Preconditioning iterative methods for the optimal control of the Stokes equation

Solving problems regarding the optimal control of partial differential equations (PDEs) – also known as PDE-constrained optimization – is a frontier area of numerical analysis. Of particular interest is the problem of flow control, where one would like to effect some desired flow by exerting, for example, an external force. The bottleneck in many current algorithms is the solution of the optimality system – a system of equations in saddle point form that is usually very large and ill-conditioned. In this paper we describe two preconditioners – a block-diagonal preconditioner for the minimal residual method and a block-lower triangular preconditioner for a non-standard conjugate gradient method – which can be effective when applied to such problems where the PDEs are the Stokes equations. We consider only distributed control here, although other problems – for example boundary control – could be treated in the same way. We give numerical results, and compare these with those obtained by solving the equivalent forward problem using similar technique

Reprocessing of radiation by multi-phase gas in Low Luminosity Accretion Flows

We discuss the role that magnetic fields in low luminosity accretion flows can play in creating and maintaining a multi-phase medium, and show that small magnetically-confined clouds or filaments of dense cold gas can dramatically reprocess the `primary' radiation from tori. In particular, radio emission would be suppressed by free-free absorption, and an extra (weak) component would appear at optical wavelengths. This is expected to be a common process in various environments in the central regions of Active Galaxies, such as broad line regions, accretion disk coronae and jets.Comment: submitted to MNRAS; 4 pages, 1 figure (MNRAS LaTex style

Chebyshev semi-iteration in Preconditioning

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. Hence a semi-iterative method, which requires eigenvalue bounds and computes an explicit polynomial, must, for just a little less computational work, give an inferior result. In this manuscript we identify a specific situation in the context of preconditioning when the Chebyshev semi-iterative method is the method of choice since it has properties which make it superior to the Conjugate Gradient method

Descriptions of reversed yielding in bending

Existence of Bauschinger effect in bending-unbending of copper beams has been shown from experiment. In modelling of the Bauschinger effect, it is shown that a significant second plastic penetration can occur with the release of the moment required for an elasticplastic bending of a beam. The theory is given for both linear and parabolic hardening material models. The elastic and plastic strains are developed from each hardening model to express the beam curvature of the unstressed neutral axis. Conditions are expressed, using the normalized stress—strain response of a rectangular beam section, for which the release is purely elastic and elastic—plastic. Under the latter the depth to which a second zone of plasticity penetrates is given. Two stress distributions: one for applying the moment and the other for its release, are sufficient to derive the residual stress. Residuals found for parabolic hardening are believed to be more realistic than those from simpler linear or perfectly plastic models, particularly, where a second penetration is evident

Limits from rapid TeV variability of Mrk 421

The extreme variability event in the TeV emission of Mrk 421, recently reported by the Whipple team, imposes the tightest limits on the typical size of the TeV emitting regions in Active Galactic Nuclei (AGN). We examine the consequences that this imposes on the bulk Lorentz factor of the emitting plasma and on the radiation fields present in the central region of this Active Nucleus. No strong evidence is found for extreme Lorentz factors. However, energetics arguments suggest that any accretion in Mrk 421 has to take place at small rates, compatible with an advection-dominated regime.Comment: 5 pages (Latex MNRAS style), revised version, submitted to MNRA

Dense, thin clouds and reprocessed radiation in the central regions of Active Galactic Nuclei

The primary radiation generated in the central continuum-forming region of Active Galactic Nuclei can be reprocessed by very dense, small-scale clouds that are optically-thin to Thomson scattering. In spite of the extreme conditions expected to prevail in this innermost, central environment, the radiative clouds can survive and maintain cool temperatures relative to the ambient emitting region by means of magnetic confinement. Motivated by these ideas, we present a detailed quantitative study of such clouds, explicitly describing the physical properties they can attain under thermal and radiative equilibrium conditions. We also discuss the thermal stability of the gas in comparison to that of other reprocessing material thought to reside at larger distances from the central source. We construct a model to predict the emergent spectra from a source region containing dense clouds which absorb and reemit the primary radiation generated therein. Our predicted spectra show the following two important results: (i) the reprocessed flux emitted at optical/UV energies is insufficient to account for the blue bump component in the observed spectra; and (ii) the amount of line radiation that is emitted is at least comparable to (and in many cases dominates) the continuum radiation. The lines are extremely broad and tend to accumulate in the extreme ultraviolet, where they form a peak much more prominent than that which is observed in the optical/UV. This result is supported by current observations, which indicate that the spectral energy distribution of radio-quiet AGN may indeed reach a maximum in the EUV band.Comment: 14 pages, 5 figures, latex, uses epsf and rotate, accepted for publication in M

WaND Briefing Note 28 Revised Options for UK Domestic Water Reduction - A Review

Demand pressure on UK water supplies is expected to increase in the next 20 years driven by increasing population, new housing development and reducing household size. Regionally and at town level migration will also affect demand particularly in the South-East which is forecast to have a larger than average growth in population and house building. The water demand moderating trends that are considered to have the greatest effect on UK consumption, in approximate order, are: 1. Metering 2. Low flush toilets 3. Normal showers 4. Efficient washing machines 5. Dishwashers 6. Cistern displacement devices (in existing homes with large cisterns) 7. Water efficient gardening measures can play an important role in reducing demand during critical drought period

The Options for UK Domestic Water Reduction: A Review

Demand pressure on UK water supplies is expected to increase in the next 20 years driven by increasing population, new housing development and reducing household size. Regionally and locally migration will also afect demand particularly in the South-East. The water reduction trends that will have the greatest reduction effect on UK consumption are: 1. For new homes; metering and new efficiencies in design and construction (e.g. low flush toilets, heating and plumbing efficiences) 2. For established housing; metering and modern washing machines

Optimal solvers for PDE-Constrained Optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other partial differential equations