Multi-level higher order QMC Galerkin discretization for affine parametric operator equations

We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [\emph{F.Y.~Kuo, Ch.~Schwab, and I.H.~Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient} (in review)] and the single level higher order analysis in [\emph{J.~Dick, F.Y.~Kuo, Q.T.~Le~Gia, D.~Nuyens, and Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations} (in review)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of {\KL} eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our \emph{a-priori} error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov-Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results

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

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

Fast QMC matrix-vector multiplication

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $\boldsymbol{y}_0, ..., \boldsymbol{y}_{N-1} \in [0,1]^s$ such that for the matrix $Y = (\boldsymbol{y}_{0}^\top, ..., \boldsymbol{y}_{N-1}^\top)^\top$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y \boldsymbol{a}^\top$, $\boldsymbol{a} \in \mathbb{R}^s$, in $\mathcal{O}(N \log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set. The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large

The future of returning genetic test results for psychiatric conditions

Background: Genome-wide association studies are rapidly advancing our understanding of the genetic architecture of complex psychiatric conditions. In order to use findings from these studies for enhanced clinical prediction, we need to gain a better understanding of the issues surrounding the return of complex genetic results. Methods: We review the current literature on genetic literacy in the population, the public’s interest in receiving genetic test results for psychiatric conditions, how individuals react to and interpret their genetic results for psychiatric conditions, and gaps in our knowledge that will be critical to address before returning genetic results for psychiatric conditions. Results: We find that in hypothetical scenarios genetic test results indicating increased risk for a psychiatric condition lowers an individual’s confidence to control behavior, reduces self-agency, and negatively impacts affect. Individuals may believe that a change in behavior is important, but there is little evidence that genetic test results indicating increased risk for a psychiatric condition are associated with behavior change. The negative impact of results indicating an increased risk may stem from common misconceptions of complex disorders that exist in approximately 25% to 35% of individuals studied. Conclusions: Individuals with these misunderstandings about the role of genetic factors in complex disorders may have a belief in genetic determinism, the idea that behaviors and characteristics are determined solely by one’s genetic information. Regardless of one’s genetic knowledge, a majority of people are interested in receiving genetic feedback for psychiatric conditions, highlighting a need for effective communication of these genetic test results.https://scholarscompass.vcu.edu/gradposters/1069/thumbnail.jp

Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assigns a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (2003) proved that the CBC algorithm achieves the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in [Kuo,2003] the error bound of the algorithm can be made independent of the dimension $d$ and we achieve the same optimal order of convergence for the function spaces from [Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to $O(d \, n \log(n))$ operations, where $n$ denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better when the weights decay slower.Comment: 13 pages, 1 figure, MCQMC2016 conference (Stanford

Cardiac Arrest from Postpartum Spontaneous Coronary Artery Dissection

<p>We present the case of a 32-year-old woman who presented to the emergency department with a witnessed cardiac arrest. She was otherwise healthy with no cardiac risk factors and had undergone an uneventful repeated cesarean section 3 days priorly. The patient underwent defibrillation, out of ventricular fibrillation to a perfusing sinus rhythm, and was taken to the catheterization laboratory where coronary angiography findings showed spontaneous dissection of the left anterior descending artery. The patient received a total of 6 stents during her hospital stay and was eventually discharged in good condition. Spontaneous coronary artery dissection is a rare entity with a predilection for pregnant or postpartum women. Early diagnosis and treatment are key for survival, and when identified early, mortality is good. [West J Emerg Med. 2011;12(4):567–570.]</p

Illness Severity among Non-English, Non-Spanish Speaking Patients in a Public Emergency Department

Background: Patients with limited English proficiency (LEP) have poor health outcomes compared to English proficient patients. Most studies on language proficiency and health disparities focus on Spanish. Objective: This study examines whether non-Spanish speaking LEP patients experience greater disparities than Spanish speaking LEP patients by comparing disease acuity and language proficiency in an emergency department. Design: This is a retrospective case-control study from November 2010 to February 2012 comparing differences between non-English non-Spanish (NENS) speaking patients to English speaking patients with differences between Spanish speaking and English speaking patients. Main outcomes: Primary endpoints include the emergency severity index (ESI) score, area of triage, days in hospital, and the rates of admission, in-hospital surgery, intensive care unit admission, and all-cause mortality. Results: Among all of the study patients, the average age was 55.1 (+/- 12.4). Comparing the NENS sample to the English sample yielded differences in surgery rates (NENS 11.3%, English 1.9%, p=0.002), admission rates (NENS 38.8%, English 24.7%, p=0.025), and days in hospital (NENS 2.49 +/-5.43, English 1.93+/-8.56, p Conclusions and relevance: We were able to demonstrate greater healthcare needs among NENS patients compared to the other two groups. The NENS patients were more likely to be admitted, have surgery, and stay longer than those speaking English or Spanish. These findings are important because they suggest further research, awareness of these disparities by healthcare providers, and public health interventions focusing on this population are warranted

Use of Short Assessment of Health Literacy for Spanish Adults (SAHLSA-50) to Determine the Health Literacy Rate of the Spanish-speaking Population in an Urban Emergency Department

Background: The Hispanic population presents a great opportunity in terms of potential improvements in clinical outcomes and cost reduction for interventions through assessing and improving health literacy. While there are various tools to assess health literacy, many do not assess comprehensive Spanish health literacy. Objectives: We sought to determine the health literacy rate of our Spanish-speaking population in the ED using the SAHLSA-50 tool. Methods: We surveyed a convenience sample of 300 patients from October to November 2012 that presented to our busy, high volume, urban ED. All subjects completed the SAHLSA-50 tool and demographic form with Spanish-speaking research assistants. Results: 63.3% were women. 8% were age 18-25, 42% were 26-40, 45% were 41-65, and 5% were 65+. 11% had less than 3 years of school, 30% had 4-6 years of school, and 59% had at least 7 years of school. Overall, 83% respondents were health literate. Those with less than 3 years of school were95% in those with 7 or more years of school. The elderly (65+) reported least years of school completed and had the lowest health literacy (56.3%). Conclusions: There was an overall health literacy rate of 83.0%. Importantly, those with lower levels of education and elderly patients were more likely to not be health literate. As a next step, targeting those with less education and the elder within the Hispanic population may yield the most impact for improving health literacy and outcomes