In multi-user multiple input multiple output （MU-MIMO） systems, the outdated channel state information at the transmit- ter caused by channel time variation has been shown to greatly reduce the achievable ergodic su...In multi-user multiple input multiple output （MU-MIMO） systems, the outdated channel state information at the transmit- ter caused by channel time variation has been shown to greatly reduce the achievable ergodic sum capacity. A simple yet effec- tive solution to this problem is presented by designing a channel extrapolator relying on Karhunen-Loeve （KL） expansion of time- varying channels. In this scheme, channel estimation is done at the base station （BS） rather than at the user terminal （UT）, which thereby dispenses the channel parameters feedback from the UT to the BS. Moreover, the inherent channel correlation and the parsimonious parameterization properties of the KL expan- sion are respectively exploited to reduce the channel mismatch error and the computational complexity. Simulations show that the presented scheme outperforms conventional schemes in terms of both channel estimation mean square error （MSE） and ergodic capacity.展开更多
This paper develops a trigonometric-basis-fimction based Karhunen-Loeve （KL） expansion for simulating random earthquake excitations with known covariance functions. The methods for determining the number of the KL t...This paper develops a trigonometric-basis-fimction based Karhunen-Loeve （KL） expansion for simulating random earthquake excitations with known covariance functions. The methods for determining the number of the KL terms and defining the involved random variables are described in detail. The simplified form of the KL expansion is given, whereby the relationship between the KL expansion and the spectral representation method is investigated and revealed. The KL expansion is of high efficiency for simulating long-term earthquake excitations in the sense that it needs a minimum number of random variables, as compared with the spectral representation method. Numerical examples demonstrate the convergence and accuracy of the KL expansion for simulating two commonly-used random earthquake excitation models and estimating linear and nonlinear random responses to the random excitations.展开更多
In this paper,we present an adaptive,analysis of variance(ANOVA)-based data-driven stochastic method(ANOVA-DSM)to study the stochastic partial differential equations(SPDEs)in the multi-query setting.Our new method int...In this paper,we present an adaptive,analysis of variance(ANOVA)-based data-driven stochastic method(ANOVA-DSM)to study the stochastic partial differential equations(SPDEs)in the multi-query setting.Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method.To handle high-dimensional stochastic problems,we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique.To improve the slow convergence of the generalized polynomial chaos(gPC)method or stochastic collocation(SC)method,we adopt the data-driven stochastic method(DSM)for speed up.An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages.In the offline stage,the original high-dimensional stochastic problem is decomposed into a series of lowdimensional stochastic subproblems,according to the ANOVA decomposition technique.Then,for each subproblem,a data-driven stochastic basis is computed using the Karhunen-Lo`eve expansion(KLE)and a two-level preconditioning optimization approach.Multiple trial functions are used to enrich the stochastic basis and improve the accuracy.In the online stage,we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline.In our ANOVA-DSM framework,solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM.An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method.To demonstrate the accuracy and efficiency of our method,numerical examples are presented for one-and two-dimensional elliptic PDEs with random coefficients.展开更多
The Karhunen-Loeve （KL） expansion and probabilistic collocation method （PCM） are combined and applied to an uncertainty analysis of rock failure behavior by integrating a self- developed numerical method （i.e., t...The Karhunen-Loeve （KL） expansion and probabilistic collocation method （PCM） are combined and applied to an uncertainty analysis of rock failure behavior by integrating a self- developed numerical method （i.e., the elastic-plastic cellular automaton （EPCA））. The results from the method developed are compared using the Monte Carlo Simulation （MCS） method. It is concluded that the method developed requires fewer collocations than MCS method to obtain very high accuracy and greatly reduces the computational cost. Based on the method, the elasto- plastic and elasto-brittle-plastic analyses of rocks under mechanical loadings are conducted to study the uncertainty in heterogeneous rock failure behaviour.展开更多
A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method(PCM).The conditional lo...A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method(PCM).The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion,in terms of some deterministic functions and a set of independent Gaussian random variables.The propagation of uncertainty in the flow simulations is carried out through the PCM,which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head.With the PCM,existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration.With illustration of several numerical examples of groundwater flow,this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data,while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.展开更多
A stochastic model was developed to simulate the flow in heterogeneous media subject to random boundary conditions. Approximate partial differential equations were derived based on the Karhunen-Loeve （KL） expansion ...A stochastic model was developed to simulate the flow in heterogeneous media subject to random boundary conditions. Approximate partial differential equations were derived based on the Karhunen-Loeve （KL） expansion and perturbation expansion. The effect of random boundary conditions on the two-dimensional flow was examined. It is shown that the proposed stochastic model is efficient to include the random boundary conditions. The random boundaries lead to the increase of head variance and velocity variance. The influence of the random boundary conditions on head uncertainty is exerted over the whole simulated region, while the randomness of the boundary conditions leads to the increase of the velocity variance in the vicinity of boundaries.展开更多
A stochastic model for saturated-unsaturated flow is developed based on the combination of the Karhunen-Loeve expansion of the input random soil properties with a perturbation method. The saturated hydraulic conductiv...A stochastic model for saturated-unsaturated flow is developed based on the combination of the Karhunen-Loeve expansion of the input random soil properties with a perturbation method. The saturated hydraulic conductivity k_ s (x) is assumed to be log-normal random functions, expressed by f(x). f(x) is decomposed as infinite series in a set of orthogonal normal random variables by the Karhunen-Loeve (KL) expansion and the pressure head is expand as polynomial chaos with the same set of orthogonal random variables. With these expansions, the stochastic saturated-unsaturated flow equation and the corresponding initial and boundary conditions are represented by a series of deterministic partial differential equations which can be solved subsequently by a suitable numerical method. Some examples are given to show the reliability and efficiency of the proposed method.展开更多
基金supported by the National Natural Science Foundation of China (6096200161071088)+2 种基金the Natural Science Foundation of Fujian Province of China (2012J05119)the Fundamental Research Funds for the Central Universities (11QZR02)the Research Fund of Guangxi Key Lab of Wireless Wideband Communication & Signal Processing (21104)
文摘In multi-user multiple input multiple output （MU-MIMO） systems, the outdated channel state information at the transmit- ter caused by channel time variation has been shown to greatly reduce the achievable ergodic sum capacity. A simple yet effec- tive solution to this problem is presented by designing a channel extrapolator relying on Karhunen-Loeve （KL） expansion of time- varying channels. In this scheme, channel estimation is done at the base station （BS） rather than at the user terminal （UT）, which thereby dispenses the channel parameters feedback from the UT to the BS. Moreover, the inherent channel correlation and the parsimonious parameterization properties of the KL expan- sion are respectively exploited to reduce the channel mismatch error and the computational complexity. Simulations show that the presented scheme outperforms conventional schemes in terms of both channel estimation mean square error （MSE） and ergodic capacity.
