scholarly journals Weak Approximations of the Wright–Fisher Process

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 125
Author(s):  
Vigirdas Mackevičius ◽  
Gabrielė Mongirdaitė
Keyword(s):  
Random Variables ◽  
Second Order ◽  
Fisher Equation ◽  
Discretization Schemes ◽  
Weak Approximations ◽  
Discretization Step ◽  
Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

scholarly journals Second-Order Weak Approximations of CKLS and CEV Processes by Discrete Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1337
Author(s):  
Gytenis Lileika ◽  
Vigirdas Mackevičius
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Second Order ◽  
First Order ◽  
Discrete Random Variables ◽  
Weak Approximations ◽  
Discretization Step ◽  
Cir Process

In this paper, we construct second-order weak split-step approximations of the CKLS and CEV processes that use generation of a three−valued random variable at each discretization step without switching to another scheme near zero, unlike other known schemes (Alfonsi, 2010; Mackevičius, 2011). To the best of our knowledge, no second-order weak approximations for the CKLS processes were constructed before. The accuracy of constructed approximations is illustrated by several simulation examples with comparison with schemes of Alfonsi in the particular case of the CIR process and our first-order approximations of the CKLS processes (Lileika– Mackevičius, 2020).

scholarly journals Weak approximations of Wright-Fisher equation

2021 ◽  
Vol 62 ◽  
pp. 23-26
Author(s):  
Gabrielė Mongirdaitė ◽  
Vigirdas Mackevičius
Keyword(s):  
Fisher Equation ◽  
Fisher Model ◽  
Weak Approximations

We construct weak approximations of the Wright-Fisher model and illustrate their accuracy by simulation examples.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

An Iterative Monte Carlo Scheme for Generating Lie Group-Valued Random Variables

1994 ◽  
Vol 26 (03) ◽  
pp. 616-628
Author(s):  
Mauro Piccioni ◽  
Sergio Scarlatti
Keyword(s):  
Monte Carlo ◽  
Lie Group ◽  
Wide Class ◽  
Approximation Scheme ◽  
Random Variables ◽  
Time Discretization ◽  
Compact Lie Group ◽  
Discretization Step ◽  
Ergodic Diffusion ◽  
Natural Way

In this paper a simple approximation scheme is proposed for the problem of generating and computing expectations of functionals of a wide class of random variables with values in a compact Lie group. The algorithm is suggested by the time-discretization of an ergodic diffusion leaving invariant the distribution of interest. It is shown to converge as the discretization step goes to zero with the iterations in a natural way.

Characterization of a class of second-order density functions

1967 ◽  
Vol 4 (1) ◽  
pp. 123-129 ◽  
Author(s):  
C. B. Mehr
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Density Functions ◽  
Linear Filtering ◽  
Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

Resonance Reliability Analysis of Aeroengine Compressor Rotor Blade

2014 ◽  
Vol 472 ◽  
pp. 79-84
Author(s):  
Hai Feng Gao ◽  
Guang Chen Bai
Keyword(s):  
Vibration Frequency ◽  
Rotor Blade ◽  
Random Variables ◽  
Second Order ◽  
Rotor Blades ◽  
Rotor Speed ◽  
Blade Vibration ◽  
Compressor Rotor ◽  
Set Up ◽  
The Impact

To describe the frequency distribution of the rotor blades and improve the optimization, resonance reliability of the rotor blades was analyzed in this paper. Considering the variety of rand-om variables, we jointly used finite element method and response surface method. The Campbell diagram was set up to describe blade resonance by analyzing the compressor rotor blade vibration characteristics. For the second-order vibration failure of the rotor blade, we considered the impact of random variables with the rotor blade material, the blade dimension and the rotor speed. The pro-bability distribution and allowable reliability of the second-order vibration frequency was calculated, and the sensitivity of the random variables influencing vibration frequency was completed. The res-ults show that the resonance reliability with the confidence level 0.95 of the rotor blade are = 0.99753 with the excited order =4 and =0.99767 with the excited order =5,and basically ag-ree with the design requirements when the rotor speed =9916.2, and the factors mainly affe-cting the distribution of the second-order vibration frequency of the blades include elastic modulus, density and the rotor speed, with the sensitivity probabilities 35.09%,34.56% and 24.15% respecti-vely.

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