Asymptotic Expansion for the Distribution of Nonlinear Boundary Crossing Time

1993 ◽  
Vol 37 (3) ◽  
pp. 560-564 ◽  
Author(s):  
F. G. Ragimov
Keyword(s):  
Asymptotic Expansion ◽  
Nonlinear Boundary ◽  
Boundary Crossing ◽  
Crossing Time

scholarly journals Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Pan Cheng ◽  
Ling Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Boundary Integral Equations ◽  
Logarithmic Singularity ◽  
Nonlinear Problems ◽  
Boundary Integral ◽  
Convergence Theory ◽  
Mechanical Quadrature ◽  
Quadrature Methods

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.

Boundary crossing for moving sums

1988 ◽  
Vol 25 (1) ◽  
pp. 81-88 ◽  
Author(s):  
J. Glaz ◽  
B. Johnson
Keyword(s):  
Standard Deviation ◽  
Random Variables ◽  
Boundary Crossing ◽  
Time Approximation ◽  
Crossing Time ◽  
Moving Sum

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m = Xj + · ·· + Xj+m–1, the jth moving sum. Let τ m = inf{j ≧ 1 : Sj,m > a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τm > n), E(τ m), and σ (τ m),the standard deviation of τm.

Boundary crossing probabilities for high-dimensional Brownian motion

2016 ◽  
Vol 53 (2) ◽  
pp. 543-553 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu
Keyword(s):  
Brownian Motion ◽  
Nonlinear Boundary ◽  
Boundary Crossing ◽  
High Dimensional ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
One Dimensional ◽  
Markov Chain Imbedding ◽  
Dimensional Boundary ◽  
Crossing Probabilities

Abstract The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

Boundary crossing for moving sums

1988 ◽  
Vol 25 (01) ◽  
pp. 81-88 ◽  
Author(s):  
J. Glaz ◽  
B. Johnson
Keyword(s):  
Standard Deviation ◽  
Random Variables ◽  
Boundary Crossing ◽  
Time Approximation ◽  
Crossing Time ◽  
Moving Sum

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m = Xj + · ·· + Xj+m –1 , the jth moving sum. Let τ m = inf{j ≧ 1 : Sj,m > a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τ m > n), E(τ m ), and σ (τ m ),the standard deviation of τ m .

scholarly journals Accuracy of multi-point boundary crossing time analysis

2011 ◽  
Vol 29 (12) ◽  
pp. 2239-2252 ◽  
Author(s):  
J. Vogt ◽  
S. Haaland ◽  
G. Paschmann
Keyword(s):  
Solar Wind ◽  
Error Analysis ◽  
Present Report ◽  
Minimum Variance ◽  
Boundary Crossing ◽  
Parameter Estimates ◽  
Cluster Data ◽  
Boundary Parameter ◽  
Crossing Time ◽  
The Individual

Abstract. Recent multi-spacecraft studies of solar wind discontinuity crossings using the timing (boundary plane triangulation) method gave boundary parameter estimates that are significantly different from those of the well-established single-spacecraft minimum variance analysis (MVA) technique. A large survey of directional discontinuities in Cluster data turned out to be particularly inconsistent in the sense that multi-point timing analyses did not identify any rotational discontinuities (RDs) whereas the MVA results of the individual spacecraft suggested that RDs form the majority of events. To make multi-spacecraft studies of discontinuity crossings more conclusive, the present report addresses the accuracy of the timing approach to boundary parameter estimation. Our error analysis is based on the reciprocal vector formalism and takes into account uncertainties both in crossing times and in the spacecraft positions. A rigorous error estimation scheme is presented for the general case of correlated crossing time errors and arbitrary spacecraft configurations. Crossing time error covariances are determined through cross correlation analyses of the residuals. The principal influence of the spacecraft array geometry on the accuracy of the timing method is illustrated using error formulas for the simplified case of mutually uncorrelated and identical errors at different spacecraft. The full error analysis procedure is demonstrated for a solar wind discontinuity as observed by the Cluster FGM instrument.

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