A variational model for infin ite perimeter segmentations based on lipschitz level set functions: Denoising while keeping finely oscillatory boundaries

We propose a new model for segmenting piecewise constant images with irregular object boundaries: a variant of the Chan-Vese model [T. F. Chan and L. A. Vese, IEEE Trans. Image Process., 10 (2000), pp. 266-277], where the length penalization of the boundaries is replaced by the area of their neighborhood of thickness e. Our aim is to keep fine details and irregularities of the boundaries while denoising additive Gaussian noise. For the numerical computation we revisit the classical BV level set formulation [S. Osher and J. A. Sethian, J. Comput. Phys., 79 (1988), pp. 12-49] considering suitable Lipschitz level set functions instead of BV ones

Reconstruction of cracks and material losses by perimeter-like penalizations and phase-field methods: numerical results

We numerically implement the variational approach for reconstruction in the inverse crack and cavity problems developed by one of the authors. The method is based on a suitably adapted free-discontinuity problem. Its main features are the use of phase-field functions to describe the defects to be reconstructed and the use of perimeter-like penalizations to regularize the ill-posed problem. The numerical implementation is based on the solution of the corresponding optimality system by a gradient method. Numerical simulations are presented to show the validity of the method.Comment: 15 pages, 12 figure

A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

A phase-field model for fractures in incompressible solids

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented

Modeling, Discretization, Optimization, and Simulation of Phase-Field Fracture Problems

This course is devoted to phase-field fracture methods. Four different sessions are centered around modeling, discretizations, solvers, adaptivity, optimization, simulations and current developments. The key focus is on research work and teaching materials concerned with the accurate, efficient and robust numerical modeling. These include relationships of model, discretization, and material parameters and their influence on discretizations and the nonlinear (Newton-type methods) and linear numerical solution. One application of such high-fidelity forward models is in optimal control, where a cost functional is minimized by controlling Neumann boundary conditions. Therein, as a side-project (which is itself novel), space-time phase-field fracture models have been developed and rigorously mathematically proved. Emphasis in the entire course is on a fruitful mixture of theory, algorithmic concepts and exercises. Besides these lecture notes, further materials are available, such as for instance the open-source libraries pfm-cracks and DOpElib. The prerequisites are lectures in continuum mechanics, introduction to numerical methods, finite elements, and numerical methods for ODEs and PDEs. In addition, functional analysis (FA) and theory of PDEs is helpful, but for most parts not necessarily mandatory. Discussions with many colleagues in our research work and funding from the German Research Foundation within the Priority Program 1962 (DFG SPP 1962) within the subproject Optimizing Fracture Propagation using a Phase-Field Approach with the project number 314067056 (D. Khimin, T. Wick), and support of the French-German University (V. Kosin) through the French-German Doctoral college ``Sophisticated Numerical and Testing Approaches" (CDFA-DFDK 19-04) is gratefully acknowledged

Image reconstruction by Mumford-Shah regularization with a priori edge information

The Mumford-Shah functional has provided an important approach for image denoising and segmentation. Recently, it has been applied to image reconstruction in fields such as X-ray tomography and electric impedance tomography. In this thesis we study the applicability of the Mumford-Shah model to a setting, where a priori edge information is available and reliable. Such a situation occurs for example in biomedical imaging, where multimodal imaging systems have received a lot of interest. The regularization terms in the Mumford-Shah functional force smoothness of the image within individual regions and simultaneously detect edges across which smoothing is prevented. We propose to divide the edge penalty into two parts depending on the a priori edge information. We investigate the proposed model for well-posedness and regularization properties under an assumption of pointwise boundedness of the underlying image. Furthermore, we present two variational approximations that allow numerical implementations. For one we prove that it Gamma converges to a special case of our proposed model, the other we motivate heuristically. The resulting algorithm alternates between an image reconstruction and an image evaluation step. We illustrate the feasibility with two numerical examples