50 research outputs found
Stabilized lowest order finite element approximation for linear three-field poroelasticity
A stabilized conforming mixed finite element method for the three-field
(displacement, fluid flux and pressure) poroelasticity problem is developed and
analyzed. We use the lowest possible approximation order, namely piecewise
constant approximation for the pressure and piecewise linear continuous
elements for the displacements and fluid flux. By applying a local pressure
jump stabilization term to the mass conservation equation we ensure stability
and avoid pressure oscillations. Importantly, the discretization leads to a
symmetric linear system. For the fully discretized problem we prove existence
and uniqueness, an energy estimate and an optimal a-priori error estimate,
including an error estimate for the divergence of the fluid flux. Numerical
experiments in 2D and 3D illustrate the convergence of the method, show the
effectiveness of the method to overcome spurious pressure oscillations, and
evaluate the added mass effect of the stabilization term.Comment: 25 page
A poroelastic model coupled to a fluid network with applications in lung modelling
Here we develop a lung ventilation model, based a continuum poroelastic
representation of lung parenchyma and a 0D airway tree flow model. For the
poroelastic approximation we design and implement a lowest order stabilised
finite element method. This component is strongly coupled to the 0D airway tree
model. The framework is applied to a realistic lung anatomical model derived
from computed tomography data and an artificially generated airway tree to
model the conducting airway region. Numerical simulations produce
physiologically realistic solutions, and demonstrate the effect of airway
constriction and reduced tissue elasticity on ventilation, tissue stress and
alveolar pressure distribution. The key advantage of the model is the ability
to provide insight into the mutual dependence between ventilation and
deformation. This is essential when studying lung diseases, such as chronic
obstructive pulmonary disease and pulmonary fibrosis. Thus the model can be
used to form a better understanding of integrated lung mechanics in both the
healthy and diseased states
A Survey of Monte Carlo Tree Search Methods
Monte Carlo tree search (MCTS) is a recently proposed search method that combines the precision of tree search with the generality of random sampling. It has received considerable interest due to its spectacular success in the difficult problem of computer Go, but has also proved beneficial in a range of other domains. This paper is a survey of the literature to date, intended to provide a snapshot of the state of the art after the first five years of MCTS research. We outline the core algorithm's derivation, impart some structure on the many variations and enhancements that have been proposed, and summarize the results from the key game and nongame domains to which MCTS methods have been applied. A number of open research questions indicate that the field is ripe for future work
Autocorrelated measurement processes and inference for ordinary differential equation models of biological systems
Ordinary differential equation models are used to describe dynamic processes
across biology. To perform likelihood-based parameter inference on these
models, it is necessary to specify a statistical process representing the
contribution of factors not explicitly included in the mathematical model. For
this, independent Gaussian noise is commonly chosen, with its use so widespread
that researchers typically provide no explicit justification for this choice.
This noise model assumes `random' latent factors affect the system in ephemeral
fashion resulting in unsystematic deviation of observables from their modelled
counterparts. However, like the deterministically modelled parts of a system,
these latent factors can have persistent effects on observables. Here, we use
experimental data from dynamical systems drawn from cardiac physiology and
electrochemistry to demonstrate that highly persistent differences between
observations and modelled quantities can occur. Considering the case when
persistent noise arises due only to measurement imperfections, we use the
Fisher information matrix to quantify how uncertainty in parameter estimates is
artificially reduced when erroneously assuming independent noise. We present a
workflow to diagnose persistent noise from model fits and describe how to
remodel accounting for correlated errors
Understanding the impact of numerical solvers on inference for differential equation models
Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers that seem sufficiently accurate for the forward problem, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the inverse problem, i.e. for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which might become jagged, causing inference algorithms to get stuck in local ‘phantom’ optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyse an ODE change point model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall run-off model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that, when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues