Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.Comment: 33 pages, 9 figures, 3 table

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T; L2(Ω)) and the higher order spaces, L∞(0, T;H1(Ω)) and H1(0, T; L2(Ω)), with optimal orders of convergence

Capturing the time-varying drivers of an epidemic using stochastic dynamical systems

Epidemics are often modelled using non-linear dynamical systems observed through partial and noisy data. In this paper, we consider stochastic extensions in order to capture unknown influences (changing behaviors, public interventions, seasonal effects etc). These models assign diffusion processes to the time-varying parameters, and our inferential procedure is based on a suitably adjusted adaptive particle MCMC algorithm. The performance of the proposed computational methods is validated on simulated data and the adopted model is applied to the 2009 H1N1 pandemic in England. In addition to estimating the effective contact rate trajectories, the methodology is applied in real time to provide evidence in related public health decisions. Diffusion driven SEIR-type models with age structure are also introduced.Comment: 21 pages, 5 figure

Duality Based A Posteriori Error Estimation for Quasi-Periodic Solutions Using Time Averages

We propose an a posteriori error estimation technique for the computation of average functionals of solutions for nonlinear time dependent problems based on duality techniques. The exact solution is assumed to have a periodic or quasi-periodic behavior favoring a fixed mesh strategy in time. We show how to circumvent the need of solving time dependent dual problems. The estimator consists of an averaged residual weighted by sensitivity factors coming from a stationary dual problem and an additional averaging error term coming from nonlinearities of the operator considered. In order to illustrate this technique the resulting adaptive algorithm is applied to several model problems: a linear scalar parabolic problem with known exact solution, the nonsteady Navier–Stokes equations with known exact solution, and finally to the well-known benchmark problem for Navier–Stokes (flow behind a cylinder) in order to verify the modeling assumptions

Goal-Oriented Adaptivity in Space-Time Finite Element Simulations of Nonstationary Incompressible Flows

Subject of this paper is the development of an a posteriori error estimator for nonstationary incompressible flow problems. The error estimator is computable and able to assess the temporal and spatial discretization errors separately. Thereby, the error is measured in an arbitrary quantity of interest because measuring errors in global norms is often of minor importance in practical applications. The basis for this is a finite element discretization in time and space. The techniques presented here also provide local error indicators which are used to adaptively refine the temporal and spatial discretization. A key ingredient in setting up an efficient discretization method is balancing the error contributions due to temporal and spatial discretization. To this end, a quantitative assessment of the individual discretization errors is required. The described method is validated by an established Navier-Stokes benchmark