Multilevel Delayed Acceptance MCMC

We develop a novel Markov chain Monte Carlo (MCMC) method that exploits a hierarchy of models of increasing complexity to efficiently generate samples from an unnormalized target distribution. Broadly, the method rewrites the Multilevel MCMC approach of Dodwell et al. (2015) in terms of the Delayed Acceptance (DA) MCMC of Christen & Fox (2005). In particular, DA is extended to use a hierarchy of models of arbitrary depth, and allow subchains of arbitrary length. We show that the algorithm satisfies detailed balance, hence is ergodic for the target distribution. Furthermore, multilevel variance reduction is derived that exploits the multiple levels and subchains, and an adaptive multilevel correction to coarse-level biases is developed. Three numerical examples of Bayesian inverse problems are presented that demonstrate the advantages of these novel methods. The software and examples are available in PyMC3.Comment: 29 pages, 12 figure

Gaussian Process Regression models for the properties of micro-tearing modes in spherical tokamak

Spherical tokamaks (STs) have many desirable features that make them an attractive choice for a future fusion power plant. Power plant viability is intrinsically related to plasma heat and particle confinement and this is often determined by the level of micro-instability driven turbulence. Accurate calculation of the properties of turbulent micro-instabilities is therefore critical for tokamak design, however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities makes repeated use of direct numerical simulations in integrated modelling workflows extremely computationally challenging and creates the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a {\it surrogate model} for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilising Gaussian Process Regression (GPR) and classification; techniques from machine learning. These two components are used in an active learning loop to maximise the efficiency of data acquisition thus minimising computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency and normalised electron heat flux; core components of a quasi-linear transport model. Five-fold cross-validation and direct validation on unseen data is used to ascertain the performance of the resulting surrogate models

Self-management including exercise, education and activity modification compared to usual care for adolescents with Osgood-Schlatter (the SOGOOD trial):protocol of a randomized controlled superiority trial

BACKGROUND: Osgood-Schlatter is the most frequent growth-related injury affecting about 10% of physically active adolescents. It can cause long-term pain and limitations in sports and physical activity, with potential sequela well into adulthood. The management of Osgood-Schlatter is very heterogeneous. Recent systematic reviews have found low level evidence for surgical intervention and injection therapies, and an absence of studies on conservative management. Recently, a novel self-management approach with exercise, education, and activity modification, demonstrated favorable outcomes for adolescents with patellofemoral pain and Osgood-Schlatter in prospective cohort studies.AIM: The aim of this trial is to assess the effectiveness of the novel self-management approach compared to usual care in improving self-reported knee-related function in sport (measured using the KOOS-child 'Sport/play' subscale) after a 5-month period.METHODS: This trial is a pragmatic, assessor-blinded, randomized controlled trial with a two-group parallel arm design, including participants aged 10-16 years diagnosed with Osgood-Schlatter. Participants will receive 3 months of treatment, consisting of either usual care or the self-management approach including exercise, education, and activity modification, followed by 2 months of self-management. Primary endpoint is the KOOS-child 'Sport/play' score at 5 months. This protocol details the planned methods and procedures.DISCUSSION: The novel approach has already shown promise in previous cohort studies. This trial will potentially provide much-needed level 1 evidence for the effectiveness of the self-management approach, representing a crucial step towards addressing the long-term pain and limitations associated with Osgood-Schlatter.TRIAL REGISTRATION: Clinicaltrials.gov: NCT05174182. Prospectively registered December 30th 2021. Date of first recruitment: January 3rd 2022. Target sample size: 130 participants.</p

Multilevel Delayed Acceptance MCMC with an Adaptive Error Model in PyMC3

Uncertainty Quantification through Markov Chain Monte Carlo (MCMC) can be prohibitively expensive for target probability densities with expensive likelihood functions, for instance when the evaluation it involves solving a Partial Differential Equation (PDE), as is the case in a wide range of engineering applications. Multilevel Delayed Acceptance (MLDA) with an Adaptive Error Model (AEM) is a novel approach, which alleviates this problem by exploiting a hierarchy of models, with increasing complexity and cost, and correcting the inexpensive models on-the-fly. The method has been integrated within the open-source probabilistic programming package PyMC3 and is available in the latest development version. In this paper, the algorithm is presented along with an illustrative example.Comment: 8 pages, 4 figures, accepted for Machine Learning for Engineering Modeling, Simulation, and Design Workshop at Neural Information Processing Systems 202