08:30
Sample-Dependent Schwartz Preconditioners for Stochastic Elliptic Equations
João Reis | INRIA Saclay / CMAP, Ecole polytechnique | France
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Authors:
João Reis | INRIA Saclay / CMAP, Ecole polytechnique | France
Olivier Le Maître | CNRS - Ecole Polytechnique | France
Pietro Marco Congedo | INRIA Saclay / CMAP, Ecole polytechnique | France
Paul Mycek | CERFACS | France
We are interested in solving stochastic elliptic equations with random field coefficients, using Monte Carlo (MC) methods. The MC method is known to be demanding as it requires solving thousands of samples to estimate well-converged statistics. The additive Schwarz method (SM) is a standard iterative domain decomposition approach that involves, at each iteration, the parallel resolution of independent local problems. The boundary values of the local problems are updated for the next iteration using the current solution over the local subdomains. The update of the boundary values has to be preconditioned, and one challenge is finding an effective preconditioner for the stochastic problem. One can use sample-independent preconditioners based, for instance, on the mean or median of the random coefficient field. These preconditioners are not adequate for high variance coefficients as they are not representative of all events. We then propose to develop sample-dependent preconditioners. Our approach determines the preconditioner in an off-line stage, from observation of a limited number of samples. It relies on a truncated Karhunen-Loeve (KL) expansion of the random coefficient field, to come up with a Polynomial Chaos (PC) expansion of the random preconditioner. The PC expansion is used in the on-line MC stage to make the approach very cost-effective. We demonstrate the potential of this parallel approach and compare it with classical sample-independent preconditioning.
08:50
New two-sided guaranteed spectral bounds for block preconditioning of stochastic Galerkin problems
Ivana Pultarova | Czech Technical University in Prague, Faculty of Civil Engineering | Czech Republic
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Authors:
Ivana Pultarova | Czech Technical University in Prague, Faculty of Civil Engineering | Czech Republic
Marie Kubinova | Institute of Geonics of the Czech Academy of Sciences | Czech Republic
Quantification of uncertainty of a numerical solution of the diffusion equation with randomly distributed data is considered. Stochastic (spectral) Galerkin method is applied, which leads to solution of a large system of linear equations in form of a sum of tensor product matrices. The whole matrix is never evaluated, still the numerical solution can be demanding due to ill-conditioning of the matrix. In our presentation we introduce a set of spectrally equivalent problems (matrices) that can be used for preconditioning. Their block diagonal forms follow from modifying only the stochastic part of the problem and allow parallel solution methods. Some of them are well known, e.g., the mean based deterministic problem (matrix). In addition, we provide a new tool for obtaining guaranteed two-sided bounds to constants of the spectral equivalence of these problems. These bounds are applicable to various distributions of parameters and depend solely on the properties of the parameter-dependent coefficient function and on the type of the approximation polynomials of stochastic variables. Moreover, the conditions on the coefficient function are only local, and therefore less restrictive than those assumed in the literature so far. Both complete and tensor product polynomials are considered for approximation of the solution. Some illustrative numerical examples are presented.
09:10
A novel preconditioning technique for stochastic Galerkin finite element discretisations
Alex Bespalov | University of Birmingham | United Kingdom
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Authors:
Alex Bespalov | University of Birmingham | United Kingdom
Daniel Loghin | University of Birmingham | United Kingdom
Rawin Youngnoi | University of Birmingham | United Kingdom
Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to sampling methods for the numerical solution of linear elliptic PDE problems with parametric or random inputs. In order to compute the stochastic Galerkin solution for a given problem, one needs to solve a large coupled system of linear equations. Thus, an efficient iterative solver is a key ingredient of any SGFEM implementation. In this talk, we introduce a new preconditioning technique for SGFEM that extends the idea of mean-based preconditioning with the aim to capture more significant components of the stochastic Galerkin matrix. We will present theoretical results, including spectral bounds for the preconditioned matrix, and report the results of numerical experiments for the model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear system) and compare our preconditioner with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based preconditioner from [C. E. Powell and H. Elman, IMA J. Numer. Anal., 29: 350–375, 2009] and the Kronecker product preconditioner from [E. Ullmann, SIAM J. Sci. Comput., 32(2): 923-946, 2010].
