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

The Summer Climate at Sam Lake, Yukon Territory

A weather station was established at Sam Lake in the British Mountains of the northern Yukon Territory in the summer of 1974. Information was collected on temperature, winds, cloud cover and precipitation. The summer conditions were found to be influenced mainly by arctic weather disturbances originating in the Beaufort Sea. A comparison of the information recorded at Sam Lake with that from coastal stations served to demonstrate the buffering effect of the mountains

A tunable cavity-locked diode laser source for terahertz photomixing

An all solid-state approach to the precise frequency synthesis and control of widely tunable terahertz radiation by differencing continuous-wave diode lasers at 850 nm is reported in this paper. The difference frequency is synthesized by three fiber-coupled external-cavity laser diodes. Two of the lasers are Pound-Drever-Hall locked to different orders of a Fabry-Perot (FP) cavity, and the third is offset-frequency locked to the second of the cavity-locked lasers using a tunable microwave oscillator. The first cavity-locked laser and the offset-locked laser produce the difference frequency, whose value is accurately determined by the sum of an integer multiple of the free spectral range of the FP cavity and the offset frequency. The dual-frequency 850-nm output of the three laser system is amplified to 500 mW through two-frequency injection seeding of a single semiconductor tapered optical amplifier. As proof of precision frequency synthesis and control of tunability, the difference frequency is converted into a terahertz wave by optical-heterodyne photomixing in low-temperature-grown GaAs and used for the spectroscopy of simple molecules. The 3-dB spectral power bandwidth of the terahertz radiation is routinely observed to be ≾1 MHz. A simple, but highly accurate, method of obtaining an absolute frequency calibration is proposed and an absolute calibration of 10^(-7) demonstrated using the known frequencies of carbon monoxide lines between 0.23-1.27 THz

Saddle point preconditioners for weak-constraint 4D-Var

Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, in order to exploit highly parallel modern computer architectures. In this setting, the choice of preconditioner is crucial to ensure fast convergence and retain the inherent parallelism of the saddle point formulation. We propose new preconditioning approaches for the model term and observation error covariance term which lead to fast convergence of preconditioned Krylov subspace methods, and many of these suggested approximations are highly parallelisable. In particular our novel approach includes model information in the model term within the preconditioner, which to our knowledge has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial, with further improvements expected if the highly parallelisable nature of the preconditioners is exploited.Comment: 28 pages, 5 figure

A Physiologically-Inspired Model of Numerical Classification Based on Graded Stimulus Coding

In most natural decision contexts, the process of selecting among competing actions takes place in the presence of informative, but potentially ambiguous, stimuli. Decisions about magnitudes – quantities like time, length, and brightness that are linearly ordered – constitute an important subclass of such decisions. It has long been known that perceptual judgments about such quantities obey Weber's Law, wherein the just-noticeable difference in a magnitude is proportional to the magnitude itself. Current physiologically inspired models of numerical classification assume discriminations are made via a labeled line code of neurons selectively tuned for numerosity, a pattern observed in the firing rates of neurons in the ventral intraparietal area (VIP) of the macaque. By contrast, neurons in the contiguous lateral intraparietal area (LIP) signal numerosity in a graded fashion, suggesting the possibility that numerical classification could be achieved in the absence of neurons tuned for number. Here, we consider the performance of a decision model based on this analog coding scheme in a paradigmatic discrimination task – numerosity bisection. We demonstrate that a basic two-neuron classifier model, derived from experimentally measured monotonic responses of LIP neurons, is sufficient to reproduce the numerosity bisection behavior of monkeys, and that the threshold of the classifier can be set by reward maximization via a simple learning rule. In addition, our model predicts deviations from Weber Law scaling of choice behavior at high numerosity. Together, these results suggest both a generic neuronal framework for magnitude-based decisions and a role for reward contingency in the classification of such stimuli

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

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