文摘This paper develops a trigonometric-basis-fimction based Karhunen-Loeve （KL） expansion for simulating random earthquake excitations with known covariance functions. The methods for determining the number of the KL terms and defining the involved random variables are described in detail. The simplified form of the KL expansion is given, whereby the relationship between the KL expansion and the spectral representation method is investigated and revealed. The KL expansion is of high efficiency for simulating long-term earthquake excitations in the sense that it needs a minimum number of random variables, as compared with the spectral representation method. Numerical examples demonstrate the convergence and accuracy of the KL expansion for simulating two commonly-used random earthquake excitation models and estimating linear and nonlinear random responses to the random excitations.
基金AFOSR MURI project under Contract FA 9550-09-1-0613,a DOE Grant DE-FG02-06ER25727,and NSF FRG Grant DMS-1159138.the U.S.Department of Energy(DOE)Office of Science Advanced Scientific Computing Research Applied Mathematics program.Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.
文摘In this paper,we present an adaptive,analysis of variance(ANOVA)-based data-driven stochastic method(ANOVA-DSM)to study the stochastic partial differential equations(SPDEs)in the multi-query setting.Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method.To handle high-dimensional stochastic problems,we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique.To improve the slow convergence of the generalized polynomial chaos(gPC)method or stochastic collocation(SC)method,we adopt the data-driven stochastic method(DSM)for speed up.An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages.In the offline stage,the original high-dimensional stochastic problem is decomposed into a series of lowdimensional stochastic subproblems,according to the ANOVA decomposition technique.Then,for each subproblem,a data-driven stochastic basis is computed using the Karhunen-Lo`eve expansion(KLE)and a two-level preconditioning optimization approach.Multiple trial functions are used to enrich the stochastic basis and improve the accuracy.In the online stage,we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline.In our ANOVA-DSM framework,solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM.An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method.To demonstrate the accuracy and efficiency of our method,numerical examples are presented for one-and two-dimensional elliptic PDEs with random coefficients.
基金supported by the National Natural Science Foundation of China(Nos.51322906 and 41272349)the National Basic Research Program of China(No.2013CB036405)Youth Innovation Promotion Association of CAS(No.2011240)
文摘The Karhunen-Loeve （KL） expansion and probabilistic collocation method （PCM） are combined and applied to an uncertainty analysis of rock failure behavior by integrating a self- developed numerical method （i.e., the elastic-plastic cellular automaton （EPCA））. The results from the method developed are compared using the Monte Carlo Simulation （MCS） method. It is concluded that the method developed requires fewer collocations than MCS method to obtain very high accuracy and greatly reduces the computational cost. Based on the method, the elasto- plastic and elasto-brittle-plastic analyses of rocks under mechanical loadings are conducted to study the uncertainty in heterogeneous rock failure behaviour.
基金the National Science and Technology Major Project of China through Grants 2011ZX05009-006 and 2011ZX05052the National Key Technology R&D Program of China(Grant No.2012BAC24B00)the National Natural Science Foundation of China(Grant No.51204008)。
文摘A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method(PCM).The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion,in terms of some deterministic functions and a set of independent Gaussian random variables.The propagation of uncertainty in the flow simulations is carried out through the PCM,which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head.With the PCM,existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration.With illustration of several numerical examples of groundwater flow,this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data,while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.
基金the National Natural Science Foundation of China ( Grant Nos. 40672164, 50379039).
文摘A stochastic model was developed to simulate the flow in heterogeneous media subject to random boundary conditions. Approximate partial differential equations were derived based on the Karhunen-Loeve （KL） expansion and perturbation expansion. The effect of random boundary conditions on the two-dimensional flow was examined. It is shown that the proposed stochastic model is efficient to include the random boundary conditions. The random boundaries lead to the increase of head variance and velocity variance. The influence of the random boundary conditions on head uncertainty is exerted over the whole simulated region, while the randomness of the boundary conditions leads to the increase of the velocity variance in the vicinity of boundaries.
文摘A stochastic model for saturated-unsaturated flow is developed based on the combination of the Karhunen-Loeve expansion of the input random soil properties with a perturbation method. The saturated hydraulic conductivity k_ s (x) is assumed to be log-normal random functions, expressed by f(x). f(x) is decomposed as infinite series in a set of orthogonal normal random variables by the Karhunen-Loeve (KL) expansion and the pressure head is expand as polynomial chaos with the same set of orthogonal random variables. With these expansions, the stochastic saturated-unsaturated flow equation and the corresponding initial and boundary conditions are represented by a series of deterministic partial differential equations which can be solved subsequently by a suitable numerical method. Some examples are given to show the reliability and efficiency of the proposed method.
