scholarly journals Lipschitz conditions for random processes from L_p(Ω) spaces of random variables

2015 ◽  
pp. 59-72
Author(s):  
Dmytro Zatula
Keyword(s):  
Random Variables ◽  
Random Processes ◽  
Lipschitz Conditions

Plate and line segment processes

1978 ◽  
Vol 15 (03) ◽  
pp. 494-501 ◽  
Author(s):  
N. A. Fava ◽  
L. A. Santaló
Keyword(s):  
Line Segment ◽  
Integral Geometry ◽  
Random Variables ◽  
Random Processes ◽  
Line Segments ◽  
Expected Values

Random processes of convex plates and line segments imbedded in R 3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.

A practical comparison between the spectral techniques in solving the SDEs

2019 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mohamed El-Beltagy
Keyword(s):  
Infinite Number ◽  
Random Variables ◽  
Nonlinear Problems ◽  
Hermite Expansion ◽  
Content Type ◽  
Spectral Techniques ◽  
Burgers Turbulence ◽  
Non Gaussian ◽  
Convergence Orders ◽  
Lipschitz Conditions

Purpose The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes. Design/methodology/approach The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques. Findings The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems. Research limitations/implications Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions. Originality/value This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Vol 143 (6) ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

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