GPU Acceleration of Partial Differential Equation Solvers

Differential equations are often directly solvable by analytical means only in their one dimensional version. Partial differential equations are generally not solvable by analytical means in two and three dimensions, with the exception of few special cases. In all other cases, numerical approximation methods need to be utilized. One of the most popular methods is the finite element method. The main areas of focus, here, are the Poisson heat equation and the plate bending equation. The purpose of this paper is to provide a quick walkthrough of the various approaches that the authors followed in pursuit of creating optimal solvers, accelerated with the use of graphical processing units, and comparing them in terms of accuracy and time efficiency with existing or self-made non-accelerated solvers.

A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol–Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge–Kutta method. The presence of chaotic limit cycles is pointed out.

Coupled consolidation of a solid, infinite cylinder using a Terzaghi formulation

Axisymmetric consolidation is a classical boundary value problem for geotechnical engineers. Under some circumstances an analysis in which the changes in pore pressure, effective stress and displacement can be uncoupled from each other is sufficient, leading to a Terzaghi formulation of the axisymmetric consolidation equation in terms of the pore pressure. However, representation of the Mandel-Cryer effect usually requires more complex, coupled, Biot formulations. A new coupled formulation for the plane strain, axisymmetric consolidation problem is presented for small, linear elastic deformations. A single, easily evaluated parameter couples changes in pore pressure to changes in effective stress, and the resulting differential equation for pore pressure dissipation is very similar to Terzaghi’s classic formulation. The governing equations are then solved using finite differences and the consolidation of a solid infinite cylinder analysed, calculating the variation with time and with radius of the excess pore pressure and the radial displacement. Comparison with a previously published semi-analytical solution indicates that the formulation successfully embodies the Mandel-Cryer effect.

Analytical description of stochastic field-Line wandering in magnetic turbulence

A nonperturbative nonlinear statistical approach is presented to describe turbulent magnetic systems embedded in a uniform mean magnetic field. A general formula in the form of an ordinary differential equation for magnetic field-line wandering (random walk) is derived. By considering the solution of this equation for different limits several new results are obtained. As an example, it is demonstrated that the stochastic wandering of magnetic field-lines in a two-component turbulence model leads to superdiffusive transport, contrary to an existing diffusive picture. The validity of quasilinear theory for field-line wandering is discussed, with respect to different turbulence geometry models, and previous diffusive results are shown to be deduced in appropriate limits.

Modelling the mechanical response of the Reed-mouthpiece-lip system of a clarinet. Part I. A one-dimensional distributed model

The motion of a clarinet reed that is clamped to a mouthpiece and supported by a lip is simulated in the time-domain using finite difference methods. The reed is modelled as a bar with non-uniform cross section, and is described using a one-dimensional, fourth-order partial differential equation. The interactions with the mouthpiece Jay and the player's lip are taken into account by incorporating conditional contact forces in the bar equation. The model is completed by clamped-free boundary conditions for the reed. An implicit finite difference method is used for discretising the system, and values for the physical parameters are chosen both from laboratory measurements and by accurate tuning of the numerical simulations. The accuracy of the numerical system is assessed through analysis of frequency warping effects and of resonance estimation. Finally, the mechanical properties of the system are studied by analysing its response to external driving forces. In particular, the effects of reed curling are investigated.

A multi-output two-stage locally regularized model construction method using the extreme learning machine

This paper investigates the construction of linear-in-the-parameters (LITP) models for multi-output regression problems. Most existing stepwise forward algorithms choose the regressor terms one by one, each time maximizing the model error reduction ratio. The drawback is that such procedures cannot guarantee a sparse model, especially under highly noisy learning conditions. The main objective of this paper is to improve the sparsity and generalization capability of a model for multi-output regression problems, while reducing the computational complexity. This is achieved by proposing a novel multi-output two-stage locally regularized model construction (MTLRMC) method using the extreme learning machine (ELM). In this new algorithm, the nonlinear parameters in each term, such as the width of the Gaussian function and the power of a polynomial term, are firstly determined by the ELM. An initial multi-output LITP model is then generated according to the termination criteria in the first stage. The significance of each selected regressor is checked and the insignificant ones are replaced at the second stage. The proposed method can produce an optimized compact model by using the regularized parameters. Further, to reduce the computational complexity, a proper regression context is used to allow fast implementation of the proposed method. Simulation results confirm the effectiveness of the proposed technique. © 2013 Elsevier B.V.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

The differential effects of chlorpromazine and haloperidol on latent inhibition in healthy volunteers.

