Quantifying Uncertainty
Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics.
vol. 202.
2005.
pp. 94--133.
Asokan, B.V. and Zabaras, N.. "A stochastic variational multiscale method for diffusion in heterogeneous random media" Journal of Computational Physics. 218
(2).
2006.
pp. 654--676.
A stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media. The exact solution of the governing equations is split into two components: a coarse-scale solution that can be captured on a coarse mesh and a subgrid solution. A localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse-scale solution. The coarse-scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model. The generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models. Finally, various numerical examples are considered for evaluating the method.
Ganapathysubramanian, B. and Zabaras, N.. "Sparse grid collocation schemes for stochastic natural convection problems" J. Comput. Phys.. 225
(1).
2007.
pp. 652--685.
In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are made.
Zabaras, N. and Ganapathysubramanian, B.. "A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach" Journal of Computational Physics. 227
(9).
2008.
pp. 4697--4735.
Experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions. Recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties. These advances raise the possibility of solving the corresponding stochastic inverse problem: the problem of designing/estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information. A scalable, parallel methodology for stochastic inverse/design problems is developed in this article. The representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology. A novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems. The stochastic inverse/design problem is transformed to a deterministic optimization problem in a larger-dimensional space that is subsequently solved using deterministic optimization algorithms. The design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems, thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes. Various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework.
Ganapathysubramanian, B. and Zabaras, N.. "A seamless approach towards stochastic modeling: Sparse grid collocation and data driven input models" Finite Elements in Analysis and Design. 44
(5).
2008.
pp. 298--320.
Many physical systems of fundamental and industrial importance are significantly affected by the underlying fluctuations/variations in boundary, initial conditions as well as variabilities in operating and surrounding conditions. There has been increasing interest in analyzing and quantifying the effects of uncertain inputs in the solution of partial differential equations that describe these physical phenomena. Such analysis naturally leads to a rigorous methodology to design/control physical processes in the presence of multiple sources of uncertainty. A general application of these ideas to many significant problems in engineering is mainly limited by two issues. The first is the significant effort required to convert complex deterministic software/legacy codes into their stochastic counterparts. The second bottleneck to the utility of stochastic modeling is the construction of realistic, viable models of the input variability. This work attempts to demystify stochastic modeling by providing easy-to-implement strategies to address these two issues. In the first part of the paper, strategies to construct realistic input models that encode the variabilities in initial and boundary conditions as well as other parameters are provided. In the second part of the paper, we review recent advances in stochastic modeling and provide a road map to trivially convert any deterministic code into its stochastic counterpart. Several illustrative examples showcasing the ease of converting deterministic codes to stochastic codes are provided.
Acharjee, S. and Zabaras, N.. "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes" Computers and Structures. 85
(5-6).
2007.
pp. 244--254.
Large deformation processes are inherently complex considering the non-linear phenomena that need to be accounted for. Stochastic analysis of these processes is a formidable task due to the numerous sources of uncertainty and the various random input parameters. As a result, uncertainty propagation using intrusive techniques requires tortuous analysis and overhaul of the internal structure of existing deterministic analysis codes. In this paper, we present an approach called non-intrusive stochastic Galerkin (NISG) method, which can be directly applied to presently available deterministic legacy software for modeling deformation processes with minimal effort for computing the complete probability distribution of the underlying stochastic processes. The method involves finite element discretization of the random support space and piecewise continuous interpolation of the probability distribution function over the support space with deterministic function evaluations at the element integration points. For the hyperelastic-viscoplastic large deformation problems considered here with varying levels of randomness in the input and boundary conditions, the NISG method provides highly accurate estimates of the statistical quantities of interest within a fraction of the time required using existing Monte Carlo methods.
Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics. 202
(1).
2005.
pp. 94--133.
An extension of the deterministic variational multiscale (VMS) approach with algebraic subgrid scale (SGS) modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic Navier-Stokes equations. The stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method (SSFEM). Generalized polynomial chaos and Karhunen-Loève expansion techniques are used for representation of uncertain quantities. The proposed stabilized method is then applied to various standard advection-diffusion and fluid-flow examples with uncertainty in essential boundary conditions. Comparisons are drawn between the numerical solutions and Monte-Carlo/analytical solutions wherever possible.
Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208
(1).
2005.
pp. 134--153.
A stabilized stochastic finite element implementation for the natural convection system of equations under Boussinesq assumptions with uncertainty in inputs is considered. The stabilized formulations are derived using the variational multiscale framework assuming a one-step trapezoidal time integration rule. The stabilization parameters are shown to be functions of the time-step size. Provision is made for explicit tracking of the subgrid-scale solution through time. A support-space/stochastic Galerkin approach and the generalized polynomial chaos expansion (GPCE) approach are considered for input-output uncertainty representation. Stochastic versions of standard Rayleigh-Benard convection problems are used to evaluate the approach. It is shown that for simulations around critical points, the GPCE approach fails to capture the highly non-linear input uncertainty propagation whereas the support-space approach gives fairly accurate results. A summary of the results and findings is provided.
Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208
(1).
2005.
pp. 134--153.
In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are mad
Ganapathysubramanian, B. and Zabaras, N.. "Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method" Journal of Computational Physics. 226
(1).
SEP 10 2007.
pp. 326--353.
In recent years, there has been intense interest in understanding various physical phenomena in random heterogeneous media. Any accurate description/simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties. An accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response. We propose a general methodology to construct a data-driven, reduced-order model to describe property variations in realistic heterogeneous media. This reduced-order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media. A decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties. A sparse-grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems. A variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems. An illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology. (C) 2007 Elsevier Inc. All rights reserved.