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

Submillimeter Spectrum of Formic Acid

We have measured new submillimeter-wave data around 600 GHz and around 1.1 THz for the 13C isotopologue of formic acid and for the two deuterium isotopomers; in each case for both the trans and cis rotamer. For cis-DCOOH and cis-HCOOD in particular only data up to 50 GHz was previously available. For all species the quality and quantity of molecular parameters has been increased providing new measured frequencies and more precise and reliable frequencies in the range of existing and near-future submillimeter and far-infrared astronomical spectroscopy instruments such as Herschel, SOFIA and ALMA

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

Floristic Composition and Conservation Status of Fens in Iowa

Over 200 extant fens of varying condition were documented during an extensive inventory conducted in Iowa between 1986 and 1991. Approximately half of the extant fens support endangered, threatened, special concern, or other rare plant species. Approximately 25 fens are outstanding conservation prospects with intact vegetation, high species richness, and rare species. Nearly 40% of all potential fen sites have been destroyed by cultivation or damage; another 30% remain unknown due to lack of a field visit, but most appear on aerial photographs to be very small, disturbed fragments. In addition to their traditionally recognized range in northwest Iowa, fens were found to be numerous and widespread in eastern Iowa. Most (95%) of the extant fens occurred on private lands; these were variously affected by grazing (65%), cropfield edge effects (33%), potential expansion of woody plants (20% ), drainage by tile lines or ditches (10% ), excavation for ponds (2%), and mining of nearby sand and gravel deposits (2%)

The bandmerged Planck Early Release Compact Source Catalogue: Probing sub-structure in the molecular gas at high Galactic latitude

The Planck Early Release Compact Source Catalogue (ERCSC) includes nine lists of highly reliable sources, individually extracted at each of the nine Planck frequency channels. To facilitate the study of the Planck sources, especially their spectral behaviour across the radio/infrared frequencies, we provide a "bandmerged" catalogue of the ERCSC sources. This catalogue consists of 15191 entries, with 79 sources detected in all nine frequency channels of Planck and 6818 sources detected in only one channel. We describe the bandmerging algorithm, including the various steps used to disentangle sources in confused regions. The multi-frequency matching allows us to develop spectral energy distributions of sources between 30 and 857 GHz, in particular across the 100 GHz band, where the energetically important CO J=1->0 line enters the Planck bandpass. We find ~3-5sigma evidence for contribution to the 100 GHz intensity from foreground CO along the line of sight to 147 sources with |b|>30 deg. The median excess contribution is 4.5+/-0.9 percent of their measured 100 GHz flux density which cannot be explained by calibration or beam uncertainties. This translates to 0.5+/-0.1 K km s^{-1} of CO which must be clumped on the scale of the Planck 100 GHz beam, i.e., ~10 arcmin. If this is due to a population of low mass (~15 Msun) molecular gas clumps, the total mass in these clumps may be more than 2000 Msun. Further, high-spatial-resolution, ground-based observations of the high-latitude sky will help shed light on the origin of this diffuse, clumpy CO emission.Comment: 15 pages, 15 figures, MNRAS in pres

Probing short-range magnetic order in a geometrically frustrated magnet by spin Seebeck effect

Competing magnetic interactions in geometrically frustrated magnets give rise to new forms of correlated matter, such as spin liquids and spin ices. Characterizing the magnetic structure of these states has been difficult due to the absence of long-range order. Here, we demonstrate that the spin Seebeck effect (SSE) is a sensitive probe of magnetic short-range order (SRO) in geometrically frustrated magnets. In low temperature (2 - 5 K) SSE measurements on a model frustrated magnet \mathrm{Gd_{3}Ga_{5}O_{12}}, we observe modulations in the spin current on top of a smooth background. By comparing to existing neutron diffraction data, we find that these modulations arise from field-induced magnetic ordering that is short-range in nature. The observed SRO is anisotropic with the direction of applied field, which is verified by theoretical calculation.Comment: 5 pages, 4 figure

Laboratory Spectroscopy of CH(+) and Isotopic CH

The A1II - X1(Epsilon) electronic band of the CH(+) ion has been used as a probe of the physical and dynamical conditions of the ISM for 65 years. In spite of being one of the first molecular species observed in the ISM and the very large number of subsequent observations with large derived column densities, the pure rotational spectra of CH+ has remained elusive in both the laboratory and in the ISM as well. We report the first laboratory measurement of the pure rotation of the CH(+) ion and discuss the detection of CH-13(+) in the ISM. Also reported are the somewhat unexpected chemical conditions that resulted in laboratory production