8 research outputs found
On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations
The KarhunenâLĂČeve (KL) decomposition provides a low-dimensional representation for random fields as it is optimal in the mean square sense. Although for many stochastic systems of practical interest, described by stochastic partial differential equations (SPDEs), solutions possess this low-dimensional character, they also have a strongly time-dependent form and to this end a fixed-in-time basis may not describe the solution in an efficient way. Motivated by this limitation of standard KL expansion, Sapsis and Lermusiaux (2009) [26] developed the dynamically orthogonal (DO) field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty âlivesâ but also the stochastic characteristics of uncertainty. Recently, Cheng et al. (2013) [28] introduced an alternative approach, the bi-orthogonal (BO) method, which performs the exact same tasks, i.e. it evolves the spatial basis and the stochastic characteristics of uncertainty. In the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other. We show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the BO and the DO solutions. We also examine a pathology of the BO equations that occurs when two eigenvalues of the solution cross, resulting in an instantaneous, infinite-speed, internal rotation of the computed spatial basis. We demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the BO approach. On the other hand, it is observed that the BO is more stable in nonlinear problems involving a relatively large number of modes. Several examples, linear and nonlinear, are presented to illustrate the DO and BO methods as well as their equivalence.National Science Foundation (U.S.) (NSF/DMS (DMS-1216437))Pacific Northwest National Laboratory (U.S.). Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4) (DOE (DE-SC0009247))United States. Dept. of Defense (OSD/MURI (FA9550-09-1-0613)
Improving predictive power of physically based rainfall-induced shallow landslide models: a probabilistic approach
Distributed models to forecast the spatial and temporal occurrence of
rainfall-induced shallow landslides are based on deterministic laws. These
models extend spatially the static stability models adopted in geotechnical
engineering, and adopt an infinite-slope geometry to balance the resisting and
the driving forces acting on the sliding mass. An infiltration model is used to
determine how rainfall changes pore-water conditions, modulating the local
stability/instability conditions. A problem with the operation of the existing
models lays in the difficulty in obtaining accurate values for the several
variables that describe the material properties of the slopes. The problem is
particularly severe when the models are applied over large areas, for which
sufficient information on the geotechnical and hydrological conditions of the
slopes is not generally available. To help solve the problem, we propose a
probabilistic Monte Carlo approach to the distributed modeling of
rainfall-induced shallow landslides. For the purpose, we have modified the
Transient Rainfall Infiltration and Grid-Based Regional Slope-Stability
Analysis (TRIGRS) code. The new code (TRIGRS-P) adopts a probabilistic approach
to compute, on a cell-by-cell basis, transient pore-pressure changes and
related changes in the factor of safety due to rainfall infiltration.
Infiltration is modeled using analytical solutions of partial differential
equations describing one-dimensional vertical flow in isotropic, homogeneous
materials. Both saturated and unsaturated soil conditions can be considered.
TRIGRS-P copes with the natural variability inherent to the mechanical and
hydrological properties of the slope materials by allowing values of the TRIGRS
model input parameters to be sampled randomly from a given probability
distribution. [..]Comment: 25 pages, 14 figures, 9 tables. Revised version; accepted for
publication in Geoscientific Model Development on 13 February 201
Stochastic methods for uncertainty quantification in radiation transport
The use of stochastic spectral expansions, specifically generalized polynomial chaos (gPC) and Karhunen-Loeve (KL) expansions, is investigated for uncertainty quantification in radiation transport. The gPC represents second-order random processes in terms of an expansion of orthogonal polynomials of random variables. The KL expansion is a Fourier-type expansion that represents a second-order random process with known covariance function in terms of a set of uncorrelated random variables and the eigenmodes of the covariance function. The flux and, in multiplying materials, the k-eigenvalue, which are the problem unknowns, are always expanded in a gPC expansion since their covariance functions are also unknown. This work assumes a single uncertain inputâthe total macroscopic cross sectionâalthough this does not represent a limitation of the approaches considered here. Two particular types of input parameter uncertainty are investigated: The cross section as a univariate Gaussian, log-normal, gamma or beta random variable, and the cross section as a spatially varying Gaussian or log-normal random process. In the first case, a gPC expansion in terms of a univariate random variable suffices, while in the second, a truncated KL expansion is first necessary followed by a gPC expansion in terms of multivariate random variables. Two solution methods are examined: The Stochastic Finite Element Method (SFEM) and the Stochastic Collocation Method (SCM). The SFEM entails taking Galerkin projections onto the orthogonal basis, which yields a system of fully-coupled equations for the PC coefficients of the flux and the k-eigenvalue. This system is linear when there is no multiplication and can be solved using Richardson iteration, employing a standard operator splitting such as block Gauss-Seidel or block Jacobi, or a Krylov iterative method, which can be preconditioned using these splittings. When multiplication is present, the SFEM system is non-linear and a Newton-Krylov method is employed to solve it. The SCM utilizes a suitable quadrature rule to compute the moments or