349 research outputs found
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
We address the denoising of images contaminated with multiplicative noise,
e.g. speckle noise. Classical ways to solve such problems are filtering,
statistical (Bayesian) methods, variational methods, and methods that convert
the multiplicative noise into additive noise (using a logarithmic function),
shrinkage of the coefficients of the log-image data in a wavelet basis or in a
frame, and transform back the result using an exponential function. We propose
a method composed of several stages: we use the log-image data and apply a
reasonable under-optimal hard-thresholding on its curvelet transform; then we
apply a variational method where we minimize a specialized criterion composed
of an data-fitting to the thresholded coefficients and a Total
Variation regularization (TV) term in the image domain; the restored image is
an exponential of the obtained minimizer, weighted in a way that the mean of
the original image is preserved. Our restored images combine the advantages of
shrinkage and variational methods and avoid their main drawbacks. For the
minimization stage, we propose a properly adapted fast minimization scheme
based on Douglas-Rachford splitting. The existence of a minimizer of our
specialized criterion being proven, we demonstrate the convergence of the
minimization scheme. The obtained numerical results outperform the main
alternative methods
Backward Stochastic Differential Equations on Manifolds
The problem of finding a martingale on a manifold with a fixed random
terminal value can be solved by considering BSDEs with a generator with
quadratic growth. We study here a generalization of these equations and we give
uniqueness and existence results in two different frameworks, using
differential geometry tools. Applications to PDEs are given, including a
certain class of Dirichlet problems on manifolds.Comment: 47 pages To be published in PTR
Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions
We perform a stochastic-homogenization analysis for composite materials
exhibiting a random microstructure. Under the assumptions of stationarity and
ergodicity, we characterize the Gamma-limit of a micromagnetic energy
functional defined on magnetizations taking value in the unit sphere, and
including both symmetric and antisymmetric exchange contributions. This
Gamma-limit corresponds to a micromagnetic energy functional with homogeneous
coefficients. We provide explicit formulas for the effective magnetic
properties of the composite material in terms of homogenization correctors.
Additionally, the variational analysis of the two exchange energy terms is
performed in the more general setting of functionals defined on manifold-valued
maps with Sobolev regularity, in the case in which the target manifold is a
bounded, orientable smooth surface with tubular neighborhood of uniform
thickness. Eventually, we present an explicit characterization of minimizers of
the effective exchange in the case of magnetic multilayers, providing
quantitative evidence of Dzyaloshinskii's predictions on the emergence of
helical structures in composite ferromagnetic materials with stochastic
microstructure
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations
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