Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: The problem of the synthesis of a feedback control, assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence, is investigated and it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets.
Abstract: In this note, linear discrete-time systems affected by both parameter and input uncertainties are considered. The problem of the synthesis of a feedback control, assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence, is investigated. It is shown that the problem has a solution if and only if there exists a certain Lyapunov function which does not belong to a preassigned class of functions (e.g., the quadratic ones), but it is determined by the target set in which ultimate boundedness is desired. One of the advantages of this approach is that we may handle systems with control constraints. No matching assumptions are made. For systems with linearly constrained uncertainties, it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets. The resulting compensator may be implemented as a linear variable-structure control. >
431 citations
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TL;DR: In this paper, an upper bound estimate of the rate of convergence to purchasing power parity was provided using a panel of 51 prices from 48 cities in the United States, and they investigated some potentially serious biases induced by i.i.d. measurement errors in the data and found their estimates to be robust to these potential biases.
Abstract: Using a panel of 51 prices from 48 cities in the United States we provide an upper bound estimate of the rate of convergence to Purchasing Power Parity. We find convergence rates substantially higher than typically found in cross-country data. We investigate some potentially serious biases induced by i.i.d. measurement errors in the data, and find our estimates to be robust to these potential biases. We also present evidence that convergence occurs faster for larger price differences. Finally, we find that rates of convergence are slower for cities farther apart. However, our estimates suggest that distance alone can only account for a small portion of the much slower convergence rates across national borders.
431 citations
TL;DR: The present paper demonstrates how the additive correction method of Settari and Aziz can be used and extended to improve the convergence rate for two- and three-dimensional problems when the coefficients are anisotropic.
Abstract: The solution of large sets of equations is required when discrete methods are used to solve fluid flow and heat transfer problems The cost of the solution often becomes prohibitive when the coefficients of the algebraic equations become strongly anisotropic or when the number of equations in the set becomes large The present paper demonstrates how the additive correction method of Settari and Aziz can be used and extended to improve the convergence rate for two- and three-dimensional problems when the coefficients are anisotropic Such methods are interpreted as simple multigrid methods With this as the basis a new general multigrid method is developed that has attractive properties The efficiency of the new method is compared to that of a conventional multigrid method, and its performance is demonstrated on other problems
429 citations
TL;DR: In this article, a direct design procedure of a full-order observer for a linear system with unknown inputs is presented, using straightforward matrix calculations; in these examples, a reduced order observer is also derived.
Abstract: A direct design procedure of a full-order observer for a linear system with unknown inputs is presented, using straightforward matrix calculations. Some examples are given; in these examples a reduced-order observer is also derived. It is shown that this may restrict the rate of convergence of some state estimates. >
428 citations
TL;DR: It is proved that the scheme converges in the strong L2 sense and its rate of convergence is derived and an L2-type regularity of the solution to such BSDEs is proved.
Abstract: In this paper we propose a numerical scheme for a class of backward stochastic differential equations (BSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^2$ sense and derive its rate of convergence. As an intermediate step we prove an $L^2$-type regularity of the solution to such BSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^2$ sense, is new. Some other features of our scheme include the following: (i) both components of the solution are approximated by step processes (i.e., piecewise constant processes); (ii) the regularity requirements on the coefficients are practically "minimum"; (iii) the dimension of the integrals involved in the approximation is independent of the partition size.
428 citations