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Topic: Tikhonov regularization

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In the News (Tue 22 Apr 14)

  Tikhonov - Wikipedia, the free encyclopedia
Nikolay Alexandrovich Tikhonov, a former Premier of the Soviet Union
Nikolay Semenovich Tikhonov, a Russian writer, a member of Serapion Brothers literatury group.
Tikhonov regularization, a method of regularization of ill-posed problems named in honour of the mathematician.
en.wikipedia.org /wiki/Tikhonov   (114 words)

 Tikhonov regularization - Wikipedia, the free encyclopedia
Tikhonov regularization is the most commonly used method of regularization of ill-posed problems.
Although at first the choice of the solution to this regularized problem may look artificial, and indeed the parameter α seems rather arbitrary, the process can be justified in a Bayesian point of view.
This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem.
en.wikipedia.org /wiki/Tikhonov_regularization   (690 words)

 References   (Site not responding. Last check: 2007-10-02)
Tikhonov regularization, least squares solution of systems of linear equations, convolution equations, final value problems, parameter identification.
The regularization toolbox provides a variety of functions for solving inverse problems, including the SVD and generalized SVD, truncated SVD solutions, Tikhonov regularization, maximum entropy regularization, and a variety of examples.
Regularization theory, Tikhonov regularization, regularization by discretization, the method of Backus and Gilbert, inverse eigenvalue problems, inverse scattering problems.
infohost.nmt.edu /~borchers/geop529/readings/node1.html   (546 words)

 Fa-Hsuan Lin : Research : Inverse : Projects - Regularization
The regularization parameter characterizes the relative weights of data and current prior error terms in the cost function to be minimized.
One choice of regularization parameter is the inverse of the SNR in the data.
The regularization parameter was proposed to be the inverse of the SNR of the inverse as
www.nmr.mgh.harvard.edu /~fhlin/research_inverse_project_regularization.htm   (1337 words)

 [No title]   (Site not responding. Last check: 2007-10-02)
The L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems by Tikhonov regularization.
Recently, inexpensively computable approximations of the L-curve and its curvature, referred to as the L-ribbon and the curvature-ribbon, respectively, were proposed for the case when the regularization operator is the identity matrix.
Tikhonov regularization of large symmetric problems (with D. Calvetti and L. Reichel), Numerical Linear Algebra(NLA), in press.
condor.depaul.edu /~ashuibi/research.html   (327 words)

regularization by discretzation, selfregularization, projection methods, Tikhonov regularization, severely ill-posed, integral equation of the first kind, logarithmic convergence rate.
We study regularization methods for the integral equation of the first kind with analytical kernel of logarithmic type.
First we consider the selfregularization of the problem by using projection methods in the sense of [9].Then we will see that the Tikhonov regularization of such methods is in accordance with a discretized version of the Tikhonov regularized solution in [1].
www.wias-berlin.de /publications/preprints/523   (131 words)

 Iterative data regularization
Though the final results of the model-space and data-space regularization are theoretically identical, the behavior of iterative gradient-based methods, such as the method of conjugate gradients, is different for the two cases.
The obvious difference is in the case where the number of model parameters is significantly larger than the number of data measurements.
In this case, the dimensions of the inverted matrix in the case of the data-space regularization are smaller that those of the model-space matrix, and the convergence of the iterative conjugate-gradient iteration is correspondingly faster.
sepwww.stanford.edu /public/docs/sep107/paper_html/node34.html   (470 words)

 A Discrepancy Principle For Tikhonov Regularization With Approximately Specified Data (ResearchIndex)   (Site not responding. Last check: 2007-10-02)
Abstract: Many discrepancy principles are known for choosing the parameter ff in the regularized operator equation (T T + ffI)x ffi ff = T y ffi, ky \Gamma y ffi k ffi, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y.
In this paper we consider a class of discrepancy principles for choosing the regularization parameter when T T and T y ffi are approximated by A n and z ffi n respectively with A n not necessarily self--adjoint.
2 Regularized Ritz approximation for Fredholm equations of the..
citeseer.ist.psu.edu /63142.html   (363 words)

