We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are Lp stable for any real number p ∈ [1,∞], and commute with the differential operators ∇, ∇×, and ∇·. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields....

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 th...

The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality o...

In this paper, we consider numerical differentiation of bivariate functions when a set of noisy data is given. A mollification method based on spanned by Legendre polynomials is proposed and the mollification parameter is chosen by a discrepancy principle. The theoretical analyses show that the smoother the genuine solution, the higher the convergence rates of the numerical solution. To get a p...

We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as limit of driven by noise that is mollified space and time. As mollification radius $ \varepsilon\to 0 $, we tune coupling constant near critical point, show single time correlation functions converge to written terms an explicit non-trivial semigroup. Our approach consists steps. First converge...