751 research outputs found
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4 Based on Generalized Multiple Fourier Series
The article is devoted to the expansions of iterated Stratonovich stochastic
integrals of multiplicities 1 to 4 on the basis of the method of generalized
multiple Fourier series that are converge in the sense of norm in Hilbert space
Mean-square convergence of the expansions for the
case of multiple Fourier-Legendre series and for the case of multiple
trigonometric Fourier series is proved. The considered expansions contain only
one operation of the limit transition in contrast to its existing analogues.
This property is very important for the mean-square approximation of iterated
stochastic integrals. The results of the article can be applied to numerical
integration of Ito stochastic differential equations with multidimensional
non-commutative noises.Comment: 50 pages. Sect. 6 is added. Minor changes along the article in the
whole. arXiv admin note: text overlap with arXiv:1712.0899
Application of ensemble transform data assimilation methods for parameter estimation in reservoir modelling
Over the years data assimilation methods have been developed to obtain
estimations of uncertain model parameters by taking into account a few
observations of a model state. The most reliable methods of MCMC are
computationally expensive. Sequential ensemble methods such as ensemble Kalman
filers and particle filters provide a favourable alternative. However, Ensemble
Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle
Filter does not have this assumption and has proven to be highly beneficial for
an initial condition estimation and a small number of parameter estimation in
chaotic dynamical systems with non-Gaussian distributions. In this paper we
employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman
Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and
2500 uncertain parameters and compare them to importance sampling (IS). We
prove that the updated parameters obtained by ETPF lie within the range of an
initial ensemble, which is not the case for ETKF. We examine the performance of
ETPF and ETKF in a twin experiment setup and observe that for a small number of
uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the
mean estimation. For a large number of uncertain parameters (2500) ETKF is
robust with respect to the initial ensemble while ETPF is sensitive due to
sampling error. Moreover, for the high-dimensional test problem ETPF gives an
increase in the root mean square error after data assimilation is performed.
This is resolved by applying distance-based localization, which however
deteriorates a posterior estimation of the leading mode by largely increasing
the variance. A possible remedy is instead of applying localization to use only
leading modes that are well estimated by ETPF, which demands a knowledge at
which mode to truncate
Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems
This work focuses on providing accurate low-cost approximations of stochastic ¿nite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments.Postprint (author's final draft
Response to Comments on PCA Based Hurst Exponent Estimator for fBm Signals Under Disturbances
In this response, we try to give a repair to our previous proof for PCA Based
Hurst Exponent Estimator for fBm Signals by using orthogonal projection.
Moreover, we answer the question raised recently: If a centered Gaussian
process admits two series expansions on different Riesz bases, we may
possibly study the asymptotic behavior of one eigenvalue sequence from the
knowledge on the asymptotic behaviors of another.Comment: This is a response for a mistake in Li Li, Jianming Hu, Yudong Chen,
Yi Zhang, PCA based Hurst exponent estimator for fBm signals under
disturbances, IEEE Transactions on Signal Processing, vol. 57, no. 7, pp.
2840-2846, 200
An efficient polynomial chaos-based proxy model for history matching and uncertainty quantification of complex geological structures
A novel polynomial chaos proxy-based history matching and uncertainty quantification
method is presented that can be employed for complex geological structures in inverse
problems. For complex geological structures, when there are many unknown geological
parameters with highly nonlinear correlations, typically more than 106 full reservoir
simulation runs might be required to accurately probe the posterior probability space
given the production history of reservoir. This is not practical for high-resolution geological
models. One solution is to use a "proxy model" that replicates the simulation
model for selected input parameters. The main advantage of the polynomial chaos
proxy compared to other proxy models and response surfaces is that it is generally
applicable and converges systematically as the order of the expansion increases. The
Cameron and Martin theorem 2.24 states that the convergence rate of the standard
polynomial chaos expansions is exponential for Gaussian random variables. To improve
the convergence rate for non-Gaussian random variables, the generalized polynomial
chaos is implemented that uses an Askey-scheme to choose the optimal basis for polynomial
chaos expansions [199]. Additionally, for the non-Gaussian distributions that
can be effectively approximated by a mixture of Gaussian distributions, we use the
mixture-modeling based clustering approach where under each cluster the polynomial
chaos proxy converges exponentially fast and the overall posterior distribution can be
estimated more efficiently using different polynomial chaos proxies.