Latent inhibition (LI) is a measure of reduced learning about a stimulus to which there has been prior exposure without any consequence. It therefore requires a comparison between a pre-exposed (PE) and a non-pre-exposed (NPE) condition. Since, in animals, LI is disrupted by amphetamines and enhanced by antipsychotics, LI disruption has been proposed as a measure of the characteristic attentional deficit in schizophrenia: the inability to ignore irrelevant stimuli. The findings in humans are, however, inconsistent. In particular, a recent investigation suggested that since haloperidol disrupted LI in healthy volunteers, and LI was normal in non-medicated patients with schizophrenia, the previous findings in schizophrenic patients were entirely due to the negative effects of their medication on LI (Williams et al., 1998). We conducted two studies of antipsychotic drug effects on auditory LI using a within-subject, parallel group design in healthy volunteers. In the first of these, single doses of haloperidol (1 mg. i.v.) were compared with paroxetine (20 mg p.o.) and placebo, and in the second, chlorpromazine (100 mg p.o.) was compared with lorazepam (2 mg. p.o.) and placebo. Eye movements, neuropsychological test performance (spatial working memory (SWM), Tower of London and intra/extra dimensional shift, from the CANTAB test battery) and visual analogue rating scales, were also included as other measures of attention and frontal lobe function. Haloperidol was associated with a non-significant reduction in LI scores, and dysphoria/akathisia (Barnes Akathisia Rating Scale) in three-quarters of the subjects. The LI finding may be explained by increased distractibility which was indicated by an increase in antisaccade directional errors in this group. In contrast, LI was significantly increased by chlorpromazine but not by an equally sedative dose of lorazepam (both drugs causing marked decreases in peak saccadic velocity). Paroxetine had no effect on LI, eye movements or CANTAB neuropsychological test performance. Haloperidol was associated with impaired SWM, which correlated with the degree of dysphoria/akathisia, but no other drug effects on CANTAB measures were detected. We conclude that the effect of antipsychotics on LI is both modality and pharmacologically dependent and that further research using a wider range of antipsychotic compounds is necessary to clarify the cognitive effects of these drugs, and to determine whether there are important differences between them.

Differential expression of components of the cardiomyocyte adrenomedullin/intermedin receptor system following blood pressure reduction in nitric oxide-deficient hypertension.

Adrenomedullin (AM) and intermedin (IMD; adrenomedulln-2) are vasodilator peptides related to calcitonin gene-related peptide (CGRP). The actions of these peptides are mediated by the calcitonin receptor-like receptor (CLR) in association with one of three receptor activity-modifying proteins. CGRP is selective for CLR/receptor activity modifying protein (RAMP)1, AM for CLR/RAMP2 and -3, and IMD acts at both CGRP and AM receptors. In a model of pressure overload induced by inhibition of nitric-oxide synthase, up-regulation of AM was observed previously in cardiomyocytes demonstrating a hypertrophic phenotype. The current objective was to examine the effects of blood pressure reduction on cardiomyocyte expression of AM and IMD and their receptor components. Nomega-nitro-L-arginine methyl ester (L-NAME) (35 mg/kg/day) was administered to rats for 8 weeks, with or without concurrent administration of hydralazine (50 mg/kg/day) and hydrochlorothiazide (7.5 mg/kg/day). In left ventricular cardiomyocytes from L-NAME-treated rats, increases (-fold) in mRNA expression were 1.6 (preproAM), 8.4 (preproIMD), 3.4 (CLR), 4.1 (RAMP1), 2.8 (RAMP2), and 4.4 (RAMP3). Hydralazine/hydrochlorothiazide normalized systolic blood pressure (BP) and abolished mRNA up-regulation of hypertrophic markers sk-alpha-actin and BNP and of preproAM, CLR, RAMP2, and RAMP3 but did not normalize cardiomyocyte width nor preproIMD or RAMP1 mRNA expression. The robust increase in IMD expression indicates an important role for this peptide in the cardiac pathology of this model but, unlike AM, IMD is not associated with pressure overload upon the myocardium. The concordance of IMD and RAMP1 up-regulation indicates a CGRP-type receptor action; considering also a lack of response to BP reduction, IMD may, like CGRP, have an anti-ischemic function.

Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

Differential Pharmacokinetics of Digoxin in Elderly Patients

Digoxin remains one of the most commonly prescribed of all cardiac medications. The main indications for digoxin usage include atrial fibrillation and heart failure; both these conditions are more prevalent in older patients. Given the aging population and the increasing incidence of heart failure we would expect prescribing of digoxin to remain as frequent or to even increase in older patients. Older patients are also more likely to develop toxicity and diagnosis of digoxin toxicity can be difficult in this group. Numerous components contribute to the development of toxicity in older patients, ranging from aging-related changes in renal function or body mass to polypharmacy and possible interactions with digoxin. It is therefore important to understand how the pharmacokinetics of digoxin may be altered in the older population. Application of basic pharmacological principles may be helpful in anticipating these problems. This review describes the pharmacokinetics of digoxin, the changes in pharmacokinetics with increasing age and how concomitant disease states or drug interactions may affect the pharmacokinetics of digoxin. Greater knowledge about the causes and prevention of digoxin toxicity should further reduce the morbidity and mortality arising from digoxin toxicity, especially in the elderly population.
For over 200 years debate has raged regarding the use of digitalis glycosides in cardiac disease. At present digoxin is the most commonly prescribed digitalis compound. This review describes the pharmacokinetics of digoxin and in particular how they are altered with increasing age. When considering the elderly population it is important to recognise the heterogeneity of response in this group, therefore there are no rules, with regards to prescribing, that can apply to the entire elderly population.