A finite element method for second order nonvariational elliptic problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form.

Competing effects of toxin-producing phytoplankton on overall plankton populations in the Bay of Bengal

The coexistence of a large number of phytoplankton species on a seemingly limited variety of resources is a classical problem in ecology, known as ‘the paradox of the plankton’. Strong fluctuations in species abundance due to the external factors or competitive interactions leading to oscillations, chaos and short-term equilibria have been cited so far to explain multi-species coexistence and biodiversity of phytoplankton. However, none of the explanations has been universally accepted. The qualitative view and statistical analysis of our field data establish two distinct roles of toxin-producing phytoplankton (TPP): toxin allelopathy weakens the interspecific competition among phytoplankton groups and the inhibition due to ingestion of toxic substances reduces the abundance of the grazer zooplankton. Structuring the overall plankton population as a combination of nontoxic phytoplankton (NTP), toxic phytoplankton, and zooplankton, here we offer a novel solution to the plankton paradox governed by the activity of TPP. We demonstrate our findings through qualitative analysis of our sample data followed by analysis of a mathematical model.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

Variational data assimilation for very large environmental problems

Variational data assimilation is commonly used in environmental forecasting to estimate the current state of the system from a model forecast and observational data. The assimilation problem can be written simply in the form of a nonlinear least squares optimization problem. However the practical solution of the problem in large systems requires many careful choices to be made in the implementation. In this article we present the theory of variational data assimilation and then discuss in detail how it is implemented in practice. Current solutions and open questions are discussed.

Regularization of descriptor systems

Implicit dynamic-algebraic equations, known in control theory as descriptor systems, arise naturally in many applications. Such systems may not be regular (often referred to as singular). In that case the equations may not have unique solutions for consistent initial conditions and arbitrary inputs and the system may not be controllable or observable. Many control systems can be regularized by proportional and/or derivative feedback.We present an overview of mathematical theory and numerical techniques for regularizing descriptor systems using feedback controls. The aim is to provide stable numerical techniques for analyzing and constructing regular control and state estimation systems and for ensuring that these systems are robust. State and output feedback designs for regularizing linear time-invariant systems are described, including methods for disturbance decoupling and mixed output problems. Extensions of these techniques to time-varying linear and nonlinear systems are discussed in the final section.

Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Efficiency of evolutionary algorithms in water network pipe sizing

The pipe sizing of water networks via evolutionary algorithms is of great interest because it allows the selection of alternative economical solutions that meet a set of design requirements. However, available evolutionary methods are numerous, and methodologies to compare the performance of these methods beyond obtaining a minimal solution for a given problem are currently lacking. A methodology to compare algorithms based on an efficiency rate (E) is presented here and applied to the pipe-sizing problem of four medium-sized benchmark networks (Hanoi, New York Tunnel, GoYang and R-9 Joao Pessoa). E numerically determines the performance of a given algorithm while also considering the quality of the obtained solution and the required computational effort. From the wide range of available evolutionary algorithms, four algorithms were selected to implement the methodology: a PseudoGenetic Algorithm (PGA), Particle Swarm Optimization (PSO), a Harmony Search and a modified Shuffled Frog Leaping Algorithm (SFLA). After more than 500,000 simulations, a statistical analysis was performed based on the specific parameters each algorithm requires to operate, and finally, E was analyzed for each network and algorithm. The efficiency measure indicated that PGA is the most efficient algorithm for problems of greater complexity and that HS is the most efficient algorithm for less complex problems. However, the main contribution of this work is that the proposed efficiency ratio provides a neutral strategy to compare optimization algorithms and may be useful in the future to select the most appropriate algorithm for different types of optimization problems.

Self-assembly of telechelic tyrosine end-capped PEO and poly(alanine) polymers in aqueous solution

