GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.

The presence of executive deficits in patients with Obstructive Sleep Apnoea

Obstructive sleep apnoea (OSA) is a chronic condition in which the upper airways collapse repeatedly during sleep, completely or partially obstructing breathing. This obstruction leads to chronic intermittent hypoxia and severe sleep fragmentation, disrupting the restorative functions of sleep. Beebe and Gozal (2002)a developed a theory which hypothesises that disruption of the restorative functions of sleep lead to a chronic low level brain damage most evident in executive functions (EF). Neuropsychological testing of EF, volumetric MRI, magnetic resonance spectroscopy, event related potentials and CSF biomarkers all provide support for this theory. Little research has been done to explore the nature of the subjective complaint and it’s impact on the activities of daily living.

Clinical factors associated with the humoral immune response to influenza vaccination in chronic obstructive pulmonary disease

Background and objective Individuals with chronic obstructive pulmonary disease (COPD) are at a high risk of developing significant complications from infection with the influenza virus. It is therefore vital to ensure that prophylaxis with the influenza vaccine is effective in COPD. The aim of this study was to assess the immunogenicity of the 2010 trivalent influenza vaccine in persons with COPD compared to healthy subjects without lung disease, and to examine clinical factors associated with the serological response to the vaccine. Methods In this observational study, 34 subjects (20 COPD, 14 healthy) received the 2010 influenza vaccine. Antibody titers at baseline and 28 days post-vaccination were measured using the hemagglutination inhibition assay (HAI) assay. Primary endpoints included seroconversion (≥4-fold increase in antibody titers from baseline) and the fold increase in antibody titer after vaccination. Results Persons with COPD mounted a significantly lower humoral immune response to the influenza vaccine compared to healthy participants. Seroconversion occurred in 90% of healthy participants, but only in 43% of COPD patients (P=0.036). Increasing age and previous influenza vaccination were associated with lower antibody responses. Antibody titers did not vary significantly with cigarette smoking, presence of other comorbid diseases, or COPD severity. Conclusion The humoral immune response to the 2010 influenza vaccine was lower in persons with COPD compared to non-COPD controls. The antibody response also declined with increasing age and in those with a history of prior vaccination.

Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

Background Biochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities. Results In this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler τ-leap, as well as two more recent τ-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments. Conclusions The Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Exocrine pancreatic dysfunction in malnourished Australian Aboriginal children

Pancreatic exocrine dysfunction has been frequently recorded in protein-energy malnutrition in underdeveloped countries. In addition, the pancreas requires optimal nutrition for enzyme synthesis and potentially correctable pancreatic enzyme insufficiency may play a role in the continuation of protein-energy malnutrition. This problem has not been previously evaluated in Australian Aborigines. We have applied a screening test for pancreatic dysfunction (human immunoreactive trypsinogen [IRT] assay) to the study of 398 infants (6-36 months) admitted to the Alice Springs Hospital over a 20-month period. All infants were assessed by anthropometric measures and were assigned to to three nutritional groups (normal, moderate or severely malnourished) and two growth groups (stunted or not stunted). Of the 198 infants who had at least a single serum cationic trypsinogen measurement taken, normal values for serum IRT (with confidence limits) were obtained from 57 children, who were normally nourished. IRT levels were significantly correlated with the degree of underweight but there was no correlation with the degree of stunting or age. Mean IRT levels for the moderate and severely underweight groups were significantly greater than the mean for the normal group (P < 0.01). Seventeen children (8.6%) had trypsinogen levels in excess of the 95th percentile for the normally nourished group, reflecting acinar cell damage or ductal obstruction. We conclude that pancreatic dysfunction may be a common and important overlooked factor contributing to ongoing malnutrition and diseases in malnourished Australian Aboriginal children.

Efficient simulation of unsaturated flow in heterogeneous soils using an exponential integrator

- Provided a practical variable-stepsize implementation of the exponential Euler method (EEM). - Introduced a new second-order variant of the scheme that enables the local error to be estimated at the cost of a single additional function evaluation. - New EEM implementation outperformed sophisticated implementations of the backward differentiation formulae (BDF) of order 2 and was competitive with BDF of order 5 for moderate to high tolerances.

Three-dimensional organization of fenestrae labyrinths in liver sinusoidal endothelial cells

- Background/Aims Liver sinusoidal endothelial cell (LSEC) fenestrae are membrane-bound pores that are grouped in sieve plates and act as a bidirectional guardian in regulating transendothelial liver transport. The high permeability of the endothelial lining is explained by the presence of fenestrae and by various membrane-bound transport vesicles. The question as to whether fenestrae relate to other transport compartments remains unclear and has been debated since their discovery almost 40 years ago. - Methods In this study, novel insights concerning the three-dimensional (3D) organization of the fenestrated cytoplasm were built on transmission electron tomographical observations on isolated and cultured whole-mount LSECs. Classical transmission electron microscopy and atomic force microscopy imaging was performed to accumulate cross-correlative structural evidence. - Results and Conclusions The data presented here indicate that different arrangements of fenestrae have to be considered: i.e. open fenestrae that lack any structural obstruction mainly located in the thin peripheral cytoplasm and complexes of multifolded fenestrae organized as labyrinth-like structures that are found in the proximity of the perinuclear area. Fenestrae in labyrinths constitute about one-third of the total LSEC porosity. The 3D reconstructions also revealed that coated pits and small membrane-bound vesicles are exclusively interspersed in the non-fenestrated cytoplasmic arms.