Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Fine sediment delivery and transfer in lowland catchments: modelling suspended sediment concentrations in response to hydrological forcing

Fine sediment delivery to and storage in stream channel reaches can disrupt aquatic habitats, impact river hydromorphology, and transfer adsorbed nutrients and pollutants from catchment slopes to the fluvial system. This paper presents a modelling toot for simulating the time-dependent response of the fine sediment system in catchments, using an integrated approach that incorporates both land phase and in-stream processes of sediment generation, storage and transfer. The performance of the model is demonstrated by applying it to simulate in-stream suspended sediment concentrations in two lowland catchments in southern England, the Enborne and the Lambourn, which exhibit contrasting hydrological and sediment responses due to differences in substrate permeability. The sediment model performs well in the Enborne catchment, where direct runoff events are frequent and peak suspended sediment concentrations can exceed 600 mg l(-1). The general trends in the in-stream concentrations in the Lambourn catchment are also reproduced by the model, although the observed concentrations are low (rarely exceeding 50 mg l(-1)) and the background variability in the concentrations is not fully characterized by the model. Direct runoff events are rare in this highly permeable catchment, resulting in a weak coupling between the sediment delivery system and the catchment hydrology. The generic performance of the model is also assessed using a generalized sensitivity analysis based on the parameter bounds identified in the catchment applications. Results indicate that the hydrological parameters contributing to the sediment response include those controlling (1) the partitioning of runoff between surface and soil zone flows and (2) the fractional loss of direct runoff volume prior to channel delivery. The principal sediment processes controlling model behaviour in the simulations are the transport capacity of direct runoff and the in-stream generation, storage and release of the fine sediment fraction. The in-stream processes appear to be important in maintaining the suspended sediment concentrations during low flows in the River Enborne and throughout much of the year in the River Lambourn. Copyright (c) 2007 John Wiley & Sons, Ltd.

Quantifying uncertainty in critical loads: (B) acidity mass balance critical loads on a sensitive site

This paper reports an uncertainty analysis of critical loads for acid deposition for a site in southern England, using the Steady State Mass Balance Model. The uncertainty bounds, distribution type and correlation structure for each of the 18 input parameters was considered explicitly, and overall uncertainty estimated by Monte Carlo methods. Estimates of deposition uncertainty were made from measured data and an atmospheric dispersion model, and hence the uncertainty in exceedance could also be calculated. The uncertainties of the calculated critical loads were generally much lower than those of the input parameters due to a "compensation of errors" mechanism - coefficients of variation ranged from 13% for CLmaxN to 37% for CL(A). With 1990 deposition, the probability that the critical load was exceeded was > 0.99; to reduce this probability to 0.50, a 63% reduction in deposition is required; to 0.05, an 82% reduction. With 1997 deposition, which was lower than that in 1990, exceedance probabilities declined and uncertainties in exceedance narrowed as deposition uncertainty had less effect. The parameters contributing most to the uncertainty in critical loads were weathering rates, base cation uptake rates, and choice of critical chemical value, indicating possible research priorities. However, the different critical load parameters were to some extent sensitive to different input parameters. The application of such probabilistic results to environmental regulation is discussed.

A well-posed integral equation formulation for three-dimensional rough surface scattering

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space L-2 (Gamma) when the scattering surface G does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, kappa, for kappa > 0, if the coupling parameter h is chosen proportional to the wave number. In the case when G is a plane, we show that the choice eta = kappa/2 is nearly optimal in terms of minimizing the condition number.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems

We consider conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle-point problems or, in particular, regularizations thereof. Such methods require preconditioners that preserve certain sub-blocks from the original systems but allow considerable flexibility for the remaining blocks. We construct a number of families of implicit factorizations that are capable of reproducing the required sub-blocks and (some) of the remainder. These generalize known implicit factorizations for the unregularized case. Improved eigenvalue clustering is possible if additionally some of the noncrucial blocks are reproduced. Numerical experiments confirm that these implicit-factorization preconditioners can be very effective in practice.

Testing the limits of quasi-geostrophic theory: application to observed laboratory flows outside the quasi-geostrophic regime

We compare laboratory observations of equilibrated baroclinic waves in the rotating two-layer annulus, with numerical simulations from a quasi-geostrophic model. The laboratory experiments lie well outside the quasi-geostrophic regime: the Rossby number reaches unity; the depth-to-width aspect ratio is large; and the fluid contains ageostrophic inertia–gravity waves. Despite being formally inapplicable, the quasi-geostrophic model captures the laboratory flows reasonably well. The model displays several systematic biases, which are consequences of its treatment of boundary layers and neglect of interfacial surface tension and which may be explained without invoking the dynamical effects of the moderate Rossby number, large aspect ratio or inertia–gravity waves. We conclude that quasi-geostrophic theory appears to continue to apply well outside its formal bounds.

