Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
4
4
2014
AN ALGORITHM FOR FAST CALCULATION OF FLOW ENSEMBLES
273-301
Nan
Jiang
Departament of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
William
Layton
Departament of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
This report presents an algorithm for computing an ensemble of p solutions of the Navier-Stokes equations. The solutions are found, at each timestep, by solving a linear system with one shared coefficient matrix and p right hand sides, reducing both storage required and computational cost of the solution process. The price that must be paid is a timestep condition involving the timestep and the size of the fluctuations about the ensemble mean. Since the method is a one step method and the timestep condition involves only known quantities, it can be imposed to adapt the next timestep. The report gives a comprehensive stability analysis, an error estimate, and some first tests.
SPARSE MULTIRESOLUTION REGRESSION FOR UNCERTAINTY PROPAGATION
303-331
Daniele
Schiavazzi
Mechanical and Aerospace Engineering Department, University of California, San Diego, California 92093, USA
Alireza
Doostan
Aerospace Engineering Sciences Department, University of Colorado, Boulder, Colorado 80309-0429, USA
Gianluca
Iaccarino
Department of Mechanical Engineering Institute for Computational Mathematical Engineering Stanford University Bldg 500, RM 500-I, Stanford CA 94305 - USA
The present work proposes a novel nonintrusive, i.e., sampling-based, framework for approximating stochastic solutions of interest admitting sparse multiresolution expansions. The coefficients of such expansions are computed via greedy approximation techniques that require a number of solution realizations smaller than the cardinality of the multiresolution basis. The effect of various random sampling strategies is investigated. The proposed methodology is verified on a number of benchmark problems involving nonsmooth stochastic responses, and is applied to quantifying the efficiency of a passive vibration control system operating under uncertainty.
TRUNCATED HIERARCHICAL PRECONDITIONING FOR THE STOCHASTIC GALERKIN FEM
333-348
Bedrich
Sousedik
UMBC
Roger
Ghanem
Department of Aerospace and Mechanical Engineering, University of Southern California, 210 KAP Hall, Los Angeles, California 90089, USA
Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems.We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications
in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain
submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.
OPTIMIZATION-BASED SAMPLING IN ENSEMBLE KALMAN FILTERING
349-364
Antti
Solonen
Lappeenranta University of Technology, Laboratory of Applied Mathematics
Alexander
Bibov
Lappeenranta University of Technology, Laboratory of Applied Mathematics
Johnathan M.
Bardsley
Department of Mathematical Sciences, The University of Montana, Missoula, Montana 59812-0864, USA
Heikki
Haario
Department of Mathematics and Physics, Lappeenranta University of Technology; Finnish Meteorological Institute, Helsinki, Finland
In the ensemble Kalman filter (EnKF), uncertainty in the state of a dynamical model is represented as samples of the
state vector. The samples are propagated forward using the evolution model, and the forecast (prior) mean and covariance matrix are estimated from the ensemble. Data assimilation is carried out by using these estimates in the Kalman filter formulas. The prior is given in the subspace spanned by the propagated ensemble, the size of which is typically much smaller than the dimension of the state space. The rank-deficiency of these covariance matrices is problematic, and, for instance, unrealistic correlations often appear between spatially distant points, and different localization or covariance tapering methods are needed to make the approach feasible in practice. In this paper, we present a novel way
to implement ensemble Kalman filtering using optimization-based sampling, in which the forecast error covariance
has full rank and the need for localization is diminished. The method is based on the randomize then optimize (RTO)
technique, where a sample from a Gaussian distribution is computed by perturbing the data and the prior, and solving
a quadratic optimization problem. We test our method in two benchmark problems: the 40-dimensional Lorenz '96
model and the 1600-dimensional two-layer quasi-geostrophic model. Results show that the performance of the method
is significantly better than that of the standard EnKF, especially with small ensemble sizes when the rank-deficiency problems in EnKF are emphasized.