399 research outputs found
Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity
This paper is concerned with the analysis and implementation of robust finite
element approximation methods for mixed formulations of linear elasticity
problems where the elastic solid is almost incompressible. Several novel a
posteriori error estimators for the energy norm of the finite element error are
proposed and analysed. We establish upper and lower bounds for the energy error
in terms of the proposed error estimators and prove that the constants in the
bounds are independent of the Lam\'{e} coefficients: thus the proposed
estimators are robust in the incompressible limit. Numerical results are
presented that validate the theoretical estimates. The software used to
generate these results is available online.Comment: 23 pages, 9 figure
Surrogate Accelerated Bayesian Inversion for the Determination of the Thermal Diffusivity of a Material
Determination of the thermal properties of a material is an important task in
many scientific and engineering applications. How a material behaves when
subjected to high or fluctuating temperatures can be critical to the safety and
longevity of a system's essential components. The laser flash experiment is a
well-established technique for indirectly measuring the thermal diffusivity,
and hence the thermal conductivity, of a material. In previous works,
optimization schemes have been used to find estimates of the thermal
conductivity and other quantities of interest which best fit a given model to
experimental data. Adopting a Bayesian approach allows for prior beliefs about
uncertain model inputs to be conditioned on experimental data to determine a
posterior distribution, but probing this distribution using sampling techniques
such as Markov chain Monte Carlo methods can be incredibly computationally
intensive. This difficulty is especially true for forward models consisting of
time-dependent partial differential equations. We pose the problem of
determining the thermal conductivity of a material via the laser flash
experiment as a Bayesian inverse problem in which the laser intensity is also
treated as uncertain. We introduce a parametric surrogate model that takes the
form of a stochastic Galerkin finite element approximation, also known as a
generalized polynomial chaos expansion, and show how it can be used to sample
efficiently from the approximate posterior distribution. This approach gives
access not only to the sought-after estimate of the thermal conductivity but
also important information about its relationship to the laser intensity, and
information for uncertainty quantification. We also investigate the effects of
the spatial profile of the laser on the estimated posterior distribution for
the thermal conductivity
Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation
Partial differential equations (PDEs) with inputs that depend on infinitely
many parameters pose serious theoretical and computational challenges.
Sophisticated numerical algorithms that automatically determine which
parameters need to be activated in the approximation space in order to estimate
a quantity of interest to a prescribed error tolerance are needed. For elliptic
PDEs with parameter-dependent coefficients, stochastic Galerkin finite element
methods (SGFEMs) have been well studied. Under certain assumptions, it can be
shown that there exists a sequence of SGFEM approximation spaces for which the
energy norm of the error decays to zero at a rate that is independent of the
number of input parameters. However, it is not clear how to adaptively
construct these spaces in a practical and computationally efficient way. We
present a new adaptive SGFEM algorithm that tackles elliptic PDEs with
parameter-dependent coefficients quickly and efficiently. We consider
approximation spaces with a multilevel structure---where each solution mode is
associated with a finite element space on a potentially different mesh---and
use an implicit a posteriori error estimation strategy to steer the adaptive
enrichment of the space. At each step, the components of the error estimator
are used to assess the potential benefits of a variety of enrichment
strategies, including whether or not to activate more parameters. No marking or
tuning parameters are required. Numerical experiments for a selection of test
problems demonstrate that the new method performs optimally in that it
generates a sequence of approximations for which the estimated energy error
decays to zero at the same rate as the error for the underlying finite element
method applied to the associated parameter-free problem.Comment: 22 page
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