Calculating extinction probabilities for the birth and death chain in a random environment

1979 ◽  
Vol 16 (4) ◽  
pp. 709-720 ◽  
Author(s):  
William C. Torrez
Keyword(s):  
State Space ◽  
Random Environment ◽  
Previous Investigation ◽  
Transition Probabilities ◽  
Finite Difference Methods ◽  
Fixed Sequence ◽  
Marginal Process ◽  
Difference Methods ◽  
Birth And Death Chain ◽  
Irreducible Markov Chain

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

Calculating extinction probabilities for the birth and death chain in a random environment

1979 ◽  
Vol 16 (04) ◽  
pp. 709-720 ◽  
Author(s):  
William C. Torrez
Keyword(s):  
State Space ◽  
Random Environment ◽  
Previous Investigation ◽  
Transition Probabilities ◽  
Finite Difference Methods ◽  
Fixed Sequence ◽  
Marginal Process ◽  
Difference Methods ◽  
Birth And Death Chain ◽  
Irreducible Markov Chain

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn ) evolving in a random environment controlled by an irreducible Markov chain (Yn ) with state space 𝒴 The process (Yn, Zn ) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn ) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn ). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn )-conditioned process (Zn ) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

A note on the convergence of fuzzy transformed finite difference methods

2020 ◽  
Vol 63 (1-2) ◽  
pp. 143-170 ◽  
Author(s):  
Amit K. Verma ◽  
Sheerin Kayenat ◽  
Gopal Jee Jha
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Difference Methods

On the relationship between the finite element and finite difference methods

1976 ◽  
Vol 10 (4) ◽  
pp. 893-923 ◽  
Author(s):  
William F. Gray ◽  
George F. Pinder
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Difference Methods ◽  
The Relationship
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