6,919 research outputs found
Iterative Updating of Model Error for Bayesian Inversion
In computational inverse problems, it is common that a detailed and accurate
forward model is approximated by a computationally less challenging substitute.
The model reduction may be necessary to meet constraints in computing time when
optimization algorithms are used to find a single estimate, or to speed up
Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use
of an approximate model introduces a discrepancy, or modeling error, that may
have a detrimental effect on the solution of the ill-posed inverse problem, or
it may severely distort the estimate of the posterior distribution. In the
Bayesian paradigm, the modeling error can be considered as a random variable,
and by using an estimate of the probability distribution of the unknown, one
may estimate the probability distribution of the modeling error and incorporate
it into the inversion. We introduce an algorithm which iterates this idea to
update the distribution of the model error, leading to a sequence of posterior
distributions that are demonstrated empirically to capture the underlying truth
with increasing accuracy. Since the algorithm is not based on rejections, it
requires only limited full model evaluations.
We show analytically that, in the linear Gaussian case, the algorithm
converges geometrically fast with respect to the number of iterations. For more
general models, we introduce particle approximations of the iteratively
generated sequence of distributions; we also prove that each element of the
sequence converges in the large particle limit. We show numerically that, as in
the linear case, rapid convergence occurs with respect to the number of
iterations. Additionally, we show through computed examples that point
estimates obtained from this iterative algorithm are superior to those obtained
by neglecting the model error.Comment: 39 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
- …