A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

Portmanteau model diagnostics and tests for nonlinearity: a comparative Monte Carlo study of two alternative methods

This paper employs an extensive Monte Carlo study to test the size and power of the BDS and close return methods of testing for departures from independent and identical distribution. It is found that the finite sample properties of the BDS test are far superior and that the close return method cannot be recommended as a model diagnostic. Neither test can be reliably used for very small samples, while the close return test has low power even at large sample sizes

Multi‐method global sensitivity analysis (MMGSA) for modelling floodplain hydrological processes

When studying hydrological processes with a numerical model, global sensitivity analysis (GSA) is essential if one is to understand the impact of model parameters and model formulation on results. However, different definitions of sensitivity can lead to a difference in the ranking of importance of the different model factors. Here we combine a fuzzy performance function with different methods of calculating global sensitivity to perform a multi-method global sensitivity analysis (MMGSA). We use an application of a finite element subsurface flow model (ESTEL-2D) on a flood inundation event on a floodplain of the River Severn to illustrate this new methodology. We demonstrate the utility of the method for model understanding and show how the prediction of state variables, such as Darcian velocity vectors, can be affected by such a MMGSA. This paper is a first attempt to use GSA with a numerically intensive hydrological model.

Multi-method global sensitivity analysis (MMGSA) for modelling floodplain hydrological processes

When studying hydrological processes with a numerical model, global sensitivity analysis (GSA) is essential if one is to understand the impact of model parameters and model formulation on results. However, different definitions of sensitivity can lead to a difference in the ranking of importance of the different model factors. Here we combine a fuzzy performance function with different methods of calculating global sensitivity to perform a multi-method global sensitivity analysis (MMGSA). We use an application of a finite element subsurface flow model (ESTEL-2D) on a flood inundation event on a floodplain of the River Severn to illustrate this new methodology. We demonstrate the utility of the method for model understanding and show how the prediction of state variables, such as Darcian velocity vectors, can be affected by such a MMGSA. This paper is a first attempt to use GSA with a numerically intensive hydrological model

Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods are consistent with the energy dissipation of the continuous PDE systems. - See more at: http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02792-0/home.html#sthash.rwTIhNWi.dpuf

Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen–Cahn/Cahn–Hilliard/Navier–Stokes–Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.

Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective

We study discontinuous Galerkin approximations of the p-biharmonic equation for p∈(1,∞) from a variational perspective. We propose a discrete variational formulation of the problem based on an appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a semicontinuity argument. We also present numerical experiments aimed at testing the robustness of the method.

Sparse space-time boundary element methods for the heat equation

The goal of this work is the efficient solution of the heat equation with Dirichlet or Neumann boundary conditions using the Boundary Elements Method (BEM). Efficiently solving the heat equation is useful, as it is a simple model problem for other types of parabolic problems. In complicated spatial domains as often found in engineering, BEM can be beneficial since only the boundary of the domain has to be discretised. This makes BEM easier than domain methods such as finite elements and finite differences, conventionally combined with time-stepping schemes to solve this problem. The contribution of this work is to further decrease the complexity of solving the heat equation, leading both to speed gains (in CPU time) as well as requiring smaller amounts of memory to solve the same problem. To do this we will combine the complexity gains of boundary reduction by integral equation formulations with a discretisation using wavelet bases. This reduces the total work to O(h

Evaluation of simulated sea-ice concentrations from sea-ice/ocean models using satellite data and polynya classification methods

Sea-ice concentrations in the Laptev Sea simulated by the coupled North Atlantic-Arctic Ocean-Sea-Ice Model and Finite Element Sea-Ice Ocean Model are evaluated using sea-ice concentrations from Advanced Microwave Scanning Radiometer-Earth Observing System satellite data and a polynya classification method for winter 2007/08. While developed to simulate largescale sea-ice conditions, both models are analysed here in terms of polynya simulation. The main modification of both models in this study is the implementation of a landfast-ice mask. Simulated sea-ice fields from different model runs are compared with emphasis placed on the impact of this prescribed landfast-ice mask. We demonstrate that sea-ice models are not able to simulate flaw polynyas realistically when used without fast-ice description. Our investigations indicate that without landfast ice and with coarse horizontal resolution the models overestimate the fraction of open water in the polynya. This is not because a realistic polynya appears but due to a larger-scale reduction of ice concentrations and smoothed ice-concentration fields. After implementation of a landfast-ice mask, the polynya location is realistically simulated but the total open-water area is still overestimated in most cases. The study shows that the fast-ice parameterization is essential for model improvements. However, further improvements are necessary in order to progress from the simulation of large-scale features in the Arctic towards a more detailed simulation of smaller-scaled features (here polynyas) in an Arctic shelf sea.

