781 research outputs found
Nonlinear Evolutionary PDE-Based Refinement of Optical Flow
The goal of this paper is to propose two nonlinear variational models for
obtaining a refined motion estimation from an image sequence. Both the proposed
models can be considered as a part of a generalized framework for an accurate
estimation of physics-based flow fields such as rotational and fluid flow. The
first model is novel in the sense that it is divided into two phases: the first
phase obtains a crude estimate of the optical flow and then the second phase
refines this estimate using additional constraints. The correctness of this
model is proved using an Evolutionary PDE approach. The second model achieves
the same refinement as the first model, but in a standard manner, using a
single functional. A special feature of our models is that they permit us to
provide efficient numerical implementations through the first-order primaldual
Chambolle-Pock scheme. Both the models are compared in the context of accurate
estimation of angle by performing an anisotropic regularization of the
divergence and curl of the flow respectively. We observe that, although both
the models obtain the same level of accuracy, the two-phase model is more
efficient. In fact, we empirically demonstrate that the single-phase and the
two-phase models have convergence rates of order and
respectively
On well-posedness of variational models of charged drops
Electrified liquids are well known to be prone to a variety of interfacial
instabilities that result in the onset of apparent interfacial singularities
and liquid fragmentation. In the case of electrically conducting liquids, one
of the basic models describing the equilibrium interfacial configurations and
the onset of instability assumes the liquid to be equipotential and interprets
those configurations as local minimizers of the energy consisting of the sum of
the surface energy and the electrostatic energy. Here we show that,
surprisingly, this classical geometric variational model is mathematically
ill-posed irrespectively of the degree to which the liquid is electrified.
Specifically, we demonstrate that an isolated spherical droplet is never a
local minimizer, no matter how small is the total charge on the droplet, since
the energy can always be lowered by a smooth, arbitrarily small distortion of
the droplet's surface. This is in sharp contrast with the experimental
observations that a critical amount of charge is needed in order to destabilize
a spherical droplet. We discuss several possible regularization mechanisms for
the considered free boundary problem and argue that well-posedness can be
restored by the inclusion of the entropic effects resulting in finite screening
of free charges.Comment: 18 pages, 2 figure
Inverse optical tomography through PDE constrained optimisation in L∞
Fluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses
non-ionising red and infrared light. Mathematically, FOT can be modelled as
an inverse parameter identification problem, associated with a coupled elliptic
system with Robin boundary conditions. Herein we utilise novel methods of
Calculus of Variations in L∞ to lay the mathematical foundations of FOT
which we pose as a PDE-constrained minimisation problem in Lp and L∞
Weighted Generalized Fractional Integration by Parts and the Euler-Lagrange Equation
Integration by parts plays a crucial role in mathematical analysis, e.g.,
during the proof of necessary optimality conditions in the calculus of
variations and optimal control. Motivated by this fact, we construct a new,
right-weighted generalized fractional derivative in the Riemann-Liouville sense
with its associated integral for the recently introduced weighted generalized
fractional derivative with Mittag-Leffler kernel. We rewrite these operators
equivalently in effective series, proving some interesting properties relating
to the left and the right fractional operators. These results permit us to
obtain the corresponding integration by parts formula. With the new general
formula, we obtain an appropriate weighted Euler-Lagrange equation for dynamic
optimization, extending those existing in the literature. We end with the
application of an optimization variational problem to the quantum mechanics
framework.Comment: This is a preprint of a paper whose final and definite form is
published Open Access in 'Axioms' at [https://doi.org/10.3390/axioms11040178
Non-local control in the conduction coefficients: well posedness and convergence to the local limit
We consider a problem of optimal distribution of conductivities in a system
governed by a non-local diffusion law. The problem stems from applications in
optimal design and more specifically topology optimization. We propose a novel
parametrization of non-local material properties. With this parametrization the
non-local diffusion law in the limit of vanishing non-local interaction
horizons converges to the famous and ubiquitously used generalized Laplacian
with SIMP (Solid Isotropic Material with Penalization) material model. The
optimal control problem for the limiting local model is typically ill-posed and
does not attain its infimum without additional regularization. Surprisingly,
its non-local counterpart attains its global minima in many practical
situations, as we demonstrate in this work. In spite of this qualitatively
different behaviour, we are able to partially characterize the relationship
between the non-local and the local optimal control problems. We also
complement our theoretical findings with numerical examples, which illustrate
the viability of our approach to optimal design practitioners
Stability of the solution set of quasi-variational inequalities and optimal control
For a class of quasi-variational inequalities (QVIs) of obstacle-type the
stability of its solution set and associated optimal control problems are
considered. These optimal control problems are non-standard in the sense that
they involve an objective with set-valued arguments. The approach to study the
solution stability is based on perturbations of minimal and maximal elements of
the solution set of the QVI with respect to {monotone} perturbations of the
forcing term. It is shown that different assumptions are required for studying
decreasing and increasing perturbations and that the optimization problem of
interest is well-posed.Comment: 29 page
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