scholarly journals Convergence rates and explicit error bounds of Hill’s method for spectra of self-adjoint differential operators

2013 ◽  
Vol 31 (1) ◽  
pp. 25-56
Author(s):  
Ken’ichiro Tanaka ◽  
Sunao Murashige
Keyword(s):  
Error Bounds ◽  
Differential Operators ◽  
Convergence Rates ◽  
Hill’S Method

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

scholarly journals Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

2019 ◽  
Vol 27 (3) ◽  
pp. 155-182 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Convergence Rates ◽  
Time Discretization ◽  
Linear Constraints ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems ◽  
Discretization Methods ◽  
Optimal Convergence

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

scholarly journals Quantile Regression Learning with Coefficient Dependent lq-Regularizer

2018 ◽  
Vol 173 ◽  
pp. 03033
Author(s):  
Pengju Sun ◽  
Meng Li ◽  
Hongwei Sun
Keyword(s):  
Quantile Regression ◽  
Error Bounds ◽  
Convergence Rates ◽  
Learning Algorithms ◽  
Concentration Inequality ◽  
Decomposition Techniques ◽  
Conditional Quantile ◽  
Pinball Loss ◽  
Regression Learning ◽  
Operator Decomposition

In this paper, We focus on conditional quantile regression learning algorithms based on the pinball loss and lq-regularizer with 1≤q≤2. Our main goal is to study the consistency of this kind of regularized quantile regression learning. With concentration inequality and operator decomposition techniques, we obtained satisfied error bounds and convergence rates.

Convergence Rates For Lavrentiev-Type Regularization In Hilbert Scales

2008 ◽  
Vol 8 (3) ◽  
pp. 279-293 ◽  
Author(s):  
M.T. NAIR ◽  
U. TAUTENHAHN
Keyword(s):  
Error Bounds ◽  
Convergence Rates ◽  
Noisy Data ◽  
Regularization Methods ◽  
Operator Equations ◽  
Optimal Error ◽  
Regularization Scheme ◽  
Data Regularization ◽  
Ill Posed ◽  
Adjoint Operators

AbstractFor solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

scholarly journals Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ruifeng Wu
Keyword(s):  
Error Estimates ◽  
Error Bounds ◽  
Nonnegative Integer ◽  
Convergence Rates ◽  
Approximation Order ◽  
Degree Polynomial ◽  
Shape Preserving ◽  
Interpolation Operators ◽  
Interpolation Polynomials ◽  
Polynomial Reproducing

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

scholarly journals Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations

2016 ◽  
Vol 16 (2) ◽  
pp. 187-212 ◽  
Author(s):  
Pierre Cantin ◽  
Alexandre Ern
Keyword(s):  
Differential Operators ◽  
Convergence Rates ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Diffusion Equations ◽  
Polyhedral Meshes ◽  
Advection Diffusion ◽  
Discrete Operators ◽  
Boundary Operators ◽  
Primal And Dual

AbstractWe devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely, primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are devised to weakly enforce Dirichlet boundary conditions. The analysis sheds new light on the theory of Friedrichs' operators at the purely algebraic level. Moreover, an extension of the stability analysis hinging on inf-sup conditions is presented to incorporate divergence-free velocity fields under some assumptions. Error bounds and convergence rates for smooth solutions are derived and numerical results are presented on three-dimensional polyhedral meshes.

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

2018 ◽  
Vol 39 (3) ◽  
pp. 1206-1245 ◽  
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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