 Numerical Solution of First-Kind Volterra Equations by Sequential Tikhonov Regularization
The method we consider is a sequential form of Tikhonov regularization that is particularly suited to problems of Volterra type.
We prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation (collocation, rectangular and midpoint quadrature), one obtains convergence of the method at an optimal rate with respect to noise in the data.
In addition we describe a fast algorithm for the implementation of sequential Tikhonov regularization and show that for small values of the regularization parameter, the method is only slightly more expensive computationally than the numerical solution of the original unregularized integral equation.
epubs.siam.org /sam-bin/dbq/article/28081   (226 words)

 [No title]   (Site not responding. Last check: 2007-10-02)
To improve estimation of potentials over conventional techniques, such as the boundary-element method (BEM), regularization and regularization parameter estimation techniques for inferring pericardial potentials using the multipole-equivalent method (MEM) were developed.
Using the MEM with modifications of constrained-least-squares (MEM-CLS) and Tikhonov regularization (MEM-TIK), potentials were inferred with relative errors (REs) less than 0.5 (50 rms error) in an eccentric-spheres model when subjected to clinically-realistic errors.
Using a single regularization parameter estimate for all frequencies, average REs were less than 0.57 using the average composite-residual-and-smoothing-operator (CRESO) parameter for MEM-TIK and less than 1.12 using the maximum parameter for BEM-TIK.
web.umr.edu /~daryl/research/dissert.abstract   (326 words)

 [No title]   (Site not responding. Last check: 2007-10-02)
Those of ordinary skill recognize that the regularization parameter, t, controls the degree of the imposed constraint and provides a balance between the accuracy and stability of the solution, while L is a regularization operator (e.
The Tikhonov regularization also requires an accurate determination of the aforementioned regularization parameter, t, which determines the constraint level.
Although the GMRes and Tikhonov reconstructions show the cooling-induced reduction in QRST integral values similar to the measured map, the GMRes reconstructs the localized minimum under the cooling probe, while the Tikhonov does not.
www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/28801.030410&ELEMENT_SET=DECL   (7626 words)

 [No title]
Nonstationary Iterated Tikhonov Regularization Martin Hanke and C.W. Groetsch Abstract A convergence rate is established for nonstationary iterated Tik- honov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iter- ation parameters.
There is a well-developed convergence theory for (2) (e.g., [5], [13]), an important ingredient of which is a strategy for relating the regularization parameter to perturbed data in such a way that as the error level diminishes to zero the approximations converge to the desired solution.
An efficient numerical implementation of nonstationary iterated Tikhonov regularization is not more expensive than using the same sequence of regular- ization parameters in a successive way for ordinary Tikhonov regularization.
www.ubka.uni-karlsruhe.de /vvv/1996/mathematik/9/9.text   (1781 words)

 [No title]
The determination of the Earth's gravitational field from GOCE measurements is a exponentially ill-posed problem which requires adequate methods of regularization.
Tikhonov scaling functions and wavelets, has been suggested as a means for the solution of the SGG-Problem e.g.
However, the main question in Tikhonov regularization is the choice of a suitable regularization parameter.
earth.esa.int /cgi-bin/confgoce.pl?abstract=42   (155 words)

 [No title]
We study regularization methods for the integral equation of the orst kind with analytical kernel of logarithmic type.
The numerical experiments concern the two kinds of regularity assumptions: First, the solution is supposed to be H10 on [0; 1] and second, the solution is supposed to be H1 in a neighborhood of one point.
The approximating sequence f(n; ff; ffi) will depend on the discretization number n, the regularization parameter ff and the noise level ffi: It belongs to a onite-dimensional space Un, that is a subspace of H10(0; 1) in the case (i) and of L2(0; 1) \ H1(O) in the cases (ii) and (iii).
www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file26.html   (2786 words)

