A seamless approach towards stochastic modeling: Sparse grid collocation and data driven input models

Ganapathysubramanian, Baskar; Zabaras, Nicholas

doi:10.1016/j.finel.2007.11.015

Cited by 17 publications

(12 citation statements)

References 56 publications

(79 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, parameters for which sufficient measurements at various spatial locations are available, can be modeled as random fields. These random fields can then be expressed in terms of random variables using KL expansion [16][17][18] or PC expansion [19][20][21]. Unfortunately, for MEMS, such detailed experimental observations regarding important design parameters such as material properties and geometrical features are not available.…”

Section: Representation Of Input Uncertaintymentioning

confidence: 99%

See 1 more Smart Citation

A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

Agarwal

Aluru

2010

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThis work presents a data-driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi-physics simulation of Micro-ElectroMechanical Systems (MEMS). The proposed framework comprises of two key steps: first, probabilistic characterization of the input uncertain parameters based on available experimental information, and second, propagation of these uncertainties through the predictive model to relevant quantities of interest. The uncertain input parameters are modeled as independent random variables, for which the distributions are estimated based on available experimental observations, using a nonparametric diffusion-mixing-based estimator, Botev (Nonparametric density estimation via diffusion mixing. Technical Report, 2007). The diffusion-based estimator derives from the analogy between the kernel density estimation (KDE) procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent. The diffusion model allows for the incorporation of the prior density and leads to an improved density estimate, in comparison with the standard KDE approach, as demonstrated through several numerical examples. Following the characterization step, the uncertainties are propagated to the output variables using the stochastic collocation approach, based on sparse grid interpolation, Smolyak (Soviet Math. Dokl. 1963; 4:240-243). The developed framework is used to study the effect of variations in Young's modulus, induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a MEMS switch.

show abstract

Section: Representation Of Input Uncertaintymentioning

confidence: 99%

“…Several researchers have employed the Karhunen-Loève (KL) expansion [16] to represent the random input parameters, modeled as random fields, in terms of a set of uncorrelated random variables (e.g. [17,18]). This approach requires prior knowledge regarding the covariance function of the random field.…”

Section: Introductionmentioning

confidence: 99%

A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

Agarwal

Aluru

2010

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…The choice of methodology and the ability to correctly characterize infinite‐dimensional random processes largely depend on the extent of available information regarding these processes. One such possible choice is the Karhunen–Lo ève (KL) expansion 32, which employs a spectral decomposition of the covariance function of the input random processes, to represent them in terms of a set of uncorrelated random variables 33, 34. Polynomial chaos expansion can also be used to represent random fields 35–37, where the coefficients of the PC modes can be determined based on the principles of maximum likelihood 36 or maximum entropy 37.…”

Section: Problem Formulationmentioning

confidence: 99%

Weighted Smolyak algorithm for solution of stochastic differential equations on non‐uniform probability measures

Agarwal

Aluru

2010

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThis paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non-uniform probability measures. Recently, there has been a growing interest in the stochastic collocation approach, which seeks to approximate the unknown stochastic solution using polynomial interpolation in the multi-dimensional random domain. Existing approaches employ sparse grid interpolation based on the Smolyak algorithm, which leads to orders of magnitude reduction in the number of support nodes as compared with usual tensor product. However, such sparse grid interpolation approaches based on piecewise linear interpolation employ uniformly sampled nodes from the random domain and do not take into account the probability measures during the construction of the sparse grids. Such a construction based on uniform sparse grids may not be ideal, especially for highly skewed or localized probability measures. To this end, this work proposes a weighted Smolyak algorithm based on piecewise linear basis functions, which incorporates information regarding non-uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard Smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution.

show abstract

“…Examples include spectral stochastic FEM , generalized polynomial chaos and sparse grid collocation (SGC) . The development of the so‐called non‐intrusive variants of these methods allow seamless integration of UQ into legacy software systems and/or black box models . We use an adaptive sparse grid collocation (ASGC) framework that seamlessly integrates around a deterministic wind turbine simulator to acquire distributional responses of wind turbine performance measures.…”

Section: Introductionmentioning

confidence: 99%

Incorporating a stochastic data‐driven inflow model for uncertainty quantification of wind turbine performance

Guo

Ganapathysubramanian

2017

Wind Energy

View full text Add to dashboard Cite

Incorporating uncertainty in wind turbine analysis and design is very necessary based on the fact that inherent variability exists in wind turbine systems. Examples of these uncertainties include fluctuations in material properties across turbine blades, variable structure parameters and stochasticity in the inflow-which is considered to be a critical factor affecting the reliability of wind turbines. However, it has been difficult to construct a low-dimensional yet accurate representation of the stochastic inflow, which precludes rigorous uncertainty propagation and quantification. Recently, we have developed a comprehensive data-driven approach [called temporal-spatial decomposition (TSD)] for constructing a stochastic, low-dimensional model that accurately represents stochastic inflow data. We leverage this approach to construct distributional forecasts of key wind turbine performance indicators. To this end, we integrated the stochastic wind model created by the TSD framework with the wind turbine solver FAST. Uncertainty propagation is performed using an adaptive sparse grid collocation approach. We investigate how the order of approximation of the stochastic model affects the quality of the predicted distribution. We observe that the probability distributions of key indicators are not necessarily Gaussian, which has implications for reliability analysis and for failure prediction. Furthermore, the distributions are sensitive to only the first few eigenmodes of the inflow wind model, which indicates that comprehensive uncertainty quantification can potentially be accomplished with moderate computational effort. The approach suggested in this paper enables seamless integration of uncertainty quantification into current deterministic codes for wind turbine simulation and has implications for the design of the next generation of wind turbines including offshore turbines.

show abstract

A seamless approach towards stochastic modeling: Sparse grid collocation and data driven input models

Cited by 17 publications

References 56 publications

A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

Weighted Smolyak algorithm for solution of stochastic differential equations on non‐uniform probability measures

Incorporating a stochastic data‐driven inflow model for uncertainty quantification of wind turbine performance

Contact Info

Product

Resources

About