Optimal importance sampling for overdamped Langevin dynamics

Calculating averages with respect to multimodal probability distributions is often necessary in applications. Markov chain Monte Carlo (MCMC) methods to this end, which are based on time averages along a realization of a Markov process ergodic with respect to the target probability distribution, are usually plagued by a large variance due to the metastability of the process. In this work, we mathematically analyze an importance sampling approach for MCMC methods that rely on the overdamped Langevin dynamics. Specifically, we study an estimator based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. The estimator we consider incorporates a reweighting term in order to rectify the bias that would otherwise be introduced by this modification of the potential. We obtain an explicit expression in dimension 1 for the biasing potential that minimizes the asymptotic variance of the estimator for a given observable, and propose a general numerical approach for approximating the optimal potential in the multi-dimensional setting. We also investigate an alternative approach where, instead of the asymptotic variance for a given observable, a weighted average of the asymptotic variances corresponding to a class of observables is minimized. Finally, we demonstrate the capabilities of the proposed method by means of numerical experiments

Optimal friction matrix for underdamped Langevin sampling

A systematic procedure for optimising the friction coefficient in underdamped Langevin dynamics as a sampling tool is given by taking the gradient of the associated asymptotic variance with respect to friction. We give an expression for this gradient in terms of the solution to an appropriate Poisson equation and show that it can be approximated by short simulations of the associated first variation/tangent process under concavity assumptions on the log density. Our algorithm is applied to the estimation of posterior means in Bayesian inference problems and reduced variance is demonstrated when compared to the original underdamped and overdamped Langevin dynamics in both full and stochastic gradient cases

Evaluation of the Training Program of the Project P.A.T.H.S.: Findings Based on the Perspective of the Participants from Different Cohorts

Subjective outcome evaluation findings based on the perspective of the participants participating in a 3-day training program of the Project P.A.T.H.S. are reported in this paper. The findings were based on the data collected from the training workshops conducted between 2005 and 2009 (N = 4.167). Results showed that the respondents had good and positive perceptions of the training program and found it very valuable, particularly with respect to training instructors and familiarization with the project. Besides, the training participants were able to acquire attitude, knowledge and skills that are conducive to the successful implementation of the program. Based on the subjective outcome evaluation findings, it is concluded that the training program was effective in helping the participants to acquire the necessary knowledge, attitudes and skills in implementing the program

Systematic Review and Meta-Analysis of Statin Use and Mortality, Intensive Care Unit Admission and Requirement for Mechanical Ventilation in COVID-19 Patients

There is mounting evidence that statin use is beneficial for COVID-19 outcomes. We performed a systematic review and meta-analysis to evaluate the association between statin use and mortality, intensive care unit (ICU) admission and mechanical ventilation in COVID-19 patients, on studies which provided covariate adjusted effect estimates, or performed propensity score matching. We searched PubMed, Embase, Web of Science and Scopus for studies and extracted odds or hazard ratios for specified outcome measures. Data synthesis was performed using a random-effects inverse variance method. Risk of bias, heterogeneity and publication bias were analyzed using standard methods. Our results show that statin use was associated with significant reductions in mortality (OR = 0.72, 95% CI: 0.67–0.77; HR = 0.74, 95% CI: 0.69, 0.79), ICU admission (OR = 0.94, 95% CI: 0.89–0.99; HR = 0.76, 95% CI: 0.60–0.96) and mechanical ventilation (OR = 0.84, 95% CI: 0.78–0.92; HR = 0.67, 95% CI: 0.47–0.97). Nevertheless, current retrospective studies are based on the antecedent use of statins prior to infection and/or continued use of statin after hospital admission. The results may not apply to the de novo commencement of statin treatment after developing COVID-19 infection. Prospective studies are lacking and necessary

Multilevel spectral coarsening for graph Laplacian problems with application to reservoir simulation

We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in FAS (full approximation scheme) for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. We illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation

Statistical Mechanics of Relativistic One-Dimensional Self-Gravitating Systems

We consider the statistical mechanics of a general relativistic one-dimensional self-gravitating system. The system consists of $N$-particles coupled to lineal gravity and can be considered as a model of $N$ relativistically interacting sheets of uniform mass. The partition function and one-particle distitrubion functions are computed to leading order in $1/c$ where $c$ is the speed of light; as $c\to\infty$ results for the non-relativistic one-dimensional self-gravitating system are recovered. We find that relativistic effects generally cause both position and momentum distribution functions to become more sharply peaked, and that the temperature of a relativistic gas is smaller than its non-relativistic counterpart at the same fixed energy. We consider the large-N limit of our results and compare this to the non-relativistic case.Comment: latex, 60 pages, 22 figure

The relative importance of head, flux, and prior information in hydraulic tomography analysis

Using cross-correlation analysis, we demonstrate that flux measurements at observation locations during hydraulic tomography (HT) surveys carry nonredundant information about heterogeneity that are complementary to head measurements at the same locations. We then hypothesize that a joint interpretation of head and flux data, even when the same observation network as head has been used, can enhance the resolution of HT estimates. Subsequently, we use numerical experiments to test this hypothesis and investigate the impact of flux conditioning and prior information (such as correlation lengths and initial mean models (i.e., uniform mean or distributed means)) on the HT estimates of a nonstationary, layered medium. We find that the addition of flux conditioning to HT analysis improves the estimates in all of the prior models tested. While prior information on geologic structures could be useful, its influence on the estimates reduces as more nonredundant data (i.e., flux) are used in the HT analysis. Lastly, recommendations for conducting HT surveys and analysis are presented

School-Related Factors in the Implementation of a Positive Youth Development Project in Hong Kong

Individual and focus group interviews were conducted to identify school-related factors that influence the process and quality of implementation of the Tier 1 Program of the Project P.A.T.H.S. in Hong Kong. Results of this case study approach showed that the program implementation quality was generally high. Factors that facilitate the implementation of the program were identified, including administrative support from the school and social work agency, presence of dedicated teachers, positive perceptions of the program among teachers, the teachers' self-disclosure, effective continuous assessment, and excellent co-teaching mode. Difficulties encountered by the school in the process of implementation were also observed. Based on the present findings, school-related process variables that facilitate or impede the implementation of positive youth development programs in the Chinese context are discussed