2,344 research outputs found
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
Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions
This paper is devoted to the construction of structure preserving stochastic
Galerkin schemes for Fokker-Planck type equations with uncertainties and
interacting with an external distribution, that we refer to as a background
distribution. The proposed methods are capable to preserve physical properties
in the approximation of statistical moments of the problem like nonnegativity,
entropy dissipation and asymptotic behaviour of the expected solution. The
introduced methods are second order accurate in the transient regimes and high
order for large times. We present applications of the developed schemes to the
case of fixed and dynamic background distribution for models of collective
behaviour
A combination technique for optimal control problems constrained by random PDEs
We present a combination technique based on mixed differences of both spatial
approximations and quadrature formulae for the stochastic variables to solve
efficiently a class of Optimal Control Problems (OCPs) constrained by random
partial differential equations. The method requires to solve the OCP for
several low-fidelity spatial grids and quadrature formulae for the objective
functional. All the computed solutions are then linearly combined to get a
final approximation which, under suitable regularity assumptions, preserves the
same accuracy of fine tensor product approximations, while drastically reducing
the computational cost. The combination technique involves only tensor product
quadrature formulae, thus the discretized OCPs preserve the convexity of the
continuous OCP. Hence, the combination technique avoids the inconveniences of
Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for
high dimensional problems. The manuscript presents an a-priori procedure to
choose the most important mixed differences and an asymptotic complexity
analysis, which states that the asymptotic complexity is exclusively determined
by the spatial solver. Numerical experiments validate the results.Comment: 25 pages, 4 figure
Goal-oriented error control of stochastic system approximations using metric-based anisotropic adaptations
International audienceThe simulation of complex nonlinear engineering systems such as compressible fluid flows may be targeted to make more efficient and accurate the approximation of a specific (scalar) quantity of interest of the system. Putting aside modeling error and parametric uncertainty, this may be achieved by combining goal-oriented error estimates and adaptive anisotropic spatial mesh refinements. To this end, an elegant and efficient framework is the one of (Riemannian) metric-based adaptation where a goal-based a priori error estimation is used as indicator for adaptivity. This work proposes a novel extension of this approach to the case of aforementioned system approximations bearing a stochastic component. In this case, an optimisation problem leading to the best control of the distinct sources of errors is formulated in the continuous framework of the Riemannian metric space. Algorithmic developments are also presented in order to quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces. The capability of the proposed method is tested on various problems including a supersonic scramjet inlet subject to geometrical and operational parametric uncertainties. It is demonstrated to accurately capture discontinuous features of stochastic compressible flows impacting pressure-related quantities of interest, while balancing computational budget and refinements in both spaces
Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs
We study sparse recovery with structured random measurement matrices having
independent, identically distributed, and uniformly bounded rows and with a
nontrivial covariance structure. This class of matrices arises from random
sampling of bounded Riesz systems and generalizes random partial Fourier
matrices. Our main result improves the currently available results for the null
space and restricted isometry properties of such random matrices. The main
novelty of our analysis is a new upper bound for the expectation of the
supremum of a Bernoulli process associated with a restricted isometry constant.
We apply our result to prove new performance guarantees for the CORSING method,
a recently introduced numerical approximation technique for partial
differential equations (PDEs) based on compressive sensing
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