9,704 research outputs found
Applications of nonlinear diffusion in image processing and computer vision
Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniques we explain the main ideas, discuss theoretical properties and present an appropriate numerical scheme. The numerical schemes are based on additive operator splittings (AOS). In contrast to traditional multiplicative splittings such as ADI, LOD or D'yakonov splittings, all axes are treated in the same manner, and additional possibilities for efficient realizations on parallel and distributed architectures appear. Geodesic active contours lead to equations that resemble mean curvature motion. For this application, a novel AOS scheme is presented that uses harmonie averaging and does not require reinitializations of the distance function in each iteration step
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
Operator splittings and spatial approximations for evolution equations
The convergence of various operator splitting procedures, such as the
sequential, the Strang and the weighted splitting, is investigated in the
presence of a spatial approximation. To this end a variant of Chernoff's
product formula is proved. The methods are applied to abstract partial delay
differential equations.Comment: to appear in J. Evol. Equations. Reviewers comments are incorporate
Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs
Operator splitting methods allow to split the operator describing a complex
dynamical system into a sequence of simpler subsystems and treat each part
independently. In the modeling of dynamical problems, systems of (possibly
coupled) differential-algebraic equations (DAEs) arise. This motivates the
application of operator splittings which are aware of the various structural
forms of DAEs. Here, we present an approach for the splitting of coupled
index-1 DAE as well as for the splitting of port-Hamiltonian DAEs, taking
advantage of the energy-conservative and energy-dissipative parts. We provide
numerical examples illustrating our second-order convergence results
Operator splitting with spatial-temporal discretization
Continuing earlier investigations, we analyze the convergence of operator
splitting procedures combined with spatial discretization and rational
approximations
A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities
It is shown that tunnelling splittings in ergodic double wells and resonant
widths in ergodic metastable wells can be approximated as easily-calculated
matrix elements involving the wavefunction in the neighbourhood of a certain
real orbit. This orbit is a continuation of the complex orbit which crosses the
barrier with minimum imaginary action. The matrix element is computed by
integrating across the orbit in a surface of section representation, and uses
only the wavefunction in the allowed region and the stability properties of the
orbit. When the real orbit is periodic, the matrix element is a natural measure
of the degree of scarring of the wavefunction. This scarring measure is
canonically invariant and independent of the choice of surface of section,
within semiclassical error. The result can alternatively be interpretated as
the autocorrelation function of the state with respect to a transfer operator
which quantises a certain complex surface of section mapping. The formula
provides an efficient numerical method to compute tunnelling rates while
avoiding the need for the exceedingly precise diagonalisation endemic to
numerical tunnelling calculations.Comment: Submitted to Annals of Physics. This work has been submitted to
Academic Press for possible publicatio
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