The self-assembly in aqueous solution of three novel telechelic conjugates comprising a central hydrophilic polymer and short (trimeric or pentameric) tyrosine end-caps has been investigated. Two of the conjugates have a central poly(oxyethylene) (polyethylene oxide, PEO) central block with different molar masses. The other conjugate has a central poly(l-alanine) (PAla) sequence in a purely amino-acid based conjugate. All three conjugates self-assemble into β-sheet based fibrillar structures, although the fibrillar morphology revealed by cryogenic-TEM is distinct for the three polymers—in particular the Tyr5-PEO6k-Tyr5 forms a population of short straight fibrils in contrast to the more diffuse fibril aggregates observed for Tyr5-PEO2k-Tyr5 and Tyr3-PAla-Tyr3. Hydrogel formation was not observed for these samples (in contrast to prior work on related systems) up to quite high concentrations, showing that it is possible to prepare solutions of peptide–polymer-peptide conjugates with hydrophobic end-caps without conformational constraints associated with hydrogelation. The Tyr5-PEO6k-Tyr5 shows significant PEO crystallization upon drying in contrast to the Tyr5-PEO2k-Tyr5 conjugate. Our findings point to the remarkable ability of short hydrophobic peptide end groups to modulate the self-assembly properties of polymers in solution in model peptide-capped “associative polymers”. Retention of fluidity at high conjugate concentration may be valuable in potential future applications of these conjugates as bioresponsive or biocompatible materials, for example exploiting the enzyme-responsiveness of the tyrosine end-groups

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Omega(0) which is interior to the physical domain Omega subset of R(n). We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Omega(0) and converges uniformly to a continuous and positive function in Omega(1) = (Omega) over bar\Omega(0). (C) 2009 Elsevier Inc. All rights reserved.

Semiclassical evolution of dissipative Markovian systems

A semiclassical approximation for an evolving density operator, driven by a `closed` Hamiltonian operator and `open` Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra `open` term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further `small-chord` approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.

Dynamics of the viscous Cahn-Hilliard equation

We study generalized viscous Cahn-Hilliard problems with nonlinearities satisfying critical growth conditions in W-0(1,p)(Omega), where Omega is a bounded smooth domain in R-n, n >= 3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p = 2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in H-0(1)(Omega) and, uniformly with respect to the viscosity parameter, L-infinity(Omega) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n = 3, 4, the attractor of the Cahn-Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation. (C) 2008 Elsevier Inc. All rights reserved.

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation -Delta u = lambda u +/- (x, u) + h(x) in a bounded domain, where f has a sublinear growth and h is an element of L-2. We find suitable conditions on f and It in order to have at least two solutions for X near to an eigenvalue of -Delta. A typical example to which our results apply is when f (x, u) behaves at infinity like a(x)vertical bar u vertical bar(q-2)u, with M > a(x) > delta > 0, and I < q < 2. (C) 2007 Elsevier Inc. All rights reserved.

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

fit the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten. this paper presents an extension of it previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier-Stokes equations. Second, it reports numerical simulation results for 1D hock tube problem, 2D impinging jet and 2D/3D broken clam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved,

Excited-state dynamics of polyfluorene derivatives in solution

The excited-state dynamics of two polyfluorene copolymers, one fully conjugated containing phenylene vinylene units alternated with 9,9`-dihexylfluorenyl groups and the other segmented by -(CH2)(8)- spacer, were studied in dilute solution of different solvents using a picosecond single-photon timing technique. The excited-state dynamics of the segmented copolymer follows the Forster resonant energy-transfer model which describes intrachain energy-transfer kinetics among random oriented chromophores. Energy transfer is confirmed by analysis of fluorescence anisotropy relaxation with the measurement of a short decay component of about 60 ps. The fluorescence decay surface of the fully conjugated copolymer is biexponential with decay times of about 470 and 900 ps, ascribed to deactivation of chain moieties containing trans and cis isomers already in a photostationary condition. Thus, energy transfer is very fast due to the conjugated nature and rigid-rod-like structure of this copolymer chain.

Load balancing solution and evaluation of F5 content switch equipment

The Thesis focused on hardware based Load balancing solution of web traffic through a load balancer F5 content switch. In this project, the implemented scenario for distributing HTTPtraffic load is based on different CPU usages (processing speed) of multiple member servers.Two widely used load balancing algorithms Round Robin (RR) and Ratio model (weighted Round Robin) are implemented through F5 load balancer. For evaluating the performance of F5 content switch, some experimental tests has been taken on implemented scenarios using RR and Ratio model load balancing algorithms. The performance is examined in terms of throughput (bits/sec) and Response time of member servers in a load balancing pool. From these experiments we have observed that Ratio Model load balancing algorithm is most suitable in the environment of load balancing servers with different CPU usages as it allows assigning the weight according to CPU usage both in static and dynamic load balancing of servers.