Two-level DDM preconditioners for positive Maxwell equations

In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations. Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement the near-kernel space made of the gradient of scalar functions. This extends the results in [2] to the variable coefficient case and non-convex domains at the expense of a larger coarse space

Impact of Vaccination and Pathogen Exposure Dosage on Shedding Kinetics of Infectious Hematopoietic Necrosis Virus (IHNV) in Rainbow Trout

Vaccine efficacy in preventing clinical disease has been well characterized. However, vaccine impacts on transmission under diversefied conditions, such as variable pathogen exposure dosages, are not fully understood. We evaluated the impacts of vaccination on disease-induced host mortality and shedding of infectious hematopoietic necrosis virus (IHNV) in Rainbow Trout Oncorhynchus mykiss. Fish, in up to three different genetic lines, were exposed to different dosages of IHNV to simulate field variability. Mortality and viral shedding of each individual fish were quantified over the course of infection. As the exposure dosage increased, mortality, number offish shedding virus,daily virus quantity shed, and total amount of virus shed also increased. Vaccination significantly reduced mortality but had a much smaller impact on shedding, such that vaccinated fish still shed significant amounts of virus, particularly at higher viral exposure dosages. These studies demonstrate that the consideration of pathogen exposure dosage and transmission are critical for robust inference of vaccine efficacy

Multipreconditioning with application to two-phase incompressible navier–stokes flow

We consider the use of multipreconditioning to solve linear systems when more than one preconditioner is available but the optimal choice is not known. In particular, we consider a selective multipreconditioned GMRES algorithm where we incorporate a weighting that allows us to prefer one preconditioner over another. Our target application lies in the simulation of incompressible two-phase flow. Since it is not always known if a preconditioner will perform well within all regimes found in a simulation, we also consider robustness of the multipreconditioning to a poorly performing preconditioner. Overall, we obtain promising results with the approach

Preconditioners for two-phase incompressible Navier-Stokes flow

We consider iterative methods for solving the linearised Navier–Stokes equations arising from two-phase flow problems and the efficient preconditioning of such systems when using mixed finite element methods. Our target application is simulation within the Proteus toolkit; in particular, we will give results for a dynamic dam-break problem in 2D. We focus on a preconditioner motivated by approximate commutators which has proved effective, displaying mesh-independent convergence for the constant coefficient single-phase Navier–Stokes equations. This approach is known as the “pressure convection–diffusion” (PCD) preconditioner [H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, second ed., Oxford University Press, 2014]. However, the original technique fails to give comparable performance in its given form when applied to variable coefficient Navier–Stokes systems such as those arising in two-phase flow models. Here we develop a generalisation of this preconditioner appropriate for two-phase flow, requiring a new form for PCD. We omit considerations of boundary conditions to focus on the key features of two-phase flow. Before considering our target application, we present numerical results within the controlled setting of a simplified problem using a variety of different mixed elements. We compare these results with those for a straightforward extension to another commutator-based method known as the “least-squares commutator” (LSC) preconditioner, a technique also discussed in the aforementioned reference. We demonstrate that favourable properties of the original PCD and LSC preconditioners (without boundary adjustments) are retained with the new preconditioners in the two-phase situation

Challenge from Transpower: Determining the effect of the aggregated behaviour of solar photovoltaic power generation and battery energy storage systems on grid exit point load in order to maintain an accurate load forecast

With limited data beyond the grid exit point (GXP) or substation level, how can Transpower determine the effect of the aggregated behaviour of solar photovoltaic power generation and battery energy storage systems on GXP load in order to maintain an accurate load forecast? In this initial study it is assumed that the GXP services a residential region. An algorithm based on non-linear programming, which minimises the financial cost to the consumer, is developed to model consumer behaviour. Input data comprises forecast energy requirements (load), solar irradiance, and pricing. Output includes both the load drawn from the grid and power returned to the grid. The algorithm presented is at the household level. The next step would be to combine the load drawn from the grid and the power returned to the grid from all the households serviced by a GXP, enabling Transpower to make load predictions. Various means of load forecasting are considered including the Holt--Winters methods which perform well for out-of-sample forecasts. Linear regression, which takes into account comparable days, solar radiation, and air temperature, yields even better performance

Inexact subdomain solves using deflated GMRES for Helmholtz problems

We examine the use of a two-level deflation preconditioner combined with GMRES to locally solve the subdomain systems arising from applying domain decomposition methods to Helmholtz problems. Our results show that the direct solution method can be replaced with an iterative approach. This will be particularly important when solving large 3D high-frequency problems as subdomain problems can be too large for direct inversion or otherwise become inefficient. We additionally show that, even with a relatively low tolerance, inexact solution of the subdomain systems does not lead to a drastic increase in the number of outer iterations. As a result, it is promising that a combination of a two-level domain decomposition preconditioner with inexact subdomain solves could provide more economical and memory efficient numerical solutions to large-scale Helmholtz problems

Inexact Subdomain Solves Using Deflated GMRES for Helmholtz Problems

In recent years, domain decomposition based preconditioners have become popular tools to solve the Helmholtz equation. Notorious for causing a variety of convergence issues, the Helmholtz equation remains a challenging PDE to solve numerically. Even for simple model problems, the resulting linear system after discretisation becomes indefinite and tailored iterative solvers are required to obtain the numerical solution efficiently. At the same time, the mesh must be kept fine enough in order to prevent numerical dispersion ‘polluting’ the solution [4]. This leads to very large linear systems, further amplifying the need to develop economical solver methodologies.Numerical Analysi

Challenge from Transpower : determining the effect of the aggregated behaviour of solar photovoltaic power generation and battery energy storage systems on grid exit point load in order to maintain an accurate load forecast

With limited data beyond the grid exit point (GXP) or substation level, how can Transpower determine the effect of the aggregated behaviour of solar photovoltaic power generation and battery energy storage systems on GXP load in order to maintain an accurate load forecast? In this initial study it is assumed that the GXP services a residential region. An algorithm based on non-linear programming, which minimises the financial cost to the consumer, is developed to model consumer behaviour. Input data comprises forecast energy requirements (load), solar irradiance, and pricing. Output includes both the load drawn from the grid and power returned to the grid. The algorithm presented is at the household level. The next step would be to combine the load drawn from the grid and the power returned to the grid from all the households serviced by a GXP, enabling Transpower to make load predictions. Various means of load forecasting are considered including the Holt--Winters methods which perform well for out-of-sample forecasts. Linear regression, which takes into account comparable days, solar radiation, and air temperature, yields even better performance