Optimal order of accuracy in Vasin's method for differentiation of noisy functions

1992 ◽  
Vol 74 (2) ◽  
pp. 373-378 ◽  
Author(s):  
C. W. Groetsch
Keyword(s):  
Optimal Order ◽  
Order Of Accuracy ◽  
Noisy Functions

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

scholarly journals COMPLEXITY ESTIMATES FOR SEVERELY ILL-POSED PROBLEMS UNDER A POSTERIORI SELECTION OF REGULARIZATION PARAMETER

2017 ◽  
Vol 22 (3) ◽  
pp. 283-299
Author(s):  
Sergii G. Solodky ◽  
Ganna L. Myleiko ◽  
Evgeniya V. Semenova
Keyword(s):  
Integral Equations ◽  
Regularization Parameter ◽  
Efficient Algorithms ◽  
Optimal Order ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Discrete Information ◽  
Ill Posed ◽  
Selection Of

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

scholarly journals An Efficient Zero-Stable Numerical Method for Fourth-Order Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
S. J. Kayode
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Numerical Method ◽  
Fourth Order ◽  
Optimal Order ◽  
Direct Solution ◽  
Order Of Accuracy ◽  
First Order ◽  
Stable Numerical Method

The purpose of this paper is to produce an efficient zero-stable numerical method with the same order of accuracy as that of the main starting values (predictors) for direct solution of fourth-order differential equations without reducing it to a system of first-order equations. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is consistent, symmetric, and of optimal order . The main predictor for the method is also consistent, symmetric, zero-stable, and of optimal order .

scholarly journals A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

2022 ◽  
Author(s):  
Poorvi Shukla ◽  
J. J. W. van der Vegt
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Mesh Size ◽  
Space Time ◽  
Basis Functions ◽  
Optimal Order ◽  
Time Step ◽  
Order Of Accuracy ◽  
Interior Penalty ◽  
Dg Method

AbstractA new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $$\Delta t$$ Δ t satisfy $$h\cong C\Delta t$$ h ≅ C Δ t , with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $$L^\infty$$ L ∞ and $$L^2$$ L 2 -norms is order $$p+1$$ p + 1 for polynomial orders $$p=1$$ p = 1 and $$p=3$$ p = 3 and order p for polynomial order $$p=2$$ p = 2 .

An algorithm for calculating Hermite-based finite difference weights

2020 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Present Author ◽  
One Dimension ◽  
Weighted Sums ◽  
Optimal Order ◽  
Order Of Accuracy ◽  
First Derivative ◽  
Matlab Code ◽  
Previous Algorithm

Abstract Finite difference (FD) formulas approximate derivatives by weighted sums of function values. Given arbitrarily distributed node locations in one-dimension, a previous algorithm by the present author (1988, Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput., 51, 699–706) provides FD weights of optimal order of accuracy for approximating any order derivative at a specified location. This algorithm is extended here to the case of finding weights to apply not only to function values but also to first derivative values in the case that these also are available. The MATLAB code for the algorithm is provided, and two examples are given to illustrate how this type of FD stencil can be applied to solving partial differential equations.

On the possibility of obtaining the optimal order of accuracy when restoring the impact by the dynamic method

2020 ◽  
pp. 147-155
Author(s):  
Andrey Yu. Vdovin ◽  
Svetlana S. Rubleva
Keyword(s):  
Approximate Solution ◽  
Dynamic Method ◽  
Numerical Differentiation ◽  
Optimal Order ◽  
Order Of Accuracy ◽  
The Impact

Yu. S. Osipov and A. V. Kryazhimsky proposed a method of dynamic regulationto restore an unknown effect in a controlled model. In the framework of this approach, inthe present work we study the property of another method based on the use of the implicitEuler method for the problem of numerical differentiation. The choice of the parameters ofthe method is indicated, which makes it possible to increase its efficiency, reduce the noiselevel of the approximate solution, and obtain the optimal order of accuracy in the metricL(T); equal to 1/2.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

2020 ◽  
Vol 26 ◽  
pp. 78
Author(s):  
Thirupathi Gudi ◽  
Ramesh Ch. Sau
Keyword(s):  
Finite Element ◽  
Control Problem ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Diffusion Problem ◽  
Discrete Control ◽  
Optimal Order ◽  
Control Constraints ◽  
Element Analysis ◽  
Dirichlet Boundary Control

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

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