• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.193-213
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• 2017

We investigate an efficient approximation of solution to stochastic Burgers equation driven by an additive space-time noise. We discuss existence and uniqueness of a solution through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we introduce the Karhunen-$Lo{\grave{e}}ve$ expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.

• Journal of the Korean Geotechnical Society
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• v.27 no.10
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• pp.93-104
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• 2011

In this paper, a numerical procedure of probabilistic steady seepage analysis that considers the spatial variability of soil permeability is presented. The procedure extends the deterministic analysis based on the finite element method to a probabilistic approach that accounts for the uncertainties and spatial variation of the soil permeability. Two-dimensional random fields are generated based on a Karhunen-Lo$\grave{e}$ve expansion in a fashion consistent with a specified marginal distribution function and an autocorrelation function. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses were performed to verify the application potential of the proposed method and to study the effects of uncertainty due to the spatial heterogeneity on the seepage behavior of soil foundation beneath water retaining structure with a single sheet pile wall. The results showed that the probabilistic framework can be used to efficiently consider the various flow patterns caused by the spatial variability of the soil permeability in seepage assessment for a soil foundation beneath water retaining structures.

• Journal of the Korean Geotechnical Society
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• v.25 no.12
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• pp.13-25
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• 2009

In this paper, a numerical procedure of probabilistic slope stability analysis that considers the spatial variability of soil properties is presented. The procedure extends the deterministic analysis based on the limit equilibrium method of slices to a probabilistic approach that accounts for the uncertainties and spatial variation of the soil parameters. Making no a priori assumptions about the critical failure surface like the Random Finite Element Method (RFEM), the approach saves the amount of solution time required to perform the analysis. Two-dimensional random fields are generated based on a Karhunen-Lo$\grave{e}$ve expansion in a fashion consistent with a specified marginal distribution function and an autocorrelation function. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses were performed to verify the application potential of the proposed method and to study the effects of uncertainty caused by the spatial heterogeneity on the stability of slope. The results show that the proposed method can efficiently consider the various failure mechanisms caused by the spatial variability of soil property in the probabilistic slope stability assessment.

• Journal of the Korean Institute of Telematics and Electronics
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• v.17 no.1
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• pp.1-9
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• 1980

In this paper several basic techniques for signal processing and analysis are surveyed. Firstly by the intervention of the uncertainty principle, an equality sign may have different degree of precision if non commutable operators are applied. Seconds y maximum entropy estimate and randam process based viewpoint must be enhanced to get rid of the well established and reigning deterministic image of science. Thirdly techniques for the analysis of a signal namely detection. ess]motion and modulation are explained as well as the positive definiteness of a covariance function, Karhunen-Loeve expansion and SVD of an image.

• Journal of the Korean Geotechnical Society
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• v.28 no.12
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• pp.65-76
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• 2012

In this study, probabilistic analysis of seepage through a two-layered soil foundation was performed. The hydraulic conductivity of soil shows significant spatial variations in different layers because of stratification; further, it varies on a smaller scale within each individual layer. Therefore, the deterministic seepage analysis method was extended to develop a probabilistic approach that accounts for the uncertainties and spatial variation of the hydraulic conductivity in a layered soil profile. Two-dimensional random fields were generated on the basis of the Karhunen-Lo$\grave{e}$ve expansion in a manner consistent with a specified marginal distribution function and an autocorrelation function for each layer. A Monte Carlo simulation was then used to determine the statistical response based on the random fields. A series of analyses were performed to verify the application potential of the proposed method and to study the effects of uncertainty due to the spatial heterogeneity on the seepage behavior of two-layered soil foundation beneath water retaining structure. The results showed that the probabilistic framework can be used to efficiently consider the various flow patterns caused by the spatial variability of the hydraulic conductivity in seepage assessment for a layered soil foundation.

• Journal of the Korean Geotechnical Society
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• v.24 no.8
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• pp.111-123
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• 2008

Geotechnical engineering problems are characterized by many sources of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the analysis. In this paper, a numerical procedure for a probabilistic analysis that considers the spatial variability of soil properties is presented to study the response of spatially random soil. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. Two-dimensional non-Gaussian random fields are generated based on a Karhunen-$Lo{\grave{e}}ve$ expansion in a fashion consistent with a specified marginal distribution function and an autocorrelation function. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses were performed to study the effects of uncertainty due to the spatial heterogeneity on the settlement and bearing capacity of a rough strip footing. The simulations provide insight into the application of uncertainty treatment to the geotechnical problem and show the importance of the spatial variability of soil properties with regard to the outcome of a probabilistic assessment.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.4
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• pp.387-407
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• 2015

This paper analyzes the $h{\times}p$ version of the finite element method for optimal control problems constrained by elliptic partial differential equations with random inputs. The main result is that the $h{\times}p$ error bound for the control problems subject to stochastic partial differential equations leads to an exponential rate of convergence with respect to p as for the corresponding direct problems. Numerical examples are used to confirm the theoretical results.