22 research outputs found
Inverse modeling of geochemical and mechanical compaction in sedimentary basins through Polynomial Chaos Expansion
We present an inverse modeling procedure for the estimation of model parameters of sedi-
mentary basins subject to compaction driven by mechanical and geochemical processes. We consider a
sandstone basin whose dynamics are governed by a set of unknown key quantities. These include geophys-
ical and geochemical system attributes as well as pressure and temperature boundary conditions. We derive
a reduced (or surrogate) model of the system behavior based on generalized Polynomial Chaos Expansion
(gPCE) approximations, which are directly linked to the variance-based Sobol indices associated with the
selected uncertain model parameters. Parameter estimation is then performed within a Maximum Likeli-
hood (ML) framework. We then study the way the ML inversion procedure can beneïŹt from the adoption of
anisotropic polynomial approximations (a-gPCE) in which the surrogate model is reïŹned only with respect
to selected parameters according to an analysis of the nonlinearity of the input-output mapping, as quanti-
ïŹed through the Sobol sensitivity indices. Results are illustrated for a one-dimensional setting involving
quartz cementation and mechanical compaction in sandstones. The reliability of gPCE and a-gPCE approxi-
mations in the context of the inverse modeling framework is assessed. The effects of (a) the strategy
employed to build the surrogate model, leading either to a gPCE or a-gPCE representation, and (b) the type
and quality of calibration data on the goodness of the parameter estimates is then explored
Uncertainty quantification in timber-like beams using sparse grids: theory and examples with off-the-shelf software utilization
When dealing with timber structures, the characteristic strength and
stiffness of the material are made highly variable and uncertain by the
unavoidable, yet hardly predictable, presence of knots and other defects. In
this work we apply the sparse grids stochastic collocation method to perform
uncertainty quantification for structural engineering in the scenario described
above. Sparse grids have been developed by the mathematical community in the
last decades and their theoretical background has been rigorously and
extensively studied. The document proposes a brief practice-oriented
introduction with minimal theoretical background, provides detailed
instructions for the use of the already implemented Sparse Grid Matlab kit
(freely available on-line) and discusses two numerical examples inspired from
timber engineering problems that highlight how sparse grids exhibit superior
performances compared to the plain Monte Carlo method. The Sparse Grid Matlab
kit requires only a few lines of code to be interfaced with any numerical
solver for mechanical problems (in this work we used an isogeometric
collocation method) and provides outputs that can be easily interpreted and
used in the engineering practice
On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods
In this work we focus on the numerical approximation of the solution
of a linear elliptic PDE with stochastic coefficients. The
problem is rewritten as a parametric PDE and the functional
dependence of the solution on the parameters is approximated by
multivariate polynomials. We first consider the Stochastic Galerkin
method, and rely on sharp estimates for the decay of the Fourier
coefficients of the spectral expansion of on an orthogonal
polynomial basis to build a sequence of polynomial subspaces that
features better convergence properties, in terms of error versus
number of degrees of freedom, than standard choices such as Total
Degree or Tensor Product subspaces.
We consider then the Stochastic Collocation method, and use the
previous estimates to introduce a new class of
Sparse Grids, based on the idea of selecting a priori the most
profitable hierarchical surpluses, that, again, features better
convergence properties compared to standard Smolyak or tensor
product grids. Numerical results show the effectiveness of
the newly introduced polynomial spaces and sparse grids
Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficientsâââ
In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids
Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison
Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods. By introducing a suitable generalization of the classical sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy vs. computational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyperbolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method. Finally, numerical results corroborate the optimality of the theoretical estimate of anisotropy ratios introduced by the authors in a previous work for the construction of anisotropic approximation spaces
ICES REPORT 09-33 - Stochastic Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison
Abstract: Much attention has recently been devoted to the development of
Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncer-
tainty quantification. An open and relevant research topic is the comparison
of these two methods. By introducing a suitable generalization of the classi-
cal sparse grid SC method, we are able to compare SG and SC on the same
underlying multivariate polynomial space in terms of accuracy versus com-
putational work. The approximation spaces considered here include isotropic
and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyper-
bolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear
elliptic SPDEs indicate a slight computational work advantage of isotropic
SC over SG, with SC-SM and SG-TD being the best choices of approximation
spaces for each method. Finally, numerical results corroborate the optimality
of the theoretical estimate of anisotropy ratios introduced by the authors in
a previous work for the construction of anisotropic approximation spaces