09:30
Stochastic Galerkin mixed finite element approximation for Biot's consolidation model with uncertain inputs
David Silvester | University of Manchester | United Kingdom
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Authors:
Arbaz Khan | University of Manchester | United Kingdom
Catherine Powell | University of Manchester | United Kingdom
David Silvester | University of Manchester | United Kingdom
Over the last couple of decades, models based on linear poroelasticity have attracted a lot of attention because of their applications in science and engineering. Although substantial work has been done in the engineering and mathematics communities on the formulation and numerical solution of deterministic poroelastic models, there has been little work to date on parameter-robust stochastic formulations of the Biot consolidation model. The aim of this talk is to discuss a novel locking-free stochastic Galerkin mixed finite element method for a new five-field Biot consolidation model with uncertain Young's modulus and hydraulic conductivity field. Working in an appropriate weighted norm, we establish well-posedness of the new formulation and develop a new preconditioner for use with the minimal residual method (MINRES).
We show that the proposed preconditioner for the discrete system is robust with respect to the discretization parameters as well as the Poisson ratio and the Biot-Willis constant. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach.
09:50
Polynomial Annihilation-Based Stochastic Galerkin Method for Discontinuous System Responses in Aerodynamic Problems
Shigetaka Kawai | The University of Tokyo | Japan
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Authors:
Shigetaka Kawai | The University of Tokyo | Japan
Akira Oyama | Institute of Space and Astronautical Science/Japan Aerospace Exploration Agency | Japan
The framework based on polynomial chaos expansion (PCE) is the most widely used for uncertainty quantification; however, it suffers from slow convergence rate when global polynomial bases are used to capture discontinuous responses in the stochastic space, which is often the case in aerodynamic problems in aerospace engineering. Although several PCE methods using piecewise smooth polynomial bases have been developed to overcome this problem in the context of intrusive methods, they may yield redundant subspaces in the stochastic space so that the efficiency of the original method is lost. In the present study, a multi-element stochastic Galerkin (SG) method based on the polynomial annihilation edge detection is developed for uncertainty quantification in aerodynamic problems. The presented method is applied to uncertainty quantification in quasi-one-dimensional nozzle flow, in which the uncertainty associated with the exit pressure of the nozzle is propagated into the inner pressure distribution. As a result, a discontinuous behavior of the solution is sharply captured by the presented method, while it deteriorates the performance of the conventional SG method. Comparison between the presented and the conventional SG methods show that the presented method is quantitatively more accurate than the conventional SG method.
10:10
Stochastic Discontinuous Galerkin Methods for Convection Diffusion Equations
Pelin Çiloğlu | Middle East Technical University | Turkey
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Authors:
Pelin Çiloğlu | Middle East Technical University | Turkey
Hamdullah Yücel | Middle East Technical University | Turkey
Many physical systems occurring in different areas of science such as fluid dynamics, oil field reservoir, and underwater pollution are modelled by partial differential equations (PDEs) together with appropriate boundary conditions. However, due to lack of knowledge about some input data or parameters of a given mathematical models, deterministic PDEs are not enough to model such real-problems. Therefore, the idea of uncertainty quantification (UQ) has become a powerful tools to model such physical problem in the last few years.
In this talk, we investigate numerical behaviors of the statistical moments of the solution for convection diffusion equations with random coefficients by using stochastic discontinuous Galerkin methods. To identify the random coefficients, the principal component analysis (PCA), i.e., linear and kernel PCA will be used in addition to the well-known technique Karhunen Loève (K-L) expansion. Since the local mass conservation play a crucial role in the real-world problems modelled by the convection diffusion equations, we use symmetric interior penalty Galerkin (SIPG) method for the spatial discretization. On the other hand, for the temporal discretization, we propose a first-order semi implicit-explicit time-stepping technique. Numerical results for both steady and unsteady cases will be provided to illustrate the efficiency of the proposed approach and to verify theoretical findings.