Results 1  10
of
28
CoherenceEnhancing Diffusion Filtering
, 1999
"... The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operato ..."
Abstract

Cited by 83 (2 self)
 Add to MetaCart
The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operator (secondmoment matrix, structure tensor). An mdimensional formulation of this method is analysed with respect to its wellposedness and scalespace properties. An efficient scheme is presented which uses a stabilization by a semiimplicit additive operator splitting (AOS), and the scalespace behaviour of this method is illustrated by applying it to both 2D and 3D images.
On the CahnHilliard equation with degenerate mobility
 SIAM J. Math. Anal
, 1996
"... An existence result for the CahnHilliard equation with a concentration dependent diffusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values \Sigma1 and it is shown that the solution is bounded by 1 in magnitude. Finally applicat ..."
Abstract

Cited by 43 (11 self)
 Add to MetaCart
An existence result for the CahnHilliard equation with a concentration dependent diffusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values \Sigma1 and it is shown that the solution is bounded by 1 in magnitude. Finally applications of our method to other degenerate fourth order parabolic equations are discussed.
Boundary case of equality in optimal Loewnertype inequalities
 TRANS. AMER. MATH. SOC
, 2004
"... We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing AbelJacobi maps from X to its Jacobi torus T b, w ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing AbelJacobi maps from X to its Jacobi torus T b, which are areadecreasing (on bdimensional areas), with respect to suitable norms. These norms are the stable norm of g, the conformally invariant norm, as well as other L pnorms. The case of equality is characterized in terms of the criticality of the lattice of deck transformations of T b, while the AbelJacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, in the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the AbelJacobi map to the maximal free abelian cover.
On Mathematical Models For Phase Separation In Elastically Stressed Solids
, 2000
"... Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. Uniqueness for homogeneous linear elasticity 26 4. Logarithmic free energy 29 4.1. A regularised problem 32 4.2. Higher integrability for the strain tensor 36 4.3. Higher integrability for the logarithmic free energy 42 4.4. Proof of the existence theorem 45 5. The sharp interface limit 46 5.1. The \Gammalimit of the elastic GinzburgLandau energies 52 5.2. EulerLagrange equation for the sharp interface functional 60 6. The GibbsThomson equation as a singular limit in the scalar case 70 7. Discussion 79 8. Appendix 81 9. Notation 86 References 90 1 1. Introduction We study a mathematical model describing phase separation in multi component alloy
An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation
 MR 99b:65141
, 1998
"... Abstract. We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No as ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
Abstract. We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape. A posteriori error estimates are given in the energy norm and the L2norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the nondiscretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay nondiscretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected. 1.
On CahnHilliard Systems with Elasticity
 Proc. Roy. Soc. Edinburgh, 133 A
"... Elastic eects can have a pronounced eect on the phase separation process in solids. The classical GinzburgLandau energy can be modi ed to account for such elastic interactions. The evolution of the system is then governed by diusion equations for the concentrations of the alloy components and by a ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Elastic eects can have a pronounced eect on the phase separation process in solids. The classical GinzburgLandau energy can be modi ed to account for such elastic interactions. The evolution of the system is then governed by diusion equations for the concentrations of the alloy components and by a quasistatic equilibrium for the mechanical part. The resulting system of equations is ellipticparabolic and can be understood as a generalisation of the CahnHilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it. 1.
A residual a posteriori error estimator for the eigenvalue problem for the LaplaceBeltrami operator
, 2005
"... The LaplaceBeltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the LaplaceBeltrami operator on subdomains of the unit sphere in R3. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
The LaplaceBeltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the LaplaceBeltrami operator on subdomains of the unit sphere in R3. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the twodimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clémenttype interpolation operator have to be introduced.
Dirichlet Forms And Markov Processes: A Generalized Framework Including Both Elliptic And Parabolic Cases
"... We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dir ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [Fu2], [M/R]) as well as time dependent Dirichlet forms (cf. [O1]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by ffexcessive functions h (htransformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, c...
Finite Element Simulation of Bone Microstructures
"... The geometric construction of finite element spaces suitable for complicated shapes or microstructured materials is discussed. As an application, the efficient computation of linearized elasticity is considered on them. Geometries are supposed to be implicitly described via 3D voxel data (e. g. µCT ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
The geometric construction of finite element spaces suitable for complicated shapes or microstructured materials is discussed. As an application, the efficient computation of linearized elasticity is considered on them. Geometries are supposed to be implicitly described via 3D voxel data (e. g. µCT scans) associated with a cubic grid. We place degrees of freedom only at the grid nodes and incorporate the complexity of the domain in the hierarchy of finite element basis functions, i. e. constructed by cut off operations at the reconstructed domain boundary. Thus, our method inherits the nestedness of uniform hexahedral grids while still being able to resolve complicated structures. In particular, the canonical coarse scales on hexahedral grid hierarchies can be used in multigrid methods. AMS Subject Classifications: 65N30, 65N55, 65N50. 1