Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

Piezoresistor-Embedded Multifunctional Magnetic Microactuators for Implantable Self-Clearing Catheter

Indwelling catheters are used widely in medicine to treat various chronic medical conditions. However, chronic implantation of catheters often leads to a premature failure due to biofilm accumulation. Previously we reported on the development of a self-clearing catheter by integrating polymer-based microscale magnetic actuators. The microactuator provides an active anti-biofouling mechanism to disrupt and remove adsorbed biofilm on demand using an externally applied stimulus. During an in vivo evaluation of self-clearing catheter, we realized that it is important to periodically monitor the performance of implanted microactuators. Here we integrate gold-based piezoresistive strain-gauge on our magnetic microactuators to directly monitor the device deflection with good sensitivity (0.035%/Deg) and linear range (±30°). With the integrated strain-gauge, we demonstrate the multi-functional capabilities of our magnetic microactuators that enable device alignment, flow-rate measurement, and obstruction detection and removal towards the development of chronically implantable self-clearing smart catheter

Asteroseismic analysis of solar-mass subgiants KIC 6442183 and KIC 11137075 observed by Kepler

Asteroseismology provides a powerful way to constrain stellar parameters. Solar-like oscillations have been observed on subgiant stars with the \emph{Kepler\/} mission. The continuous and high-precision time series enables us to carry out a detailed asteroseismic study for these stars. We carry out data processing of two subgiants of spectral type G: KIC 6442183 and KIC 11137075 observed with the \emph{Kepler} mission, and perform seismic analysis for the two evolved stars. We estimate the values of global asteroseismic parameters: $\Delta\nu=64.9\pm 0.2$ $\mu$Hz and $\nu_{\rm max}=1225 \pm 17$ $\mu$Hz for KIC 6442183, $\Delta\nu=65.5\pm 0.2$ $\mu$Hz and $\nu_{\rm max}=1171 \pm 8$ $\mu$Hz for KIC 11137075, respectively. In addition, we extract the individual mode frequencies of the two stars. We compare stellar models and observations, including mode frequencies and mode inertias. The mode inertias of mixed modes, which are sensitive to the stellar interior, are used to constrain stellar models. We define a quantity $d\nu_{\rm m-p}$ that measures the difference between the mixed modes and the expected pure pressure modes, which is related to the inertia ratio of mixed modes to radial modes. Asteroseismic together with spectroscopic constraints provide the estimations of the stellar parameters: $M = 1.04_{-0.04}^{+0.01} M_{\odot}$, $R = 1.66_{-0.02}^{+0.01} R_{\odot}$ and $t=8.65_{-0.06}^{+1.12}$ Gyr for KIC 6442183, and $M = 1.00_{-0.01}^{+0.01} M_{\odot}$, $R = 1.63_{-0.01}^{+0.01} R_{\odot}$ and $t=10.36_{-0.20}^{+0.01}$ Gyr for KIC 11137075. Either mode inertias or $d\nu_{\rm m-p}$ could be used to constrain stellar models.Comment: 9 pages, 8 figures, 5 tables A&A accepte

Vitamin D prescribing habits and clinical outcome in pediatric patients with inflammatory bowel disease

INTRODUCTION: The inflammation observed in patients with IBD can negatively impact the intake or absorption of vitamin D. This can increase the risk of disease relapse, impact patients’ quality of life, and increase the risk of IBD related surgeries. In addition to the traditional observation that vitamin D deficiency may be a comorbid manifestation of IBD, there is now growing evidence pointing to serum vitamin D levels as a pathogenic factor contributing to the initiation and propagation of mucosal inflammation in patients with IBD. It is well-established that variation in clinical practice leads to less optimal outcomes in any clinical setting. The relative scarcity of clinical and translational studies is even more pronounced in the pediatric population. OBJECTIVES: The primary objective of this study is to quantify the prevalence of clinician assessment of vitamin D levels in pediatric patients with IBD. We will also look at this behavior in subpopulations and compare their vitamin D status. It is secondary for this study to also describe variations in physician practices with respect to the testing and treatment of vitamin D deficiency at a single tertiary care IBD Center. METHODS: We conducted a retrospective cohort study on consecutive patients with UC, CD, and ID, that were followed in the ambulatory program in the Center for Inflammatory Bowel Disease at Boston Children’s Hospital from 1/1/2014 to 12/31/2014. We identified 498 patients and collected their demographic information, serologic testing, and physician prescribing behavior. RESULTS: Out of the entire population, 64% of the patients were vitamin D deficient (vitamin D level below 32 ng/ml). 24% of the patients received vitamin D supplementation. Vitamin D deficiency was less prevalent in patients with UC than those with CD, with an OR of 0.64 (95% CI 0.43-0.94). Out of the ones receiving supplementation, 37% of them were deficient. In terms of physician practice trends, 62% of the patients were not formally prescribed supplementation. 14.5% of those who were prescribed supplementation were receiving 50,000 IU weekly, and the rest receiving 400 – 2,000 IU daily. Patients with vitamin D levels below 20 ng/ml were more likely to receive the high dose vitamin D prescription (OR 11.5) than those with levels between 20 and 30 ng/ml (OR 5.7). CONCLUSIONS: Our study suggests that despite high prevalence of vitamin D deficiency in pediatric patients with IBD, there is a lack of consensus with respect to the assessment of vitamin D levels and consistency in prescribing vitamin D supplementation. With the potential role that vitamin D plays in IBD pathology and suggestions of the therapeutic effects of vitamin D supplementation, further studies are needed to explore this area