Probabilistic simulations of crop yield over western India using the DEMETER seasonal hindcast ensembles

Process-based integrated modelling of weather and crop yield over large areas is becoming an important research topic. The production of the DEMETER ensemble hindcasts of weather allows this work to be carried out in a probabilistic framework. In this study, ensembles of crop yield (groundnut, Arachis hypogaea L.) were produced for 10 2.5 degrees x 2.5 degrees grid cells in western India using the DEMETER ensembles and the general large-area model (GLAM) for annual crops. Four key issues are addressed by this study. First, crop model calibration methods for use with weather ensemble data are assessed. Calibration using yield ensembles was more successful than calibration using reanalysis data (the European Centre for Medium-Range Weather Forecasts 40-yr reanalysis, ERA40). Secondly, the potential for probabilistic forecasting of crop failure is examined. The hindcasts show skill in the prediction of crop failure, with more severe failures being more predictable. Thirdly, the use of yield ensemble means to predict interannual variability in crop yield is examined and their skill assessed relative to baseline simulations using ERA40. The accuracy of multi-model yield ensemble means is equal to or greater than the accuracy using ERA40. Fourthly, the impact of two key uncertainties, sowing window and spatial scale, is briefly examined. The impact of uncertainty in the sowing window is greater with ERA40 than with the multi-model yield ensemble mean. Subgrid heterogeneity affects model accuracy: where correlations are low on the grid scale, they may be significantly positive on the subgrid scale. The implications of the results of this study for yield forecasting on seasonal time-scales are as follows. (i) There is the potential for probabilistic forecasting of crop failure (defined by a threshold yield value); forecasting of yield terciles shows less potential. (ii) Any improvement in the skill of climate models has the potential to translate into improved deterministic yield prediction. (iii) Whilst model input uncertainties are important, uncertainty in the sowing window may not require specific modelling. The implications of the results of this study for yield forecasting on multidecadal (climate change) time-scales are as follows. (i) The skill in the ensemble mean suggests that the perturbation, within uncertainty bounds, of crop and climate parameters, could potentially average out some of the errors associated with mean yield prediction. (ii) For a given technology trend, decadal fluctuations in the yield-gap parameter used by GLAM may be relatively small, implying some predictability on those time-scales.

OTHR impulsive interference suppression in strong clutter background

External interferences can severely degrade the performance of an Over-the-horizon radar (OTHR), so suppression of external interferences in strong clutter environment is the prerequisite for the target detection. The traditional suppression solutions usually began with clutter suppression in either time or frequency domain, followed by the interference detection and suppression. Based on this traditional solution, this paper proposes a method characterized by joint clutter suppression and interference detection: by analyzing eigenvalues in a short-time moving window centered at different time position, Clutter is suppressed by discarding the maximum three eigenvalues at every time position and meanwhile detection is achieved by analyzing the remained eigenvalues at different position. Then, restoration is achieved by forward-backward linear prediction using interference-free data surrounding the interference position. In the numeric computation, the eigenvalue decomposition (EVD) is replaced by values decomposition (SVD) based on the equivalence of these two processing. Data processing and experimental results show its efficiency of noise floor falling down about 10-20 dB.

Pricing inefficiencies in private real estate markets using total return swaps

Efficient markets should guarantee the existence of zero spreads for total return swaps. However, real estate markets have recorded values that are significantly different from zero in both directions. Possible explanations might suggest non-rational behaviour by inexperienced market players or unusual features of the underlying asset market. We find that institutional characteristics in the underlying market lead to market inefficiencies and, hence, to the creation of a rational trading window with upper and lower bounds within which transactions do not offer arbitrage opportunities. Given the existence of this rational trading window, we also argue that the observed spreads can substantially be explained by trading imbalances due to the limited liquidity of a newly formed market and/or to the effect of market sentiment, complementing explanations based on the lag between underlying market returns and index returns.

Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.

Baroclinic stationary waves in aquaplanet models

An aquaplanet model is used to study the nature of the highly persistent low-frequency waves that have been observed in models forced by zonally symmetric boundary conditions. Using the Hayashi spectral analysis of the extratropical waves, the authors find that a quasi-stationary wave 5 belongs to a wave packet obeying a well-defined dispersion relation with eastward group velocity. The components of the dispersion relation with k ≥ 5 baroclinically convert eddy available potential energy into eddy kinetic energy, whereas those with k < 5 are baroclinically neutral. In agreement with Green’s model of baroclinic instability, wave 5 is weakly unstable, and the inverse energy cascade, which had been previously proposed as a main forcing for this type of wave, only acts as a positive feedback on its predominantly baroclinic energetics. The quasi-stationary wave is reinforced by a phase lock to an analogous pattern in the tropical convection, which provides further amplification to the wave. It is also found that the Pedlosky bounds on the phase speed of unstable waves provide guidance in explaining the latitudinal structure of the energy conversion, which is shown to be more enhanced where the zonal westerly surface wind is weaker. The wave’s energy is then trapped in the waveguide created by the upper tropospheric jet stream. In agreement with Green’s theory, as the equator-to-pole SST difference is reduced, the stationary marginally stable component shifts toward higher wavenumbers, while wave 5 becomes neutral and westward propagating. Some properties of the aquaplanet quasi-stationary waves are found to be in interesting agreement with a low frequency wave observed by Salby during December–February in the Southern Hemisphere so that this perspective on low frequency variability, apart from its value in terms of basic geophysical fluid dynamics, might be of specific interest for studying the earth’s atmosphere.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.