56,066 research outputs found
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
control, double obstacles, necessary optimality condition
Description and Realization for a Class of Irrational Transfer Functions
This paper proposes an exact description scheme which is an extension to the
well-established frequency distributed model method for a class of irrational
transfer functions. The method relaxes the constraints on the zero initial
instant by introducing the generalized Laplace transform, which provides a wide
range of applicability. With the discretization of continuous frequency band,
the infinite dimensional equivalent model is approximated by a finite
dimensional one. Finally, a fair comparison to the well-known Charef method is
presented, demonstrating its added value with respect to the state of art.Comment: 9 pages, 9 figure
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
In the recent paper Well-posedness and regularity for a generalized
fractional CahnHilliard system, the same authors derived general
well-posedness and regularity results for a rather general system of
evolutionary operator equations having the structure of a CahnHilliard
system. The operators appearing in the system equations were fractional
versions in the spectral sense of general linear operators A and B having
compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process
modeled by the CahnHilliard system could be of a very general type that
includes standard physically meaningful cases such as polynomial,
logarithmic, and double obstacle nonlinearities. In the subsequent paper
Optimal distributed control of a generalized fractional CahnHilliard system
(Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the
same authors, an analysis of distributed optimal control problems was
performed for such evolutionary systems, where only the differentiable case
of certain polynomial and logarithmic double-well potentials could be
admitted. Results concerning existence of optimizers and first-order
necessary optimality conditions were derived, where more restrictive
conditions on the operators A and B had to be assumed in order to be able to
show differentiability properties for the associated control-to-state
operator. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called deep quench method. This
technique, in which the nondifferentiable double obstacle nonlinearity is
approximated by differentiable logarithmic nonlinearities, was first
developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper A
deep quench approach to the optimal control of an AllenCahn equation with
dynamic boundary conditions and double obstacles (Appl. Math. Optim. 71
(2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal
control problems with double obstacle potentials in the framework of systems
of CahnHilliard type. We first give a general convergence analysis of the
deep quench approximation that includes an error estimate and then
demonstrate that its use leads in the double obstacle case to appropriate
first-order necessary optimality conditions in terms of a variational
inequality and the associated adjoint state system
Deep quench approximation and optimal control of general Cahn--Hilliard systems with fractional operators and double obstacle potentials
In the recent paper ''Well-posedness and regularity for a generalized fractional Cahn--Hilliard system'', the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn--Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn--Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper ''Optimal distributed control of a generalized fractional Cahn--Hilliard system'' (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic double-well potentials could be admitted. Results concerning existence of optimizers and first-order necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated control-to-state operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called ''deep quench'' method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper ''A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles'' (Appl. Math. Optim. 71 (2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of Cahn--Hilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system
Exchange-Correlation Energy from Pairing Matrix Fluctuation and the Particle-Particle Random Phase Approximation
We formulate an adiabatic connection for the exchange-correlation energy in
terms of pairing matrix fluctuation. This connection opens new channels for
density functional approximations based on pairing interactions. Even the
simplest approximation to the pairing matrix fluctuation, the particle-particle
Random Phase Approximation (pp-RPA), has some highly desirable properties. It
has no delocalization error with a nearly linear energy behavior for systems
with fractional charges, describes van der Waals interactions similarly and
thermodynamic properties significantly better than particle-hole RPA, and
eliminates static correlation error for single-bond systems. Most
significantly, the pp-RPA is the first known functional that has an explicit
and closed-form dependence on the occupied and unoccupied orbitals and captures
the energy derivative discontinuity in strongly correlated systems. These
findings illlustrate the potential of including pairing interactions within a
density functional framework
Stabilising Model Predictive Control for Discrete-time Fractional-order Systems
In this paper we propose a model predictive control scheme for constrained
fractional-order discrete-time systems. We prove that all constraints are
satisfied at all time instants and we prescribe conditions for the origin to be
an asymptotically stable equilibrium point of the controlled system. We employ
a finite-dimensional approximation of the original infinite-dimensional
dynamics for which the approximation error can become arbitrarily small. We use
the approximate dynamics to design a tube-based model predictive controller
which steers the system state to a neighbourhood of the origin of controlled
size. We finally derive stability conditions for the MPC-controlled system
which are computationally tractable and account for the infinite dimensional
nature of the fractional-order system and the state and input constraints. The
proposed control methodology guarantees asymptotic stability of the
discrete-time fractional order system, satisfaction of the prescribed
constraints and recursive feasibility
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
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