The main disadvantage of the polynomial chaos proxy is that for high-dimensional problems,
the number of the polynomial chaos terms increases drastically as the order of the
polynomial chaos expansions increases. Although different non-intrusive methods have
been developed in the literature to address this issue, still a large number of simulation
runs is required to compute high-order terms of the polynomial chaos expansions. This
work resolves this issue by proposing the reduced-terms polynomial chaos expansion
which preserves only the relevant terms in the polynomial chaos representation. We
demonstrated that the sparsity pattern in the polynomial chaos expansion, when used
with the Karhunen-Loéve decomposition method or kernel PCA, can be systematically
captured.
A probabilistic framework based on the polynomial chaos proxy is also suggested in the
context of the Bayesian model selection to study the plausibility of different geological
interpretations of the sedimentary environments. The proposed surrogate-accelerated
Bayesian inverse analysis can be coherently used in practical reservoir optimization
workflows and uncertainty assessments
Karhunen-Loeve representation of stochastic ocean waves
A new stochastic representation of a seastate is developed based on the Karhunen–Loeve spectral decomposition of stochastic signals and the use of Slepian prolate spheroidal wave functions with a tunable bandwidth parameter. The new representation allows the description of stochastic ocean waves in terms of a few independent sources of uncertainty when the traditional representation of a seastate in terms of Fourier series requires an order of magnitude more independent components. The new representation leads to parsimonious stochastic models of the ambient wave kinematics and of the nonlinear loads and responses of ships and offshore platforms. The use of the new representation is discussed for the derivation of critical wave episodes, the derivation of up-crossing rates of nonlinear loads and responses and the joint stochastic representation of correlated wave and wind profiles for use in the design of fixed or floating offshore wind turbines. The forecasting is also discussed of wave elevation records and vessel responses for use in energy yield enhancement of compliant floating wind turbines.ALSTOM (Firm)Ente nazionale per l'energia elettricab_TE
Characteristic eddy decomposition of turbulence in a channel
The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions, a generaliztion of the shot-noise expansion is used to determine the characteristic eddies. In this expansion, the Fourier coefficients of the characteristic eddy cannot be obtained from the second-order statistics. Three different techniques are used to determine the phases of these coefficients. They are based on: (1) the bispectrum, (2) a spatial compactness requirement, and (3) a functional continuity argument. Results from these three techniques are found to be similar in most respects. The implications of these techniques and the shot-noise expansion are discussed. The dominant eddy is found to contribute as much as 76 percent to the turbulent kinetic energy. In both 2D and 3D, the characteristic eddies consist of an ejection region straddled by streamwise vortices that leave the wall in the very short streamwise distance of about 100 wall units
A spectral-based numerical method for Kolmogorov equations in Hilbert spaces
We propose a numerical solution for the solution of the
Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial
differential equations in Hilbert spaces.
The method is based on the spectral decomposition of the Ornstein-Uhlenbeck
semigroup associated to the Kolmogorov equation. This allows us to write the
solution of the Kolmogorov equation as a deterministic version of the
Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov
equation as a infinite system of ordinary differential equations, and by
truncation it we set a linear finite system of differential equations. The
solution of such system allow us to build an approximation to the solution of
the Kolmogorov equations. We test the numerical method with the Kolmogorov
equations associated with a stochastic diffusion equation, a Fisher-KPP
stochastic equation and a stochastic Burgers Eq. in dimension 1.Comment: 28 pages, 10 figure
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