80 research outputs found
A combined first and second order variational approach for image reconstruction
In this paper we study a variational problem in the space of functions of
bounded Hessian. Our model constitutes a straightforward higher-order extension
of the well known ROF functional (total variation minimisation) to which we add
a non-smooth second order regulariser. It combines convex functions of the
total variation and the total variation of the first derivatives. In what
follows, we prove existence and uniqueness of minimisers of the combined model
and present the numerical solution of the corresponding discretised problem by
employing the split Bregman method. The paper is furnished with applications of
our model to image denoising, deblurring as well as image inpainting. The
obtained numerical results are compared with results obtained from total
generalised variation (TGV), infimal convolution and Euler's elastica, three
other state of the art higher-order models. The numerical discussion confirms
that the proposed higher-order model competes with models of its kind in
avoiding the creation of undesirable artifacts and blocky-like structures in
the reconstructed images -- a known disadvantage of the ROF model -- while
being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure
New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries
The field of mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a flourishing area of research. Many such systems arise from mathematical models in material science, fluid dynamics and biology, for example phase separation in alloys, epitaxial growth, dynamics of multiphase fluids, evolution of cell membranes and in industrial processes such as crystal growth. The governing equations for the dynamics of the interfaces in many of these applications involve surface tension expressed in terms of the mean curvature and a driving force. Here the forcing terms depend on variables that are solutions of additional partial differential equations which hold either on the interface itself or in the surrounding bulk regions. Often in applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimization problems with partial differential equation constraints including free boundaries. Because of the maturity of the field of computational free boundary problems it is now timely to consider such control problems
Schnelle Löser für Partielle Differentialgleichungen
This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds
A linear, second-order, energy stable, fully adaptive finite-element method for phase-field modeling of wetting phenomena
We propose a new numerical method to solve the Cahn-Hilliard equation coupled with non-linear wetting boundary conditions. We show that the method is mass-conservative and that the discrete solution satisfies a discrete energy law similar to the one satisfied by the exact solution. We perform several tests inspired by realistic situations to verify the accuracy and performance of the method: wetting of a chemically heterogeneous substrate in three dimensions, wetting-driven nucleation in a complex two dimensional domain and three-dimensional diffusion through a porous medium
Geometric partial differential equations: Surface and bulk processes
The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems
A Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients
In this paper, we develop a gradient recovery based linear (GRBL) finite
element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second
order elliptic equations in non-divergence form. The elliptic equation is
casted into a symmetric non-divergence weak formulation, in which second order
derivatives of the unknown function are involved. We use gradient and Hessian
recovery operators to calculate the second order derivatives of linear finite
element approximations. Although, thanks to low degrees of freedom (DOF) of
linear elements, the implementation of the proposed schemes is easy and
straightforward, the performances of the methods are competitive. The unique
solvability and the seminorm error estimate of the GRBL scheme are
rigorously proved. Optimal error estimates in both the norm and the
seminorm have been proved when the coefficient is diagonal, which have been
confirmed by numerical experiments. Superconvergence in errors has also been
observed. Moreover, our methods can handle computational domains with curved
boundaries without loss of accuracy from approximation of boundaries. Finally,
the proposed numerical methods have been successfully applied to solve fully
nonlinear Monge-Amp\`{e}re equations
- …