scholarly journals On the Local Regularization of Inverse Problems of Volterra Type

1995 ◽  
Author(s):  
Patricia K. Lamm
Keyword(s):  
Tikhonov Regularization ◽  
Volterra Operator ◽  
Dimensional Regularization ◽  
Convergence Theory ◽  
True Solution ◽  
Infinite Dimensional ◽  
Local Regularization ◽  
Regularization Parameters ◽  
Regularization Problem ◽  
Local Procedure

Abstract We consider a local regularization method for the solution of first-kind Volterra integral equations with convolution kernel. The local regularization is based on a splitting of the original Volterra operator into “local” and “global” parts, and a use of Tikhonov regularization to stabilize the inversion of the local operator only. The regularization parameters for the local procedure include the standard Tikhonov parameter, as well as a parameter that represents the length of the local regularization interval. We present a convergence theory for the infinite-dimensional regularization problem and show that the regularized solutions converge to the true solution as the regularization parameters go to zero (in a prescribed way). In addition, we show how numerical implementation of the ideas of local regularization can lead to the notion of “sequential Tikhonov regularization” for Volterra problems; this approach has been shown in (Lamm and Eldén, 1995) to be just as effective as Tikhonov regularization, but to be much more efficient computationally.

A new choice rule for regularization parameters in Tikhonov regularization

2011 ◽  
Vol 90 (10) ◽  
pp. 1521-1544 ◽  
Author(s):  
Kazufumi Ito ◽  
Bangti Jin ◽  
Jun Zou
Keyword(s):  
Tikhonov Regularization ◽  
Choice Rule ◽  
Regularization Parameters

scholarly journals A Novel Method for Invariant Image Reconstruction

2021 ◽  
Vol 11 (1) ◽  
pp. 69-80
Author(s):  
Mirosław Pawlak ◽  
Gurmukh Singh Panesar ◽  
Marcin Korytkowski
Keyword(s):  
Image Reconstruction ◽  
Symmetry Axis ◽  
Bilateral Symmetry ◽  
Reconstruction Accuracy ◽  
Invariance Properties ◽  
Regularization Parameters ◽  
Regularization Problem ◽  
Symmetry Constraints ◽  
Novel Method ◽  
Constraints Model

AbstractIn this paper we propose a novel method for invariant image reconstruction with the properly selected degree of symmetry. We make use of Zernike radial moments to represent an image due to their invariance properties to isometry transformations and the ability to uniquely represent the salient features of the image. The regularized ridge regression estimation strategy under symmetry constraints for estimating Zernike moments is proposed. This extended regularization problem allows us to enforces the bilateral symmetry in the reconstructed object. This is achieved by the proper choice of two regularization parameters controlling the level of reconstruction accuracy and the acceptable degree of symmetry. As a byproduct of our studies we propose an algorithm for estimating an angle of the symmetry axis which in turn is used to determine the possible asymmetry present in the image. The proposed image recovery under the symmetry constraints model is tested in a number of experiments involving image reconstruction and symmetry estimation.

scholarly journals Numerical Solutions of Stochastic Functional Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 141-161 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Convergence Theory ◽  
Stochastic Functional Differential Equations ◽  
True Solution ◽  
Linear Growth Condition ◽  
Mean Square Convergence ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

Scattered data fitting by minimal surface

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Yong-Xia Hao ◽  
Dianchen Lu
Keyword(s):  
Tikhonov Regularization ◽  
Numerical Experiments ◽  
Data Fitting ◽  
Scattered Data ◽  
B Spline ◽  
Scattered Data Fitting ◽  
Area Minimization ◽  
Regularization Parameters ◽  
The Given ◽  
Minimal Area

AbstractThe goal of this paper is to develop a computational model for obtaining the fitting surface to the given scattered data with minimal area. The basic idea of the model is to utilize the B-spline and area minimization. The model is turned into a Tikhonov regularization model finally. By choosing the regularization parameters with the L-curve criterion and the GCV method, respectively, numerical experiments indicate that the model can provide an acceptable compromise between the minimization of the data mismatch term and the area of the surface.

scholarly journals DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION

2017 ◽  
Vol 22 (2) ◽  
pp. 202-212
Author(s):  
Teresa Reginska
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Discrepancy Principle ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Data Error ◽  
Two Parameter ◽  
Regularized Solution

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.

COMMON ANALYTIC BASIS OF THE ζ-FUNCTION AND THE DIMENSIONAL REGULARIZATION SCHEMES

1989 ◽  
Vol 04 (09) ◽  
pp. 863-867
Author(s):  
DAKSH LOHIYA
Keyword(s):  
Analytic Continuation ◽  
Dimensional Regularization ◽  
Zeta Functions ◽  
Infinite Dimensional ◽  
Analytic Methods

The analytic continuation invoked in the theory of generalized zeta functions associated with infinite-dimensional operators is shown to be equivalent in structure to the basic analytic methods deployed in dimensional regularization.

The Steady Inverse Heat Conduction Problem: A Comparison of Methods With Parameter Selection

2001 ◽  
Vol 123 (4) ◽  
pp. 633-644 ◽  
Author(s):  
Robert Throne ◽  
Lorraine Olson
Keyword(s):  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Heat Conduction Problem ◽  
Regularization Parameter ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
True Solution ◽  
Sensitivity Method ◽  
Inverse Boundary Value Problems ◽  
Steady Heat Conduction

In the past we have developed the Generalized Eigensystem GESL techniques for solving inverse boundary value problems in steady heat conduction, and found that these vector expansion methods often give superior results to those obtained with standard Tikhonov regularization methods. However, these earlier comparisons were based on the optimal results for each method, which required that we know the true solution to set the value of the regularization parameter (t) for Tikhonov regularization and the number of mode clusters Nclusters for GESL. In this paper we introduce a sensor sensitivity method for estimating appropriate values of Nclusters for GESL. We compare those results with Tikhonov regularization using the Combined Residual and Smoothing Operator (CRESO) to estimate the appropriate values of t. We find that both methods are quite effective at estimating the appropriate parameters, and that GESL often gives superior results to Tikhonov regularization even when Nclusters is estimated from measured data.

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