On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems

The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 (SIAM. J. Matrix Anal. & Appl., v.32) using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology, and present numerical results to demonstrate the performance of the four preconditioners in practice

A Radial Basis Function Method for Solving PDE Constrained Optimization Problems

In this article, we apply the theory of meshfree methods to the problem of PDE constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and the Neumann boundary control problem, both involving Poisson's equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modication to guarantee invertibility. We implement these methods using MATLAB, and produce numerical results to demonstrate the methods' capability. We also comment on the methods' effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended

Preconditioned iterative methods for Navier-Stokes control problems

PDE-constrained optimization problems are a class of problems which have attracted much recent attention in scientific computing and applied science. In this paper, we discuss preconditioned iterative methods for a class of Navier-Stokes control problems, one of the main problems of this type in the field of fluid dynamics. Having detailed the Oseen-type iteration we use to solve the problems and derived the structure of the matrix system to be solved at each step, we utilize the theory of saddle point systems to develop efficient preconditioned iterative solution techniques for these problems. We also require theory of solving convection-diffusion control problems, as well as a commutator argument to justify one of the components of the preconditioner

Community Response Strategies for Environmental Problems of Water Supply and Wastewater Disposal in Fairbanks, Alaska

This report examines the history of the response strategies of the Fairbanks, Alaska, community to problems of water supply and wastewater disposal. Fairbanks is significant since it is the largest settlement in the northern subarctic and arctic regions of North America. Today, the City of Fairbanks and the surrounding urban area have a combined population of over 40,000

Environmental quality conditions in Fairbanks, Alaska, 1972

Published by The Institute of Water Resources and The Institute of Social, Economic and Government Research Fairbanks, AlaskaThis study represents a starting point for investigating the nature and interconnectivity of environmental quality problems in Fairbanks in the 1970's. Since the Fairbanks flood of 1967, no detailed survey of environmental quality conditions has been conducted despite the impact of the flood, the considerable expansion of the city limits, and the population expansion (anticipated and actual) associated with the oil pipeline. The study focuses on selective aspects of environmental quality of continuing and increasing concern to Fairbanks area residents and also to the city and borough governments. Specifically, the issues analyzed are (1) the environmental setting of the area, (2) structures, especially housing conditions, (3) premise conditions, and (4) waste control. Much of the data was derived from a program called NEEDS, an acronym for Neighborhood Environmental Evaluation and Decision System. NEEDS was developed by the Bureau of Community Environmental Management of the Department of Health, Education, and Welfare for rapid gathering of environmental, health, and social information in urban areas.1 The NEEDS survey design consists of two separate stages. Stage I is concerned with collecting general environmental quality information to determine geographically where the most pronounced environmental health problems exist in a given urban area. Stage II consists of detailed interviews with residents of the identified "problem areas" to determine the exact nature of existing health and environmental problems, e.g., housing, health, availability of services, and attitudes regarding existing government (local, state, and federal) programs. With this information, local officials could begin to reorganize existing programs and/or develop new programs to solve some of the interrelated environmental quality problems in the disadvantaged sections of their cities.The work upon which this report is based was supported by funds provided by the State of Alaska, the University of Alaska at Fairbanks, the United States Public Health Service, and the Office of Water Research and Technology

Transport properties of N2 gas at cryogenic temperatures

The viscosity and thermal conductivity of nitrogen gas for the temperature range 5 K - 135 K have been computed from the second Chapman-Enskog approximation. Quantum effects, which become appreciable at the lower temperatures, are included by utilizing collision integrals based on quantum theory. A Lennard-Jones (12-6) potential was assumed. The computations yield viscosities about 20 percent lower than those predicted for the high end of this temperature range by the method of corresponding states, but the agreement is excellent when the computed values are compared with existing experimental data

Estimating the power spectrum covariance matrix with fewer mock samples

The covariance matrices of power-spectrum (P(k)) measurements from galaxy surveys are difficult to compute theoretically. The current best practice is to estimate covariance matrices by computing a sample covariance of a large number of mock catalogues. The next generation of galaxy surveys will require thousands of large volume mocks to determine the covariance matrices to desired accuracy. The errors in the inverse covariance matrix are larger and scale with the number of P(k) bins, making the problem even more acute. We develop a method of estimating covariance matrices using a theoretically justified, few-parameter model, calibrated with mock catalogues. Using a set of 600 BOSS DR11 mock catalogues, we show that a seven parameter model is sufficient to fit the covariance matrix of BOSS DR11 P(k) measurements. The covariance computed with this method is better than the sample covariance at any number of mocks and only ~100 mocks are required for it to fully converge and the inverse covariance matrix converges at the same rate. This method should work equally well for the next generation of galaxy surveys, although a demand for higher accuracy may require adding extra parameters to the fitting function.Comment: 7 pages, 7 figure

A rational deferred correction approach to parabolic optimal control problems

The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies

A Detection of the Baryon Acoustic Oscillation Features in the SDSS BOSS DR12 Galaxy Bispectrum

We present the first high significance detection ($4.1\sigma$) of the Baryon Acoustic Oscillations (BAO) feature in the galaxy bispectrum of the twelfth data release (DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS sample ($0.43 \leq z \leq 0.7$). We measured the scale dilation parameter, $\alpha$, using the power spectrum, bispectrum, and both simultaneously for DR12, plus 2048 MultiDark-PATCHY mocks in the North and South Galactic Caps (NGC and SGC, respectively), and the volume weighted averages of those two samples (N+SGC). The fitting to the mocks validated our analysis pipeline, yielding values consistent with the mock cosmology. By fitting to the power spectrum and bispectrum separately, we tested the robustness of our results, finding consistent values from the NGC, SGC and N+SGC in all cases. We found $D_{\mathrm{V}} = 2032 \pm 24 (\mathrm{stat.}) \pm 15 (\mathrm{sys.})$ Mpc, $D_{\mathrm{V}} = 2038 \pm 55 (\mathrm{stat.}) \pm 15 (\mathrm{sys.})$ Mpc, and $D_{\mathrm{V}} = 2031 \pm 22 (\mathrm{stat.}) \pm 10 (\mathrm{sys.})$ Mpc from the N+SGC power spectrum, bispectrum and simultaneous fitting, respectively.Comment: Submitted to Monthly Notices of the Royal Astronomical Society. 13 pages, 11 figure