The Effects of Different Loading Times on the Bone Response Around Dental Implants: A Histomorphometric Study in Dogs

Purpose: The aim of this study was to evaluate, through histomorphometric analysis, the effect that different loading times would have on the bone response around implants. Materials and Methods: Three Replace Select implants were placed on each side of the mandible in eight dogs (n = 48 implants). One pair of implants was selected for an immediate loading protocol (IL). After 7 days, the second pair of implants received prostheses for an early loading protocol (EL). Fourteen days after implant placement, the third pair of implants received prostheses for advanced early loading (AEL). Following 12 weeks of prosthetics, counted following the positioning of the metallic crowns for the AEL group, the animals were sacrificed and the specimens were prepared for histomorphometric analysis. The differences between loading time in the following parameters were evaluated through analysis of variance: bone-to-implant contact, bone density, and crestal bone loss. Results: The mean percentage of bone-to-implant contact for IL was 77.9% +/- 1.71%, for EL it was 79.25% +/- 2.11%, and for AEL it was 79.42% +/- 1.49%. The mean percentage of bone density for IL was 69.97% +/- 3.81%, for EL it was 69.23% +/- 5.68%, and for AEL it was 69.19% +/- 2.90%. Mean crestal bone loss was 1.57 +/- 0.22 mm for IL, 1.23 +/- 0.19 mm for EL, and 1.17 +/- 0.32 mm for AEL. There was no statistical difference for any of the parameters evaluated (P > .05). Conclusion: Different early loading times did not seem to significantly affect the bone response around dental implants. INT J ORAL MAXILLOFAC IMPLANTS 2010;25:473-481

Effect of Three Methods for Cleaning Dentures on Biofilms Formed In Vitro on Acrylic Resin

Purpose: The aim of this study was to evaluate the effect of three denture hygiene methods against different microbial biofilms formed on acrylic resin specimens. Materials and methods: The set (sterile stainless steel basket and specimens) was contaminated (37 degrees C for 48 hours) by a microbial inoculum with 106 colony-forming units (CFU)/ml (standard strains: Staphylococcus aureus, Streptococcus mutans, Escherichia coli, Candida albicans, Pseudomonas aeruginosa, and Enterococcus faecalis; field strains: S. mutans, C. albicans, C. glabrata, and C. tropicalis). After inoculation, specimens were cleansed by the following methods: (1) chemical: immersion in an alkaline peroxide solution (Bonyplus tablets) for 5 minutes; (2) mechanical: brushing with a dentifrice for removable prostheses (Dentu Creme) for 20 seconds; and (3) a combination of chemical and mechanical methods. Specimens were applied onto a Petri plate with appropriate culture medium for 10 minutes. Afterward, the specimens were removed and the plates incubated at 37 degrees C for 48 hours. Results: Chemical, mechanical, and combination methods showed no significant difference in the reduction of CFU for S. aureus, S. mutans (ATCC and field strain), and P. aeruginosa. Mechanical and combination methods were similar and more effective than the chemical method for E. faecalis, C. albicans (ATCC and field strain), and C. glabrata. The combination method was better than the chemical method for E. coli and C. tropicalis, and the mechanical method showed intermediate results. Conclusion: The three denture hygiene methods showed different effects depending on the type of microbial biofilms formed on acrylic base resin specimens.

Comparison of Neck Skin Excision and Whole Carcass Rinse Sampling Methods for Microbiological Evaluation of Broiler Carcasses before and after Immersion Chilling

Sampling protocols for detecting Salmonella on poultry differ among various countries. In the United States, the U.S. Department of Agriculture Food Safety and Inspection Service dictates that whole broiler carcasses should be rinsed with 400 ml of 1% buffered peptone water, whereas in the European Union 25-g samples composed of neck skin from three carcasses are evaluated. The purpose of this study was to evaluate a whole carcass rinse (WCR) and a neck skin excision (NS) procedure for Salmonella and Escherichia coli isolation from the same broiler carcass. Carcasses were obtained from three broiler processing plants. The skin around the neck area was aseptically removed and bagged separately from the carcass, and microbiological analysis was performed. The corresponding carcass was bagged and a WCR sample was evaluated. No significant difference (alpha <= 0.05) in Salmonella prevalence was found between the samples processed by the two methods, but both procedures produced many false-negative Salmonella results. Prechill, 37% (66 carcasses), 28% (50 carcasses), and 51% (91 carcasses) of the 180 carcasses examined were positive for Salmonella by WCR, NS, and both procedures combined, respectively. Postchill, 3% (5 carcasses), 7% (12 carcasses), and 10% (17 carcasses) of the 177 carcasses examined were positive for Salmonella by the WCR, NS, and combination of both procedures, respectively. Prechill, E. coli plus coliform counts were 3.0 and 2.6 log CFU/ml by the WCR and NS methods, respectively. Postchill. E. coli plus coliform counts were 1.7 and 1.4 log CFU/ml by the WCR and NS methods, respectively.