Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
On an interpolatory method for high dimensional integration
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
Understanding Molecular Simulation
Understanding Molecular Simulation
Introduction to Algorithms
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation
Journal of Scientific Computing
Modeling uncertainty in flow simulations via generalized polynomial chaos
Journal of Computational Physics
Stochastic Solutions for the Two-Dimensional Advection-Diffusion Equation
SIAM Journal on Scientific Computing
Using stochastic analysis to capture unstable equilibrium in natural convection
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB
ACM Transactions on Mathematical Software (TOMS)
Stochastic analysis of transport in tubes with rough walls
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
A non-linear dimension reduction methodology for generating data-driven stochastic input models
Journal of Computational Physics
Performance evaluation of generalized polynomial chaos
ICCS'03 Proceedings of the 2003 international conference on Computational science
Geodesic entropic graphs for dimension and entropy estimation in manifold learning
IEEE Transactions on Signal Processing
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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.