems in order to make them well-posed. Ill-posed problems are problems where noise get magnified during the solution process. An example of this is how measurement errors increases with differentiation. To correct this we use mollification. Mollification is a regularization method that uses integration or weighted average to even out a noisy function. The different types of error that occurs when mollifying are the truncation error and the propagated data error. We are going to calculate these errors and see what affects them. An other thing worth investigating is the ability to differentiate a mollified function even if the function itself can not be differentiated. An application to mollification is a blood vessel problem in biometrics where the goal is to calculate the elasticity of the blood vessel’s wall. To do this measurements from the blood and the blood vessel are required, as well as equations for the calculations. The model used for the calculations is ill-posed with regard to specific variables which is why we want to apply mollification. Here we are also going to take a look at how the noise level affects the final result as well as the mollification radius.