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

A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property

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

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

We express the asymptotics of the remainders of the partial sums {s_n} of the generalized hypergeometric function q+1_F_q through an inverse power series z^n n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k} may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z=1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added several references, added comparison to other methods, and added discussion of recursion stabilit

From the animal house to the field : are there consistent individual differences in immunological profile in wild populations of field voles (Microtus agrestis)?

Inbred mouse strains, living in simple laboratory environments far removed from nature, have been shown to vary consistently in their immune response. However, wildlife populations are typically outbreeding and face a multiplicity of challenges, parasitological and otherwise. In this study we seek evidence of consistent difference in immunological profile amongst individuals in the wild. We apply a novel method in this context, using longitudinal (repeated capture) data from natural populations of field voles, Microtus agrestis, on a range of life history and infection metrics, and on gene expression levels. We focus on three immune genes, IFN-γ, Gata3, and IL-10, representing respectively the Th1, Th2 and regulatory elements of the immune response. Our results show that there was clear evidence of consistent differences between individuals in their typical level of expression of at least one immune gene, and at most all three immune genes, after other measured sources of variation had been taken into account. Furthermore, individuals that responded to changing circumstances by increasing expression levels of Gata3 had a correlated increase in expression levels of IFN-γ. Our work stresses the importance of acknowledging immunological variation amongst individuals in studies of parasitological and infectious disease risk in wildlife populations

Traumatic brain injury as a risk factor for Alzheimer disease. Comparison of two retrospective autopsy cohorts with evaluation of ApoE genotype

BACKGROUND AND PURPOSE: The impact of traumatic brain injury (TBI) on the pathogenesis of Alzheimer disease (AD) is still controversial. The aim of our retrospective autopsy study was to assess the impact of TBE and ApoE allele frequency on the development of AD. MATERIAL AND METHODS: We examined 1. the incidence of AD pathology (Braak stageing, CERAD, NIA-Reagan Institute criteria) in 58 consecutive patients (mean age ± SD 77.0 ± 6.8 years) with residual closed TBI lesions, and 2. the frequency of TBI residuals in 57 age-matched autopsy proven AD cases. In both series, ApoE was evaluated from archival paraffin-embedded brain material. RESULTS: 1. TBE series: 12.1 % showed definite and 10.3% probable AD (mean age 77.6 and 75.2 years), only 2/13 with ApoEε3/4. From 45 (77.6%) non-AD cases (mean age 78.2 years), 3 had ApoEε3/4. The prevalence of 22.4% AD in this small autopsy cohort was significantly higher than 3.3% in a recent large clinical series and 14% in the general population over age 70. 2. In the AD cohort with ApoEε4 allele frequency of 30% similar to other AD series, residuals of closed TBI were seen in 4 brains (7%) (mean age ± SD 78.2 ± 6.4), all lacking the ApoEε4 allele. TBI incidence was slightly lower than 8.5% in the clinical MIRAGE study. CONCLUSIONS: The results of this first retrospective autopsy study of TBI, ApoEε allele frequency, and AD confirm clinical studies suggesting severe TBI to be a risk factor for the development AD higher in subjects lacking ApoEε4 alleles. Further studies in larger autopsy series are needed to elucidate the relationship between TBI, genetic predisposition, and AD

The macroecology of phylogenetically structured hummingbird-plant networks

Aim To investigate the association between species richness, species' phylogenetic signal, insularity and historical and current climate with hummingbird-plant network structure. Location 54 communities along a c. 10,000 kilometer latitudinal gradient across the Americas (39ºN - 32ºS), ranging from sea level to c. 3700 m asl, located on the mainland and on islands, and covering a wide range of climate regimes. Methods We measured null-modeled corrected complementary specialization and bipartite modularity (compartmentalization) in networks of quantitative interactions between hummingbird and plant species. Using an ordinary least squares multi-model approach, we examined the influence of species richness, phylogenetic signal, insularity, and current and historical climate conditions on network structure. Results Phylogenetically-related species, especially plants, showed a tendency to interact with a similar array of partners. The spatial variation in network structure exhibited a constant association with species' phylogeny (R2=0.18-0.19). Species richness and environmental factors showed the strongest associations with network structure (R2=0.20-0.44; R2138 =0.32-0.45, respectively). Specifically, higher levels of complementary specialization and modularity were associated to species-rich communities and communities in which closely-related hummingbirds visited distinct sets of flowering species. On the mainland, warmer temperatures and higher historical temperature stability associated to higher levels of complementary specialization. Main conclusions Previous macroecological studies of interaction networks have highlighted the importance of environment and species richness in determining network structure. Here, for the first time, we report an association between species phylogenetic signal and network structure at macroecological scale. Specifically, null model corrected complementary specialization and modularity exhibited a positive association with species richness and a negative association with hummingbird phylogenetic signal, indicating that both high richness and high inter-specific competition among closely-related 150 hummingbirds exhibit important relationships with specialization in hummingbird-plant networks. Our results document how species richness, phylogenetic signal and climate associate with network structure in complex ways at macroecological scale

Healthcare professionals' intentions to use wiki-based reminders to promote best practices in trauma care: a survey protocol

<p>Abstract</p> <p>Background</p> <p>Healthcare professionals are increasingly using wikis as collaborative tools to create, synthesize, share, and disseminate knowledge in healthcare. Because wikis depend on collaborators to keep content up-to-date, healthcare professionals who use wikis must adopt behaviors that foster this collaboration. This protocol describes the methods we will use to develop and test the metrological qualities of a questionnaire that will assess healthcare professionals' intentions and the determinants of those intentions to use wiki-based reminders that promote best practices in trauma care.</p> <p>Methods</p> <p>Using the Theory of Planned Behavior, we will conduct semi-structured interviews of healthcare professionals to identify salient beliefs that may affect their future use of wikis. These beliefs will inform our questionnaire on intended behavior. A test-retest of the survey will verify the questionnaire's stability over time. We will interview 50 healthcare professionals (25 physicians and 25 allied health professionals) working in the emergency departments of three trauma centers in Quebec, Canada. We will analyze the content of the interviews and construct and pilot a questionnaire. We will then test the revised questionnaire with 30 healthcare professionals (15 physicians and 15 allied health professionals) and retest it two weeks later. We will assess the internal consistency of the questionnaire constructs using Cronbach's alpha coefficients and determine their stability with the intra-class correlation (ICC).</p> <p>Discussion</p> <p>To our knowledge, this study will be the first to develop and test a theory-based survey that measures healthcare professionals' intentions to use a wiki-based intervention. This study will identify professionals' salient beliefs qualitatively and will quantify the psychometric capacities of the questionnaire based on those beliefs.</p