# On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type

@article{Lamm2005OnLR, title={On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type}, author={Patricia K. Lamm and Zhewei Dai}, journal={Inverse Problems}, year={2005}, volume={21}, pages={1773 - 1790} }

Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, … in Lamm (2005 Inverse Problems 21 785–803). In this paper, we enlarge the family of convergent local…

## 15 Citations

### Local regularization of nonlinear Volterra equations of Hammerstein type

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- 2010

The method of local regularization has been shown to be an effective tool for the reconstruction of solutions of linear and nonlinear inverse problems, especially those problems with special…

### Full convergence of sequential local regularization methods for Volterra inverse problems

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Local regularization methods for ill-posed linear Volterra equations have been shown to be efficient regularization procedures preserving the causal structure of the Volterra problem and allowing for…

### A generalized approach to local regularization of linear Volterra problems in Lp spaces

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- 2011

A generalized version of local regularization is developed. The convergence criteria set forth in the generalization are shown to be realized naturally by a particular local regularization scheme…

### Local Regularization for the Nonlinear Inverse Autoconvolution Problem

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This theory provides regularization methods that preserve the causal nature of the autoconvolution problem, allowing for fast sequential numerical solution and proves the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero.

### Local regularization for n-dimensional integral equations with applications to image processing

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- 2007

We examine the method of local regularization for the solution of linear first-kind integral equations on . We provide a theoretical analysis of the method and prove that the regularized solutions…

### Statistical Multiresolution Estimators in Linear Inverse Problems - Foundations and Algorithmic Aspects

- Computer Science, Mathematics
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This dissertation introduces a novel algorithmic framework to computeSMR-estimators in practice and is the first that allows for computation of SMR-ESTimators for (possibly ill-posed) inverse problems and can also be combined with a variety of penalty functions.

### Estimation metrics and optimal regularization in a Tikhonov digital filter for the inverse heat conduction problem

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### Theory and numerical solution of Volterra functional integral equations

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(I.2) C.T.H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math., 125 (2000), 217-249. (I.3) P.R. Beesack, Comparison theorems and integral inequalities for…

### A DISCREPANCY PRINCIPLE FOR PARAMETER SELECTION IN LOCAL REGULARIZATION OF LINEAR VOLTERRA INVERSE PROBLEMS

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A DISCREPANCY PRINCIPLE FOR PARAMETER SELECTION IN LOCAL REGULARIZATION OF LINEAR VOLTERRA INVERSE PROBLEMS

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