A New Approximation of the Schur Complement in Preconditioners for PDE Constrained Optimization

Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients respectively; with appropriate approximation blocks these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and of the Tikhonov regularization parameter \beta that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and \beta-independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds which verify the effectiveness of the approximation, and present numerical results which show that these new preconditioners work well in practice

Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems

Robust Iterative Solution of a Class of Time-Dependent Optimal Control Problems

The fast iterative solution of optimal control problems, and in particular PDE-constrained optimization problems, has become an active area of research in applied mathematics and numerical analysis. In this paper, we consider the solution of a class of time-dependent PDE-constrained optimization problems, specifically the distributed control of the heat equation. We develop a strategy to approximate the (1,1)-block and Schur complement of the saddle point system that results from solving this problem, and therefore derive a block diagonal preconditioner to be used within the MINRES algorithm. We present numerical results to demonstrate that this approach yields a robust solver with respect to step-size and regularization parameter

An Introduction to Chemical Pharmacology. By Hugh McGuigan, Ph.D., M.D., Professor of Pharmacology, University of Illinois. 8 vo. xii + 418 pages. Cloth $4.00. P. Blakiston's Son & Co., Philadelphia

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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

An Empirically Based Calculation of the Extragalactic Infrared Background

Using the excellent observed correlations among various infrared wavebands with 12 and 60 micron luminosities, we calculate the 2-300 micron spectra of galaxies as a function of luminosity. We then use 12 micron and 60 micron galaxy luminosity functions derived from IRAS data, together with recent data on the redshift evolution of galaxy emissivity, to derive a new, empirically based IR background spectrum from stellar and dust emission in galaxies. Our best estimate for the IR background is of order 2-3 nW/m^2/sr with a peak around 200 microns reaching 6-8 nW/m^2/sr. Our empirically derived background spectrum is fairly flat in the mid-IR, as opposed to spectra based on modeling with discrete temperatures which exhibit a "valley" in the mid-IR. We also derive a conservative lower limit to the IR background which is more than a factor of 2 lower than our derived flux.Comment: 14 pages AASTeX, 2 .ps figures, the Astrophysical Journal, in pres

Asking the experts : developing and validating parental diaries to assess children's minor injuries

The methodological issues involved in parental reporting of events in children's everyday lives are discussed with reference to the development and validation of an incident diary, collecting concurrent data on minor injuries in a community study of children under eight years old. Eighty-two mothers participated in a comparison over nine days of daily telephone interviews and structured incident diaries. Telephone methods resulted in more missing data, and participants in both groups expressed a preference for the diary method. This diary was then validated on a sample of 56 preschool and school-aged children by comparing injury recording by a research health visitor with that of their mothers. Each failed to report some injuries, but there was good agreement overall, and in descriptive data on injuries reported by both. Parental diaries have the potential to provide rich data, of acceptable validity, on minor events in everyday life

Results of winglet development studies for DC-10 derivatives

The results of investigations into the application of winglets to the DC-10 aircraft are presented. The DC-10 winglet configuration was developed and its cruise performance determined in a previous investigation. This study included high speed and low speed wind tunnel tests to evaluate aerodynamic characteristics, and a subsonic flutter wind tunnel test with accompanying analysis and evaluation of results. Additionally, a configuration integration study employed the results of the wind tunnel studies to determine the overall impact of the installation of winglets on the DC-10 aircraft. Conclusions derived from the high speed and low speed tests indicate that the winglets had no significant effects on the DC-10 stability characteristics or high speed buffet. It was determined that winglets had a minimal effect on aircraft lift characteristics and improved the low speed aircraft drag under high lift conditions. The winglets affected the DC-10 flutter characteristics by reducing the flutter speed of the basic critical mode and introducing a new critical mode involving outer wing torsion and longitudinal bending. The overall impact of winglets was determined to be of sufficient benefit to merit flight evaluation