PC coefficients of the flux and k-eigenvalue, and thus involves the solution of a system of independent deterministic transport equations. The accuracy and efficiency of the two methods are compared and contrasted. Both are shown to accurately compute the PC coefficients of the unknown, and numerical proof is provided that the two methods are in fact equivalent in certain cases. The PC coefficients are used to compute the moments and probability density functions of the unknowns, which are shown to be accurate by comparing with Monte Carlo results. An analytic diffusion analysis, corroborated by numerical results, reveals that the random transport equation is well approximated by a deterministic diffusion equation when the medium is diffusive with respect to the average cross section but without constraint on the amplitude of the random fluctuations. Our work shows that stochastic spectral expansions are a viable alternative to random sampling-based uncertainty quantification techniques since both provide a complete characterization of the distribution of the flux and the k-eigenvalue. Furthermore, it is demonstrated that, unlike perturbation methods, SFEM and SCM can handle large parameter uncertainty
Stability analysis and control of stochastic dynamic systems using polynomial chaos
Recently, there has been a growing interest in analyzing stability and developing
controls for stochastic dynamic systems. This interest arises out of a need to develop
robust control strategies for systems with uncertain dynamics. While traditional
robust control techniques ensure robustness, these techniques can be conservative as
they do not utilize the risk associated with the uncertainty variation. To improve
controller performance, it is possible to include the probability of each parameter
value in the control design. In this manner, risk can be taken for parameter values
with low probability and performance can be improved for those of higher probability.
To accomplish this, one must solve the resulting stability and control problems
for the associated stochastic system. In general, this is accomplished using sampling
based methods by creating a grid of parameter values and solving the problem for
each associated parameter. This can lead to problems that are difficult to solve and
may possess no analytical solution.
The novelty of this dissertation is the utilization of non-sampling based methods
to solve stochastic stability and optimal control problems. The polynomial chaos expansion
is able to approximate the evolution of the uncertainty in state trajectories
induced by stochastic system uncertainty with arbitrary accuracy. This approximation
is used to transform the stochastic dynamic system into a deterministic system
that can be analyzed in an analytical framework. In this dissertation, we describe the generalized polynomial chaos expansion and
present a framework for transforming stochastic systems into deterministic systems.
We present conditions for analyzing the stability of the resulting systems. In addition,
a framework for solving L2 optimal control problems is presented. For linear systems,
feedback laws for the infinite-horizon L2 optimal control problem are presented. A
framework for solving finite-horizon optimal control problems with time-correlated
stochastic forcing is also presented. The stochastic receding horizon control problem
is also solved using the new deterministic framework. Results are presented that
demonstrate the links between stability of the original stochastic system and the
approximate system determined from the polynomial chaos approximation. The solutions
of these stochastic stability and control problems are illustrated throughout
with examples
Stochastic finite element modelling of elementary random media.
Following a stochastic approach, this thesis presents a numerical framework for elastostatics of random media. Firstly, after a mathematically rigorous investigation of the popular white noise model in an engineering context, the smooth spatial stochastic dependence between material properties is identified as a fundamental feature of practical random media. Based on the recognition of the probabilistic essence of practical random media and driven by engineering simulation requirements, a comprehensive random medium model, namely elementary random media (ERM), is consequently defined and its macro-scale properties including stationarity, smoothness and principles for material measurements are systematically explored. Moreover, an explicit representation scheme, namely the Fourier-Karhunen-Loeve (F-K-L) representation, is developed for the general elastic tensor of ERM by combining the spectral representation theory of wide-sense stationary stochastic fields and the standard dimensionality reduction technology of principal component analysis. Then, based on the concept of ERM and the F-K-L representation for its random elastic tensor, the stochastic partial differential equations regarding elastostatics of random media are formulated and further discretized, in a similar fashion as for the standard finite element method, to obtain a stochastic system of linear algebraic equations. For the solution of the resulting stochastic linear algebraic system, two different numerical techniques, i.e. the joint diagonalization solution strategy and the directed Monte Carlo simulation strategy, are developed. Original contributions include the theoretical analysis of practical random medium modelling, establishment of the ERM model and its F-K-L representation, and development of the numerical solvers for the stochastic linear algebraic system. In particular, for computational challenges arising from the proposed framework, two novel numerical algorithms are developed: (a) a quadrature algorithm for multidimensional oscillatory functions, which reduces the computational cost of the F-K-L representation by up to several orders of magnitude; and (b) a Jacobi-like joint diagonalization solution method for relatively small mesh structures, which can effectively solve the associated stochastic linear algebraic system with a large number of random variables