14,165 research outputs found
Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario
A variety of methods is available to quantify uncertainties arising with\-in
the modeling of flow and transport in carbon dioxide storage, but there is a
lack of thorough comparisons. Usually, raw data from such storage sites can
hardly be described by theoretical statistical distributions since only very
limited data is available. Hence, exact information on distribution shapes for
all uncertain parameters is very rare in realistic applications. We discuss and
compare four different methods tested for data-driven uncertainty
quantification based on a benchmark scenario of carbon dioxide storage. In the
benchmark, for which we provide data and code, carbon dioxide is injected into
a saline aquifer modeled by the nonlinear capillarity-free fractional flow
formulation for two incompressible fluid phases, namely carbon dioxide and
brine. To cover different aspects of uncertainty quantification, we incorporate
various sources of uncertainty such as uncertainty of boundary conditions, of
conceptual model definitions and of material properties. We consider recent
versions of the following non-intrusive and intrusive uncertainty
quantification methods: arbitary polynomial chaos, spatially adaptive sparse
grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The
performance of each approach is demonstrated assessing expectation value and
standard deviation of the carbon dioxide saturation against a reference
statistic based on Monte Carlo sampling. We compare the convergence of all
methods reporting on accuracy with respect to the number of model runs and
resolution. Finally we offer suggestions about the methods' advantages and
disadvantages that can guide the modeler for uncertainty quantification in
carbon dioxide storage and beyond
Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins
In this work we propose an Uncertainty Quantification methodology for
sedimentary basins evolution under mechanical and geochemical compaction
processes, which we model as a coupled, time-dependent, non-linear,
monodimensional (depth-only) system of PDEs with uncertain parameters. While in
previous works (Formaggia et al. 2013, Porta et al., 2014) we assumed a
simplified depositional history with only one material, in this work we
consider multi-layered basins, in which each layer is characterized by a
different material, and hence by different properties. This setting requires
several improvements with respect to our earlier works, both concerning the
deterministic solver and the stochastic discretization. On the deterministic
side, we replace the previous fixed-point iterative solver with a more
efficient Newton solver at each step of the time-discretization. On the
stochastic side, the multi-layered structure gives rise to discontinuities in
the dependence of the state variables on the uncertain parameters, that need an
appropriate treatment for surrogate modeling techniques, such as sparse grids,
to be effective. We propose an innovative methodology to this end which relies
on a change of coordinate system to align the discontinuities of the target
function within the random parameter space. The reference coordinate system is
built upon exploiting physical features of the problem at hand. We employ the
locations of material interfaces, which display a smooth dependence on the
random parameters and are therefore amenable to sparse grid polynomial
approximations. We showcase the capabilities of our numerical methodologies
through two synthetic test cases. In particular, we show that our methodology
reproduces with high accuracy multi-modal probability density functions
displayed by target state variables (e.g., porosity).Comment: 25 pages, 30 figure
Efficient cosmological parameter sampling using sparse grids
We present a novel method to significantly speed up cosmological parameter
sampling. The method relies on constructing an interpolation of the
CMB-log-likelihood based on sparse grids, which is used as a shortcut for the
likelihood-evaluation. We obtain excellent results over a large region in
parameter space, comprising about 25 log-likelihoods around the peak, and we
reproduce the one-dimensional projections of the likelihood almost perfectly.
In speed and accuracy, our technique is competitive to existing approaches to
accelerate parameter estimation based on polynomial interpolation or neural
networks, while having some advantages over them. In our method, there is no
danger of creating unphysical wiggles as it can be the case for polynomial fits
of a high degree. Furthermore, we do not require a long training time as for
neural networks, but the construction of the interpolation is determined by the
time it takes to evaluate the likelihood at the sampling points, which can be
parallelised to an arbitrary degree. Our approach is completely general, and it
can adaptively exploit the properties of the underlying function. We can thus
apply it to any problem where an accurate interpolation of a function is
needed.Comment: Submitted to MNRAS, 13 pages, 13 figure
Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations
This paper addresses the estimation of uncertain distributed diffusion
coefficients in elliptic systems based on noisy measurements of the model
output. We formulate the parameter identification problem as an infinite
dimensional constrained optimization problem for which we establish existence
of minimizers as well as first order necessary conditions. A spectral
approximation of the uncertain observations allows us to estimate the infinite
dimensional problem by a smooth, albeit high dimensional, deterministic
optimization problem, the so-called finite noise problem in the space of
functions with bounded mixed derivatives. We prove convergence of finite noise
minimizers to the appropriate infinite dimensional ones, and devise a
stochastic augmented Lagrangian method for locating these numerically. Lastly,
we illustrate our method with three numerical examples
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
- …