A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.     

Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

A system of conservation laws including a stiff relaxation term; the 2D case

We analyze a system of conservation laws in two space dimensions with a stiff relaxation term. A semi-implicit finite difference method approximating the system is studied and an error bound of order measured inL ¹ is derived. This error bound is independent of the relaxation time > 0. Furthermore, it is proved that the solutions of the system converge towards the solution of an equilibrium model as the relaxation time tends to zero, and that the rate of convergence measured inL ¹ is of order . Finally, we present some numerical illustrations.This research has been supported by the Norwegian Research Council (NFR), program no. STP 110673/420, at the Department of Applied Mathematics, SINTEF, Oslo, Norway     

Crosslinking of PVA and glutaraldehyde in water monitored by viscosity and pulse field gradient NMR: a comparative study

The crosslinking of poly(vinyl alcohol) (PVA) with glutaraldehyde at 80 °C was characterized by viscosity and pulse field gradient (PFG) nuclear magnetic resonance (NMR) techniques. NMR signified an initial dormant period of approximately 6 hr, in which the self‐diffusion coefficient of PVA was found to be constant and independent of time. During the next 7 hr (the “primary” gel period), this induction period was succeeded by a fast decay of the self‐diffusion coefficient of rate (9.13 ± 0.45) × 10⁻⁵ sec⁻¹ followed by a slower decay rate of (3.22 ± 0.30) × 10⁻⁵ sec⁻¹ (the “secondary” gel period). The viscosity of the solution showed the same time behavior, i.e. an initial dormant period (∼ 6 hrs), followed by a fast increase of the viscosity for the next 7 –8 hr. During the “secondary” gel regime, the viscosity became too large to be reliably determined. However, within the time regime where both techniques produced reliable data, they gave identical information regarding the kinetics of the gel process, suggesting that PFG NMR enables in situ monitoring of gelation within porous materials (for instance sandstone). Copyright © 2000 John Wiley & Sons, Ltd.     

A numerical study of optimized sparse preconditioners

Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n ^1/4) andO(n ^1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.Supported by The Norwegian Research Council for Science and the Humanities (NAVF) under grants no. 413.90/002 and 412.93/005.Supported by The Royal Norwegian Council for Scientific and Industrial Research (NTNF) through program no. STP.28402: Toolkits in industrial mathematics.     

On the use of the resting potential and level set methods for identifying ischemic heart disease: An inverse problem

The electrical activity in the heart is modeled by a complex, nonlinear, fully coupled system of differential equations. Several scientists have studied how this model, referred to as the bidomain model, can be modified to incorporate the effect of heart infarctions on simulated ECG (electrocardiogram) recordings.We are concerned with the associated inverse problem; how can we use ECG recordings and mathematical models to identify the position, size and shape of heart infarctions? Due to the extreme CPU efforts needed to solve the bidomain equations, this model, in its full complexity, is not well-suited for this kind of problems. In this paper we show how biological knowledge about the resting potential in the heart and level set techniques can be combined to derive a suitable stationary model, expressed in terms of an elliptic PDE, for such applications. This approach leads to a nonlinear ill-posed minimization problem, which we propose to regularize and solve with a simple iterative scheme.Finally, our theoretical findings are illuminated through a series of computer simulations for an experimental setup involving a realistic heart in torso geometry. More specifically, experiments with synthetic ECG recordings, produced by solving the bidomain model, indicate that our method manages to identify the physical characteristics of the ischemic region(s) in the heart. Furthermore, the ill-posed nature of this inverse problem is explored, i.e. several quantitative issues of our scheme are explored.     

Lessons from the first international proficiency test for the detection of spores from the honey bee pathogen Paenibacillus larvae

We report on the first proficiency test in the context of honey bee disease testing on a broad international scale. Honey samples were distributed to 12 participating laboratories for isolation and identification of spores of the etiological agent of American foulbrood, Paenibacillus larvae. Of the 11 laboratories responding to this proficiency testing event, only 6 (54%) provided data that were in full agreement with the previously determined status of the honey samples; eight (72%) laboratories fell within the threshold for qualification as defined a priori in this proficiency test (=level of agreement of at least 90%). Some lessons can now be drawn from the organizational point of view and will certainly improve new initiatives to be taken.     

Forming MOFs into Spheres by Use of Molecular Gastronomy Methods

A novel method utilizing hydrocolloids to prepare nicely shaped spheres of metal–organic frameworks (MOFs) has been developed. Microcrystalline CPO‐27‐Ni particles are dispersed in either alginate or chitosan solutions, which are added dropwise to solutions containing, respectively, either divalent group 2 cations or base that act as gelling agents. Well‐shaped spheres are immediately formed, which can be dried into spheres containing mainly MOF (>95 wt %). The spheronizing procedures have been optimized with respect to maximum specific surface area, shape, and particle density of the final sphere. At optimal conditions, well‐shaped 2.5–3.5 mm diameter CPO‐27‐Ni spheres with weight‐specific surface areas <10 % lower than the nonformulated CPO‐27‐Ni precursor, and having sphere densities in the range 0.8 to 0.9 g cm^?3 and particle crushing strengths above 20 N, can be obtained. The spheres are well suited for use in fixed‐bed catalytic or adsorption processes.