Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Analysis of circulant embedding methods for sampling stationary random fields

In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, positive definiteness is guaranteed when the random field is sampled on a cube of size order $(1 + \nu^{1/2} \log h_0^{-1})$ times larger than the size of the physical domain. (Here $h_0$ is the mesh spacing of the regular grid and $\nu$ the Mat\'ern smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when $h_0$ and $\nu$ are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Lo\`{e}ve eigenvalues of the covariance operator

Spasticity of the gastrosoleus muscle is related to the development of reduced passive dorsiflexion of the ankle in children with cerebral palsy: A registry analysis of 2,796 examinations in 355 children

Background and purpose Spasticity and muscle contracture are two common manifestations of cerebral palsy (CP). A spastic muscle may inhibit growth in length of the muscle, but the importance of this relationship is not known. In 1994, a register and a healthcare program for children with CP in southern Sweden were initiated. The child's muscle tone according to the Ashworth scale and the ankle range of motion (ROM) is measured annually during the entire growth period. We have used these data to analyze the relationship between spasticity and ROM of the gastrosoleus muscle. Patients and methods All measurements in the total population of children with CP aged 0-18 years during the period January 1995 through June 2008 were analyzed. The study was based on 2,796 examinations in 355 children. In the statistical analysis, the effect of muscle tone on ROM was estimated using a random effects model. Results The range of dorsiflexion of the ankle joint decreased in the total material by mean 19 (95% CI: 14-24) degrees during the first 18 years of life. There was a statistically significant association between the ROM and the child's level of spasticity during the year preceding the ROM measurement. Interpretation Spasticity is related to the development of muscle contracture. In the treatment of children with CP, the spasticity, contracture, and strength of the gastrosoleus muscle must be considered together

Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661

Development of spasticity with age in a total population of children with cerebral palsy

<p>Abstract</p> <p>Background</p> <p>The development of spasticity with age in children with cerebral palsy (CP) has, to our knowledge, not been studied before. In 1994, a register and a health care program for children with CP in southern Sweden were initiated. In the programme the child's muscle tone according to the modified Ashworth scale is measured twice a year until six years of age, then once a year. We have used this data to analyse the development of spasticity with age in a total population of children with cerebral palsy.</p> <p>Methods</p> <p>All measurements of muscle tone in the gastrocnemius-soleus muscle in all children with CP from 0 to 15 years during the period 1995–2006 were analysed. The CP subtypes were classified according to the Surveillance of Cerebral Palsy in Europe network system. Using these criteria, the study was based on 6218 examinations in 547 children. For the statistical analysis the Ashworth scale was dichotomized. The levels 0–1 were gathered in one category and levels 2–4 in the other. The pattern of development with age was evaluated using piecewise logistic regression in combination with Akaike's An Information Criterion.</p> <p>Results</p> <p>In the total sample the degree of muscle tone increased up to 4 years of age. After 4 years of age the muscle tone decreased each year up to 12 years of age. A similar development was seen when excluding the children operated with selective dorsal rhizotomy, intrathecal baclofen pump or tendo Achilles lengthening. At 4 years of age about 47% of the children had spasticity in their gastro-soleus muscle graded as Ashworth 2–4. After 12 years of age 23% of the children had that level of spasticity. The CP subtypes spastic bilateral and spastic unilateral CP showed the same pattern as the total sample. Children with dyskinetic type of CP showed an increasing muscle tone up to age 6, followed by a decreasing pattern up to age 15.</p> <p>Conclusion</p> <p>In children with CP, the muscle tone as measured with the Ashworth scale increases up to 4 years of age and then decreases up to 12 years of age. The same tendency is seen in all spastic subtypes. The findings may have implications both for clinical judgement and for research studies on spasticity treatment.</p

Building the Field of Health Policy and Systems Research: An Agenda for Action

In the final article in a series addressing the current challenges and opportunities for the development of Health Policy and Systems Research (HPSR), Sara Bennett and colleagues lay out an agenda for action moving forward