 SCAR » Report 23
The dependence of the regularization parameter on the variance of the studying field and the variance of the noise is considered.
is the Tikhonov regularization parameter (Neyman, 1979; Moritz, 1980; Marchenko and Tartachynska, 2003).
was done by the regularization method at the adopted grids points with the resolution (2' x4') and (3' x 3'), completely filled all marine part of the studying area.
www.scar.org /publications/reports/23/marchenko   (1937 words)

 Rama CONT: Model Calibration and Inverse problems in financial modeling.
Along the way we will cover some aspects of the theory of regularization of ill-posed inverse problems, an active branch of applied mathematics.
Determination of the regularization parameter: a priori, a posteriori and error-free parameter choice.
Regularization by relative entropy minimization (Stutzer, Buchen and Kelly, Avellaneda).
www.cmap.polytechnique.fr /~rama/teaching/princeton.html   (1014 words)

 Hydrodynamic Size-Distribution Analysis
The function that we calculate by regularization is the one just on the edge of this circle, within the subset of smooth and positive functions.
These are, for example, the apparent sedimentation coefficient distribution ls-g*(s) (which for fundamental reasons is diffusionally broadened and therefore cannot have sharp peaks), and the study of broad distributions of synthetic polymers (where we may know that the statistical nature of the synthesis leads to a quasi-continuous size-distribution without sharp peaks).
A value of 0.51 will cause very little regularization; values of 0.68 to 0.90 would correspond to commonly used confidence levels (usually, with 50 scans or more the chi-square increase corresponding to a probability of 0.7 is of the order of 0.1%), while values close to 0.99 would cause very high regularization.
www.nih.gov /od/ors/dbeps/PBR/AUCtutorials/sizedistributions.htm   (4384 words)

 Stackelberg problems: Tikhonov regularization and subgame perfect equilibria
To overcome the numerical difficulties, due to the non-uniqueness of the best reply, regularization methods have been suggested to tackle this problem like Tikhonov regularization (Loridan and Morgan (1992) and the approaches followed by Dempe (Dempe (1996) to solve the strong Stackelberg problem and by Molodtsov (1976), Soholovic (1970) and Loridan and Morgan (1992, 1996).
A partial answer to this question was given in Loridan and Morgan (1992) for the Tikhonov regularization: they defined the so-called ``lower Stackelberg equilibrium pair'' and shown that any limit point obtained via Tikhonov regularization, when the regularization parameter goes to zero, is a lower Stackelberg equilibrium pair.
Let's notice that Tikhonov regularization method applies to a wider class of problems than Dempe's and Molodtsov's approach, since it is not required any kind of strong convexity assumption for the leader's functional in order to obtain the uniqueness of the best-reply.
www.dima.unige.it /~patrone/abstract/Stackreg.htm   (853 words)

 Technical Report (Michael Ulbrich)   (Site not responding. Last check: 2007-10-02)
We propose and analyze a generalization of the Tikhonov regularization for nonlinear ill-posed operator equations F(y)=z.
Hereby, regularization functionals of the form R(y)_W^2 with linear or nonlinear operator R are considered.
Under the additional assumption that the regularization space W is embedded in a Hilbert scale, we present error estimates if the regularization parameter is chosen either according to the discrepancy principle of Morozov or by the usual a-priori strategy.
www-m1.ma.tum.de /m1/personen/mulbrich/papers/tikreg.html   (182 words)

 Applications of the Modified Discrepancy Principle to Tikhonov Regularization of Nonlinear Ill-Posed Problems
Applications of the Modified Discrepancy Principle to Tikhonov Regularization of Nonlinear Ill-Posed Problems: SIAM Journal on Numerical Analysis Vol.
In this paper, we consider the finite-dimensional approximations of Tikhonov regularization for nonlinear ill-posed problems with approximately given right-hand sides.
We propose an a posteriori parameter choice strategy, which is a modified form of Morozov's discrepancy principle, to choose the regularization parameter.
epubs.siam.org /sam-bin/dbq/article/31547   (176 words)

