Approximations of analytic and differentiable semigroups — Rate of convergence with nonsmooth initial conditions

1984 ◽  
pp. 123-138 ◽  
Author(s):  
I. Lasiecka
Keyword(s):  
Rate Of Convergence ◽  
Initial Conditions

ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS

2003 ◽  
Vol 13 (05) ◽  
pp. 613-644 ◽  
Author(s):  
ESPEN ROBSTAD JAKOBSEN
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Optimal Stopping ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
Approximation Schemes ◽  
Dynamic Programming Principle ◽  
Bellman Equations ◽  
Difficult Time ◽  
Difference Methods

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

scholarly journals UNIT ROOT AND COINTEGRATING LIMIT THEORY WHEN INITIALIZATION IS IN THE INFINITE PAST

2009 ◽  
Vol 25 (6) ◽  
pp. 1682-1715 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Tassos Magdalinos
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Rate Of Convergence ◽  
Unit Root ◽  
Initial Conditions ◽  
Cauchy Distribution ◽  
Least Squares Estimator ◽  
Initial Condition ◽  
Limit Theory ◽  
Regression Theory

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

Focusing of weak shock waves at an arête

1978 ◽  
Vol 88 (2) ◽  
pp. 209-222 ◽  
Author(s):  
M. S. Cramer ◽  
A. R. Seebass
Keyword(s):  
Shock Waves ◽  
Rate Of Convergence ◽  
Wave Front ◽  
Initial Conditions ◽  
Pressure Coefficient ◽  
Weak Shock ◽  
Linear Solution ◽  
Initial Strength ◽  
Maximum Value

The focusing of very weak and slightly concave symmetrical shock waves is examined. The equation that describes this focusing is derived and the resulting similitude discussed. The initial conditions come from a formal matching of this nonlinear description with the linear solution. The maximum value of the pressure coefficient is shown to be proportional to the two-thirds power of both the initial strength of the wave front and a parameter characterizing its rate of convergence.

CLOSE-PACKING OF GROWING DISCS

1988 ◽  
Vol 02 (11n12) ◽  
pp. 1245-1252 ◽  
Author(s):  
L.A. BURSILL ◽  
FAN XUDONG
Keyword(s):  
Rate Of Convergence ◽  
Initial Conditions ◽  
Close Packing ◽  
Lucas Numbers ◽  
Growth Centre ◽  
Initial Placement ◽  
Fibonacci And Lucas Numbers

Spiral lattices are derived by allowing growing discs to aggregate under a close-packing rule. Both Fibonacci and Lucas numbers of visible spirals arise naturally, dependent only on the choice of growth centre. Both the rate of convergence towards an ideal spiral, and chirality, are determined by the initial placement of the first few discs (initial conditions). Thus the appearance of spiral packings is no more or less mysterious than the appearance of hexagonal packed arrays of equal discs.

scholarly journals Global Dynamics of a Higher Order Difference Equation with a Quadratic Term

2020 ◽  
Author(s):  
Erkan Taşdemir
Keyword(s):  
Difference Equation ◽  
Rate Of Convergence ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Quadratic Term ◽  
Global Dynamics ◽  
Order Difference ◽  
Convergence Of Solutions ◽  
Order Difference Equation

In this paper, we investigate the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n})/(x_{n-m}²)) with A,B and initial conditions are positive numbers, and m∈{2,3,⋯}. Especially we study the boundedness, periodicity, semi-cycles, global asymptotically stability and rate of convergence of solutions of related higher order difference equations.

scholarly journals Global Dynamics of a Higher Order Difference Equation with Quadratic Term

2020 ◽  
Author(s):  
Erkan Taşdemir
Keyword(s):  
Difference Equation ◽  
Rate Of Convergence ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Quadratic Term ◽  
Global Dynamics ◽  
Order Difference ◽  
Order Difference Equation ◽  
Global Asymptotically Stability

In this paper, we investigate the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n})/(x_{n-m}²)) with A,B and initial conditions are positive numbers. Especially we study the boundedness, periodicity, global asymptotically stability and rate of convergence of related higher order difference equations.

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.

Assessment of Embedding Dimension and Sensitivity Initial Conditions Based on Statistical Tests

2006 ◽  
Author(s):  
Xavier Rifa-Ros ◽  
Manel Viader-Junyen
Keyword(s):  
Initial Conditions ◽  
Statistical Tests ◽  
Embedding Dimension
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