Convex Regularization Method for Solving Cauchy Problem of the Helmholtz Equation

In this paper, we introduce the Convex Regularization Method (CRM) for regularizing the (instability) solution of the Helmholtz equation with Cauchy data. The CRM makes it possible for the solution of Helmholtz equation to depend continuously on the small perturbations in the Cauchy data. In addition, the numerical computation of the reg- ularized Helmholtz equation with Cauchy data is stable, accurate and gives high rate of convergence of solution in Hilbert space. Undoubtedly, the error estimated analysis associated with CRM is minimal.Mathematics Subject Classi cation: 44B28; 44B30Keywords: Convex Regularization Method, ill-posed Helmholtz equation with Cauchy data, stable solutio

Joint Image Reconstruction and Segmentation Using the Potts Model

We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from $7$ angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data

Approximation of mild solutions of the linear and nonlinear elliptic equations

In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form $$\frac{\partial^{2}}{\partial t^{2}}u\left(t\right)=\mathcal{A}u\left(t\right)+f\left(t,u\left(t\right)\right),\quad t\in\left[0,T\right],$$ where $\mathcal{A}$ is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.Comment: 29 pages, 16 figures, July 201