 Fast CG-Based Methods for Tikhonov-Phillips Regularization - Frommer, Maass (ResearchIndex)   (Site not responding. Last check: 2007-10-02)
Abstract: Tikhonov-Phillips regularization is one of the bestknown regularization methods for inverse problems.
A posteriori criteria for determining the regularization parameter ff require solving (A A+ ffI)x = A y ffi () for different values of ff.
0.6: Morozov's Discrepancy Principle for Tikhonov regularization of..
citeseer.ist.psu.edu /82287.html   (705 words)

 Regularized Resolvent Transform -- Jianhan Chen   (Site not responding. Last check: 2007-10-02)
The Regularized Resolvent Transfrom (RRT) is a new method recently developed for high resolution spectral estimation.
A regularization parameter is then introduced when applying Tikhonov regularization.
The variable "q" is the regularization parameter, and the numbers shown are normalized according to the norm of the complex signal.
www.physics.uci.edu /~jianhanc/rrt.html   (310 words)

 [No title]   (Site not responding. Last check: 2007-10-02)
A damage detection algorithm is presented based on updating a finite element model with measured eigenfrequencies and mode shapes.
Tikhonov regularization and truncated singular value decomposition are commonly used regularization techniques for linear problems.
We present an iterative Gauss-Newton algorithm with Tikhonov regularization that includes line search and constraints on the update paramters to improve convergence.
www.sem.org /APP-CONF-List2-Abstract.asp?PaperNo=206   (175 words)

 Tikhonov - TheBestLinks.com - Tychonoff, Mathematician, Russia, Tychonoff space, ...   (Site not responding. Last check: 2007-10-02)
Tikhonov - TheBestLinks.com - Tychonoff, Mathematician, Russia, Tychonoff space,...
Tychonoff, Tikhonov, Mathematician, Russia, Tychonoff space, Premier of the...
Nikolay Aleksandrovich Tikhonov, former Premier of the Soviet Union
www.thebestlinks.com /Tychonoff.html   (143 words)

Applies Tikhonov regularization on the data and noise covariance.
Tikhonov regularization causes bias on the solution, but reduces the affect of noise.
Specify the Regularization parameter lambda as a percentage of the maximum singular value of the covariance matrix.
neuroimage.usc.edu /brainstorm/GUI_DataViewer_LCMVBF.htm   (327 words)

 Tikhonov's regularization and multiresolution analysis   (Site not responding. Last check: 2007-10-02)
We have in fact a multiresolution Tikhonov's regularization.
This method has the advantage to furnish a solution quickly, but optimal regularization parameters cannot be found directly, and several tests are generally necessary before finding an acceptable solution.
Hovewer, the method can be interesting if we need to deconvolve a big number of images with the same noise characteristics.
www.eso.org /projects/esomidas/doc/user/98NOV/vol2_html/node338.html   (239 words)

 [No title]   (Site not responding. Last check: 2007-10-02)
ISBN 0-8247-6986-4 C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, London, 1984.
C.W. Groetsch and J.T. King, The saturation phenomena for Tikhonov regularization, Journal of the Australian Mathematical Society, (Series A) 35 (1983), 254-262, with J. King.
C.W. Groetsch, Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind, Integral Equations and Operator Theory 13(1990), 67-75.
math.uc.edu /~groetsch/pubs.doc   (1910 words)

 Indmath Info Server
Scherzer, H.W. Engl, and K. Kunisch, Optimal a-posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J.Num.Anal.
H.W. Engl and P. Manselli, Stability estimates and regularization for an inverse heat conduction problem in semi-infinite and finite time intervals, Numer.Funct.Anal.
H.W. Engl and A. Neubauer, Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales, Proc.of the Amer.Math.Soc.
www.indmath.uni-linz.ac.at /publications/old   (1705 words)

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