1,847 research outputs found
Sufficient conditions for unique global solutions in optimal control of semilinear equations with nonlinearity
We consider a semilinear elliptic optimal control problem possibly
subject to control and/or state constraints. Generalizing previous work we
provide a condition which guarantees that a solution of the necessary first
order conditions is a global minimum. A similiar result also holds at the
discrete level where the corresponding condition can be evaluated explicitly.
Our investigations are motivated by G\"unter Leugering, who raised the question
whether our previous results can be extended to the nonlinearity
. We develop a corresponding analysis and present several
numerical test examples demonstrating its usefulness in practice
Fast iterative solution of reaction-diffusion control problems arising from chemical processes
PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix system, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects in our solvers are saddle point theory, mass matrix representation and effective Schur complement approximation, as well as the outer (Newton) iteration to take account of the nonlinearity of the underlying PDEs
Regularization-robust preconditioners for time-dependent PDE constrained optimization problems
In this article, we motivate, derive and test �effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two diff�erent functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the eff�ectiveness of our preconditioners in each case is an eff�ective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are eff�ective for a wide range of regularization parameter values, as well as mesh sizes and time-steps
A principled approach to programming with nested types in Haskell
Initial algebra semantics is one of the cornerstones of the theory of modern functional programming languages. For each inductive data type, it provides a Church encoding for that type, a build combinator which constructs data of that type, a fold combinator which encapsulates structured recursion over data of that type, and a fold/build rule which optimises modular programs by eliminating from them data constructed using the buildcombinator, and immediately consumed using the foldcombinator, for that type. It has long been thought that initial algebra semantics is not expressive enough to provide a similar foundation for programming with nested types in Haskell. Specifically, the standard folds derived from initial algebra semantics have been considered too weak to capture commonly occurring patterns of recursion over data of nested types in Haskell, and no build combinators or fold/build rules have until now been defined for nested types. This paper shows that standard folds are, in fact, sufficiently expressive for programming with nested types in Haskell. It also defines buildcombinators and fold/build fusion rules for nested types. It thus shows how initial algebra semantics provides a principled, expressive, and elegant foundation for programming with nested types in Haskell
Time dilation in dynamic visual display
How does the brain estimate time? This old question has led to many biological and psychological models of time perception (R. A. Block, 1989; P. Fraisse, 1963; J. Gibbon, 1977; D. L. I. Zakay, 1989). Because time cannot be directly measured at a given moment, it has been proposed that the brain estimates time based on the number of changes in an event (S. W. Brown, 1995; P. Fraisse, 1963; W. D. Poynter, 1989). Consistent with this idea, dynamic visual stimuli are known to lengthen perceived time (J. F. Brown, 1931; S. Goldstone & W. T. Lhamon, 1974; W. T. Lhamon & S. Goldstone, 1974, C. O. Z. Roelofs & W. P. C. Zeeman, 1951). However, the kind of information that constitutes the basis for time perception remains unresolved. Here, we show that the temporal frequency of a stimulus serves as the “clock” for perceived duration. Other aspects of changes, such as speed or coherence, were found to be inconsequential. Time dilation saturated at a temporal frequency of 4–8 Hz. These results suggest that the clock governing perceived time has its basis at early processing stages. The possible links between models of time perception and neurophysiological functions of early visual areas are discussed
Gravitational waves from stochastic relativistic sources: primordial turbulence and magnetic fields
The power spectrum of a homogeneous and isotropic stochastic variable,
characterized by a finite correlation length, does in general not vanish on
scales larger than the correlation scale. If the variable is a divergence free
vector field, we demonstrate that its power spectrum is blue on large scales.
Accounting for this fact, we compute the gravitational waves induced by an
incompressible turbulent fluid and by a causal magnetic field present in the
early universe. The gravitational wave power spectra show common features: they
are both blue on large scales, and peak at the correlation scale. However, the
magnetic field can be treated as a coherent source and it is active for a long
time. This results in a very effective conversion of magnetic energy in
gravitational wave energy at horizon crossing. Turbulence instead acts as a
source for gravitational waves over a time interval much shorter than a Hubble
time, and the conversion into gravitational wave energy is much less effective.
We also derive a strong constraint on the amplitude of a primordial magnetic
field when the correlation length is much smaller than the horizon.Comment: Replaced with revised version accepted for publication in Phys Rev
On the Nature of Incompressible Magnetohydrodynamic Turbulence
A novel model of incompressible magnetohydrodynamic turbulence in the
presence of a strong external magnetic field is proposed for explanation of
recent numerical results. According to the proposed model, in the presence of
the strong external magnetic field, incompressible magnetohydrodynamic
turbulence becomes nonlocal in the sense that low frequency modes cause
decorrelation of interacting high frequency modes from the inertial interval.
It is shown that the obtained nonlocal spectrum of the inertial range of
incompressible magnetohydrodynamic turbulence represents an anisotropic
analogue of Kraichnan's nonlocal spectrum of hydrodynamic turbulence. Based on
the analysis performed in the framework of the weak coupling approximation,
which represents one of the equivalent formulations of the direct interaction
approximation, it is shown that incompressible magnetohydrodynamic turbulence
could be both local and nonlocal and therefore anisotropic analogues of both
the Kolmogorov and Kraichnan spectra are realizable in incompressible
magnetohydrodynamic turbulence.Comment: Physics of Plasmas (Accepted). A small chapter added about 2D MHD
turbulenc
Coherence-controlled transparency and far-from-degenerate parametric gain in a strongly-absorbing Doppler-broadened medium
An inversionless gain of anti-Stokes radiation above the oscillation
threshold in an optically-dense far-from-degenerate double-Lambda
Doppler-broadened medium accompanied by Stokes gain is predicted. The outcomes
are illustrated with numerical simulations applied to sodium dimer vapor.
Optical switching from absorption to gain via transparency controlled by a
small variation of the medium and of the driving radiation parameters which are
at a level less than one photon per molecule is shown. Related video/audio
clips see in: A.K. Popov, S.A. Myslivets, and T.F. George, Optics Express Vol.
7, No 3, 148 (2000)(http://epubs.osa.org/oearchive/source/22947.htm) or
download: http://kirensky.krasn.ru/popov/opa/opa.htmComment: 4 pages, 3 eps figures, to be published in Optics Letters, vol.25, No
18 (2000), minor style changes and reference correctio
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