RESUMEN
Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalizing the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).
RESUMEN
A standard approach to computing expectations with respect to a given target measure is to introduce an overdamped Langevin equation which is reversible with respect to the target distribution, and to approximate the expectation by a time-averaging estimator. As has been noted in recent papers [30, 37, 61, 72], introducing an appropriately chosen nonreversible component to the dynamics is beneficial, both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution. In this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility. Our theoretical findings are supported by numerical simulations.
RESUMEN
As people live longer, so cardiologists are having to manage coronary artery disease in progressively older patients with more severe coronary lesions. The authors tried to determine the feasibility and results of coronary angiography in a retrospective study of 65 patients (44 men and 21 women) over 75 years of age (range 75 to 84 years) with coronary artery disease (excluding valvular heart disease). The study period was 22 months. The commonest indication was unstable or invalidating angina resistant to medical therapy (42 of the 65 cases). Twenty-two patients underwent coronary angiography in the context of myocardial infarction complicated in 3 cases by septal rupture. With the exception of these 3 patients, two of whom underwent surgery, 39 of the 62 remaining patients were judged to be candidates for myocardial revascularisation (63%); 37 underwent a revascularisation procedure (60%), 20 by percutaneous transluminal coronary angioplasty and 17 by coronary bypass surgery. The primary success rate of angioplasty was 90% (18 out of 20). There were no deaths in this group. Two patients who were referred for surgery died, an operative mortality of 12% (2 out of 17). All patients were followed up: 56 out of the 65 were still alive at the time of enquiry, after an average period of 18 months. All surviving patients who were successfully revascularised (by angioplasty or surgery) were pauci- or asymptomatic. Although there is an increased mortality related to revascularisation of elderly patients, this would seem to be acceptable given the quality of the medium term clinical results.