scholarly journals Rate of convergence of a two-scale expansion for some “weakly” stochastic homogenization problems

2012 ◽  
Vol 80 (3-4) ◽  
pp. 237-267
Author(s):  
Claude Le Bris ◽  
Frédéric Legoll ◽  
Florian Thomines
Keyword(s):  
Rate Of Convergence ◽  
Stochastic Homogenization ◽  
Scale Expansion ◽  
Homogenization Problems

HOMOGENIZATION OF FIRST-ORDER EQUATIONS WITH u/∊-PERIODIC HAMILTONIAN: RATE OF CONVERGENCE AS ∊ → 0 AND NUMERICAL METHODS

2011 ◽  
Vol 21 (06) ◽  
pp. 1317-1353 ◽  
Author(s):  
YVES ACHDOU ◽  
STEFANIA PATRIZI
Keyword(s):  
Numerical Methods ◽  
Rate Of Convergence ◽  
Effective Hamiltonian ◽  
First Order ◽  
Jacobi Equations ◽  
Homogenization Problems

We consider homogenization problems for first-order Hamilton–Jacobi equations with u∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

Products of 2 × 2 stochastic matrices with random entries

1986 ◽  
Vol 23 (04) ◽  
pp. 1019-1024
Author(s):  
Walter Van Assche
Keyword(s):  
Rate Of Convergence ◽  
Beta Distribution ◽  
Stopping Time ◽  
Stochastic Matrices

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

2020 ◽  
Vol 20 (4) ◽  
pp. 783-798
Author(s):  
Shukai Du ◽  
Nailin Du
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Convergence Conditions ◽  
Principal Angles ◽  
Factorization Formula ◽  
Ill Posed

AbstractWe give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between {T^{\dagger}T(X_{n})} and {T^{*}T(X_{n})}, and the principal angles between {X_{n}\cap(\mathcal{N}(T)\cap X_{n})^{\perp}} and {(\mathcal{N}(T)+X_{n})\cap\mathcal{N}(T)^{\perp}}. At the end, we consider several specific cases and examples to further illustrate our theorems.

scholarly journals Self-exciting multifractional processes

2021 ◽  
Vol 58 (1) ◽  
pp. 22-41
Author(s):  
Fabian A. Harang ◽  
Marc Lagunas-Merino ◽  
Salvador Ortiz-Latorre
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Rate Of Convergence ◽  
Existence And Uniqueness ◽  
Volterra Equation ◽  
The Self ◽  
The Past ◽  
Stochastic Volterra Equation ◽  
Multifractional Brownian Motion ◽  
Self Organizing

AbstractWe propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all $p\geq1$. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

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