Implicit Error Bounds for Picard Iterations on Hilbert Spaces

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

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Unified error analysis for nonconforming space discretizations of wave-type equations

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry036 ◽

2018 ◽

Vol 39 (3) ◽

pp. 1206-1245 ◽

Cited By ~ 4

Author(s):

David Hipp ◽

Marlis Hochbruck ◽

Christian Stohrer

Keyword(s):

Wave Equation ◽

Error Analysis ◽

Hilbert Spaces ◽

Error Bounds ◽

Wave Equations ◽

Evolution Equations ◽

Convergence Rates ◽

Linear Wave ◽

Element Discretization ◽

Continuous Space

Abstract This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

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New Čebyšev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

Chinese Journal of Mathematics ◽

10.1155/2014/363050 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Mohammad W. Alomari

Keyword(s):

Hilbert Spaces ◽

Error Bounds ◽

Čebyšev Functional ◽

Adjoint Operators

Several new error bounds for the Čebyšev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well.

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Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

Mathematica Pannonica ◽

10.1556/314.2021.00011 ◽

2021 ◽

Author(s):

Silvestru Sever Dragomir

Keyword(s):

Hilbert Space ◽

Power Function ◽

Hilbert Spaces ◽

Error Bounds ◽

Real Function ◽

Positive Measure ◽

Integral Transform ◽

Positive Function ◽

Complex Hilbert Space ◽

Second Derivative

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.

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Difference gap functions and global error bounds for random mixed equilibrium problems

Filomat ◽

10.2298/fil2008739h ◽

2020 ◽

Vol 34 (8) ◽

pp. 2739-2761

Author(s):

Nguyen Hung ◽

Xiaolong Qin ◽

Vo Tam ◽

Jen-Chih Yao

Keyword(s):

Hilbert Spaces ◽

Error Bounds ◽

Equilibrium Problems ◽

Gap Function ◽

Global Error ◽

Gap Functions ◽

Mixed Equilibrium Problems ◽

Mixed Equilibrium ◽

The Difference ◽

Regularized Gap Functions

The aim of this paper is to study the difference gap (in short, D-gap) function and error bounds for a class of the random mixed equilibrium problems in real Hilbert spaces. Firstly, we consider regularized gap functions of the Fukushima type and Moreau-Yosida type. Then difference gap functions are established by using these terms of regularized gap functions. Finally, the global error bounds for random mixed equilibrium problems are also developed. The results obtained in this paper are new and extend some corresponding known results in literatures. Some examples are given for the illustration of our results.

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