Sufficient conditions for unique global solutions in optimal control of semilinear equations with C1−C^1-nonlinearity

We consider a $C^1-$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 $\phi(s)=s|s|$. 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

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