33 research outputs found
Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems
Polynomial chaos expansions are used to reduce the computational cost in the
Bayesian solutions of inverse problems by creating a surrogate posterior that
can be evaluated inexpensively. We show, by analysis and example, that when the
data contain significant information beyond what is assumed in the prior, the
surrogate posterior can be very different from the posterior, and the resulting
estimates become inaccurate. One can improve the accuracy by adaptively
increasing the order of the polynomial chaos, but the cost may increase too
fast for this to be cost effective compared to Monte Carlo sampling without a
surrogate posterior
Parameter estimation by implicit sampling
Implicit sampling is a weighted sampling method that is used in data
assimilation, where one sequentially updates estimates of the state of a
stochastic model based on a stream of noisy or incomplete data. Here we
describe how to use implicit sampling in parameter estimation problems, where
the goal is to find parameters of a numerical model, e.g.~a partial
differential equation (PDE), such that the output of the numerical model is
compatible with (noisy) data. We use the Bayesian approach to parameter
estimation, in which a posterior probability density describes the probability
of the parameter conditioned on data and compute an empirical estimate of this
posterior with implicit sampling. Our approach generates independent samples,
so that some of the practical difficulties one encounters with Markov Chain
Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples,
are avoided. We describe a new implementation of implicit sampling for
parameter estimation problems that makes use of multiple grids (coarse to fine)
and BFGS optimization coupled to adjoint equations for the required gradient
calculations. The implementation is "dimension independent", in the sense that
a well-defined finite dimensional subspace is sampled as the mesh used for
discretization of the PDE is refined. We illustrate the algorithm with an
example where we estimate a diffusion coefficient in an elliptic equation from
sparse and noisy pressure measurements. In the example, dimension\slash
mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions
Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation
Implicit particle filtering is a sequential Monte Carlo method for data
assim- ilation, designed to keep the number of particles manageable by
focussing attention on regions of large probability. These regions are found by
min- imizing, for each particle, a scalar function F of the state variables.
Some previous implementations of the implicit filter rely on finding the
Hessians of these functions. The calculation of the Hessians can be cumbersome
if the state dimension is large or if the underlying physics are such that
derivatives of F are difficult to calculate. This is the case in many
geophysical applica- tions, in particular for models with partial noise, i.e.
with a singular state covariance matrix. Examples of models with partial noise
include stochastic partial differential equations driven by spatially smooth
noise processes and models for which uncertain dynamic equations are
supplemented by con- servation laws with zero uncertainty. We make the implicit
particle filter applicable to such situations by combining gradient descent
minimization with random maps and show that the filter is efficient, accurate
and reliable because it operates in a subspace whose dimension is smaller than
the state dimension. As an example, we assimilate data for a system of
nonlinear partial differential equations that appears in models of
geomagnetism
Implicit particle methods and their connection with variational data assimilation
The implicit particle filter is a sequential Monte Carlo method for data
assimilation that guides the particles to the high-probability regions via a
sequence of steps that includes minimizations. We present a new and more
general derivation of this approach and extend the method to particle smoothing
as well as to data assimilation for perfect models. We show that the
minimizations required by implicit particle methods are similar to the ones one
encounters in variational data assimilation and explore the connection of
implicit particle methods with variational data assimilation. In particular, we
argue that existing variational codes can be converted into implicit particle
methods at a low cost, often yielding better estimates, that are also equipped
with quantitative measures of the uncertainty. A detailed example is presented