Comments, with reply, on "Computational algorithms for pole assignment" by N.K. Nichols

Some points of the paper by N.K. Nichols (see ibid., vol.AC-31, p.643-5, 1986), concerning the robust pole assignment of linear multiinput systems, are clarified. It is stressed that the minimization of the condition number of the closed-loop eigenvector matrix does not necessarily lead to robustness of the pole assignment. It is shown why the computational method, which Nichols claims is robust, is in fact numerically unstable with respect to the determination of the gain matrix. In replying, Nichols presents arguments to support the choice of the conditioning of the closed-loop poles as a measure of robustness and to show that the methods of J Kautsky, N. K. Nichols and P. VanDooren (1985) are stable in the sense that they produce accurate solutions to well-conditioned problems.

A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations

In this paper a cell by cell anisotropic adaptive mesh technique is added to an existing staggered mesh Lagrange plus remap finite element ALE code for the solution of the Euler equations. The quadrilateral finite elements may be subdivided isotropically or anisotropically and a hierarchical data structure is employed. An efficient computational method is proposed, which only solves on the finest level of resolution that exists for each part of the domain with disjoint or hanging nodes being used at resolution transitions. The Lagrangian, equipotential mesh relaxation and advection (solution remapping) steps are generalised so that they may be applied on the dynamic mesh. It is shown that for a radial Sod problem and a two-dimensional Riemann problem the anisotropic adaptive mesh method runs over eight times faster.

National and sub-national assessments of soil organic carbon stocks and changes: The GEFSOC modelling system

Soil organic carbon (SOC) plays a vital role in ecosystem function, determining soil fertility, water holding capacity and susceptibility to land degradation. In addition, SOC is related to atmospheric CO, levels with soils having the potential for C release or sequestration, depending on land use, land management and climate. The United Nations Convention on Climate Change and its Kyoto Protocol, and other United Nations Conventions to Combat Desertification and on Biodiversity all recognize the importance of SOC and point to the need for quantification of SOC stocks and changes. An understanding of SOC stocks and changes at the national and regional scale is necessary to further our understanding of the global C cycle, to assess the responses of terrestrial ecosystems to climate change and to aid policy makers in making land use/management decisions. Several studies have considered SOC stocks at the plot scale, but these are site specific and of limited value in making inferences about larger areas. Some studies have used empirical methods to estimate SOC stocks and changes at the regional scale, but such studies are limited in their ability to project future changes, and most have been carried out using temperate data sets. The computational method outlined by the Intergovernmental Panel on Climate Change (IPCC) has been used to estimate SOC stock changes at the regional scale in several studies, including a recent study considering five contrasting eco regions. This 'one step' approach fails to account for the dynamic manner in which SOC changes are likely to occur following changes in land use and land management. A dynamic modelling approach allows estimates to be made in a manner that accounts for the underlying processes leading to SOC change. Ecosystem models, designed for site scale applications can be linked to spatial databases, giving spatially explicit results that allow geographic areas of change in SOC stocks to be identified. Some studies have used variations on this approach to estimate SOC stock changes at the sub-national and national scale for areas of the USA and Europe and at the watershed scale for areas of Mexico and Cuba. However, a need remained for a national and regional scale, spatially explicit system that is generically applicable and can be applied to as wide a range of soil types, climates and land uses as possible. The Global Environment Facility Soil Organic Carbon (GEFSOC) Modelling System was developed in response to this need. The GEFSOC system allows estimates of SOC stocks and changes to be made for diverse conditions, providing essential information for countries wishing to take part in an emerging C market, and bringing us closer to an understanding of the future role of soils in the global C cycle. (C) 2007 Elsevier B.V. All rights reserved.

Structure-based energetics of protein interfaces guides foot-and-mouth disease virus vaccine design

Virus capsids are primed for disassembly, yet capsid integrity is key to generating a protective immune response. Foot-and-mouth disease virus (FMDV) capsids comprise identical pentameric protein subunits held together by tenuous noncovalent interactions and are often unstable. Chemically inactivated or recombinant empty capsids, which could form the basis of future vaccines, are even less stable than live virus. Here we devised a computational method to assess the relative stability of protein-protein interfaces and used it to design improved candidate vaccines for two poorly stable, but globally important, serotypes of FMDV: O and SAT2. We used a restrained molecular dynamics strategy to rank mutations predicted to strengthen the pentamer interfaces and applied the results to produce stabilized capsids. Structural analyses and stability assays confirmed the predictions, and vaccinated animals generated improved neutralizing-antibody responses to stabilized particles compared to parental viruses and wild-type capsids.

A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

Finding the smallest eigenvalue by the inverse Monte Carlo method with refinement

Finding the smallest eigenvalue of a given square matrix A of order n is computationally very intensive problem. The most popular method for this problem is the Inverse Power Method which uses LU-decomposition and forward and backward solving of the factored system at every iteration step. An alternative to this method is the Resolvent Monte Carlo method which uses representation of the resolvent matrix [I -qA](-m) as a series and then performs Monte Carlo iterations (random walks) on the elements of the matrix. This leads to great savings in computations, but the method has many restrictions and a very slow convergence. In this paper we propose a method that includes fast Monte Carlo procedure for finding the inverse matrix, refinement procedure to improve approximation of the inverse if necessary, and Monte Carlo power iterations to compute the smallest eigenvalue. We provide not only theoretical estimations about accuracy and convergence but also results from numerical tests performed on a number of test matrices.

A swarm intelligence method for feature tracking in AMV derivation

Feature tracking is a key step in the derivation of Atmospheric Motion Vectors (AMV). Most operational derivation processes use some template matching technique, such as Euclidean distance or cross-correlation, for the tracking step. As this step is very expensive computationally, often shortrange forecasts generated by Numerical Weather Prediction (NWP) systems are used to reduce the search area. Alternatives, such as optical flow methods, have been explored, with the aim of improving the number and quality of the vectors generated and the computational efficiency of the process. This paper will present the research carried out to apply Stochastic Diffusion Search, a generic search technique in the Swarm Intelligence family, to feature tracking in the context of AMV derivation. The method will be described, and we will present initial results, with Euclidean distance as reference.

Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

An efficient numerical method for compressible flows of a real gas using arithmetic averaging

An efficient numerical method is presented for the solution of the Euler equations governing the compressible flow of a real gas. The scheme is based on the approximate solution of a specially constructed set of linearised Riemann problems. An average of the flow variables across the interface between cells is required, and this is chosen to be the arithmetic mean for computational efficiency, which is in contrast to the usual square root averaging. The scheme is applied to a test problem for five different equations of state.

Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing

We study the numerical efficiency of solving the self-consistent field theory (SCFT) for periodic block-copolymer morphologies by combining the spectral method with Anderson mixing. Using AB diblock-copolymer melts as an example, we demonstrate that this approach can be orders of magnitude faster than competing methods, permitting precise calculations with relatively little computational cost. Moreover, our results raise significant doubts that the gyroid (G) phase extends to infinite $\chi N$. With the increased precision, we are also able to resolve subtle free-energy differences, allowing us to investigate the layer stacking in the perforated-lamellar (PL) phase and the lattice arrangement of the close-packed spherical (S$_{cp}$) phase. Furthermore, our study sheds light on the existence of the newly discovered Fddd (O$^{70}$) morphology, showing that conformational asymmetry has a significant effect on its stability.