140 research outputs found
Analysis of circulant embedding methods for sampling stationary random fields
A standard problem in uncertainty quantification and in computational statistics is the sampling of stationary Gaussian random fields with given covariance in a d-dimensional (physical) domain. In many applications it is sufficient to perform the sampling on a regular grid on a cube enclosing the physical domain, in which case the corresponding covariance matrix is nested block Toeplitz. After extension to a nested block circulant matrix, this can be diagonalized by FFT—the “circulant embedding method.” Provided the circulant matrix is positive definite, this provides a finite expansion of the field in terms of a deterministic basis, with coefficients given by i.i.d. standard normals. In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Matérn covariance, and prove (for fixed correlation length) that, as h 0 ? 0, positive definiteness is guaranteed when the random field is sampled on a cube of size order (1 + ? 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 ? the Matérn smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when h 0 and ? 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–Loève eigenvalues of the covariance operator. The method analyzed here complements the numerical experiments for uncertainty quantification in porous media problems in an earlier paper by the same authors in J. Comput. Phys., 230 (2011), pp. 3668–3694. </p
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 and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
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 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 , positive definiteness is guaranteed when
the random field is sampled on a cube of size order times larger than the size of the physical domain. (Here is
the mesh spacing of the regular grid and the Mat\'ern smoothness
parameter.) We show that the sampling cube can become smaller as the
correlation length decreases when and 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
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