1,547 research outputs found
An adaptive regularization method in Banach spaces
This paper considers optimization of nonconvex functionals in smooth infinite dimensional spaces. It is first proved that functionals in a class containing multivariate polynomials augmented with a sufficiently smooth regularization can be minimized by a simple linesearch-based algorithm. Sufficient smoothness depends on gradients satisfying a novel two-terms generalized Lipschitz condition. A first-order adaptive regularization method applicable to functionals with β-Hölder continuous derivatives is then proposed, that uses the linesearch approach to compute a suitable trial step. It is shown to find an ϵ-approximate first-order point in at most (Formula presented.) evaluations of the functional and its first p derivatives.</p
A Posteriori Error Control for the Binary Mumford-Shah Model
The binary Mumford-Shah model is a widespread tool for image segmentation and
can be considered as a basic model in shape optimization with a broad range of
applications in computer vision, ranging from basic segmentation and labeling
to object reconstruction. This paper presents robust a posteriori error
estimates for a natural error quantity, namely the area of the non properly
segmented region. To this end, a suitable strictly convex and non-constrained
relaxation of the originally non-convex functional is investigated and Repin's
functional approach for a posteriori error estimation is used to control the
numerical error for the relaxed problem in the -norm. In combination with
a suitable cut out argument, a fully practical estimate for the area mismatch
is derived. This estimate is incorporated in an adaptive meshing strategy. Two
different adaptive primal-dual finite element schemes, and the most frequently
used finite difference discretization are investigated and compared. Numerical
experiments show qualitative and quantitative properties of the estimates and
demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl
Kohn-Sham theory with paramagnetic currents: compatibility and functional differentiability
Recent work has established Moreau-Yosida regularization as a mathematical
tool to achieve rigorous functional differentiability in density-functional
theory. In this article, we extend this tool to paramagnetic
current-density-functional theory, the most common density-functional framework
for magnetic field effects. The extension includes a well-defined Kohn-Sham
iteration scheme with a partial convergence result. To this end, we rely on a
formulation of Moreau-Yosida regularization for reflexive and strictly convex
function spaces. The optimal -characterization of the paramagnetic current
density is derived from the -representability conditions.
A crucial prerequisite for the convex formulation of paramagnetic
current-density-functional theory, termed compatibility between function spaces
for the particle density and the current density, is pointed out and analyzed.
Several results about compatible function spaces are given, including their
recursive construction. The regularized, exact functionals are calculated
numerically for a Kohn-Sham iteration on a quantum ring, illustrating their
performance for different regularization parameters
Confidence driven TGV fusion
We introduce a novel model for spatially varying variational data fusion,
driven by point-wise confidence values. The proposed model allows for the joint
estimation of the data and the confidence values based on the spatial coherence
of the data. We discuss the main properties of the introduced model as well as
suitable algorithms for estimating the solution of the corresponding biconvex
minimization problem and their convergence. The performance of the proposed
model is evaluated considering the problem of depth image fusion by using both
synthetic and real data from publicly available datasets
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