The Stability of Behavioral PLS Results in Ill-Posed Neuroimaging Problems

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

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Convergence and stability analysis of the half thresholding based few-view CT reconstruction

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0003 ◽

2020 ◽

Vol 28 (6) ◽

pp. 829-847

Author(s):

Hua Huang ◽

Chengwu Lu ◽

Lingli Zhang ◽

Weiwei Wang

Keyword(s):

Noise Suppression ◽

Reconstructed Image ◽

Reference Image ◽

Projection Data ◽

Ct Reconstruction ◽

Reconstruction Algorithms ◽

Ill Posed ◽

Thresholding Algorithm ◽

The Stability ◽

Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

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Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2021-0038 ◽

2021 ◽

Vol 24 (3) ◽

pp. 895-922

Author(s):

Platon G. Surkov

Keyword(s):

Control Theory ◽

Fractional Derivative ◽

Approximate Calculation ◽

Classical Problem ◽

Tikhonov Regularization Method ◽

Inaccurate Data ◽

Local Modification ◽

Ill Posed ◽

Computational Errors ◽

The Stability

Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

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Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Mathematics ◽

10.3390/math8010048 ◽

2020 ◽

Vol 8 (1) ◽

pp. 48

Author(s):

Hongwu Zhang ◽

Xiaoju Zhang

Keyword(s):

Cauchy Problem ◽

A Priori ◽

Type Equation ◽

Conditional Stability ◽

Elliptic Type ◽

Type Method ◽

The Cauchy Problem ◽

Ill Posed ◽

The Stability ◽

Regularized Solution

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

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Wavelet-Galerkin Method for Identifying an Unknown Source Term in a Heat Equation

Mathematical Problems in Engineering ◽

10.1155/2012/904183 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

Fangfang Dou

Keyword(s):

Heat Equation ◽

Galerkin Method ◽

Identification Problem ◽

Stability And Convergence ◽

Wavelet Galerkin Method ◽

Unknown Source ◽

Frequency Components ◽

Ill Posed ◽

Meyer Wavelets ◽

The Stability

We consider the problem of identification of the unknown source in a heat equation. The problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in the basis of the Meyer wavelets, high-frequency components can be filtered away. Under the additional assumptions concerning the smoothness of the solution, we discuss the stability and convergence of a wavelet-Galerkin method for the source identification problem. Numerical examples are presented to verify the efficiency and accuracy of the method.

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A NEW OPTIMIZED RFM OF HIGH-RESOLUTION SATELLITE IMAGERY

ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ◽

10.5194/isprsarchives-xli-b3-65-2016 ◽

2016 ◽

Vol XLI-B3 ◽

pp. 65-69 ◽

Cited By ~ 2

Author(s):

C. Li ◽

X. J. Liu ◽

T. Deng

Keyword(s):

High Resolution ◽

Constant Term ◽

Satellite Observation ◽

Least Square ◽

Orthogonal Distance Regression ◽

Ill Posed ◽

High Resolution Satellite Imagery ◽

The Stability ◽

Spot 5 ◽

Cross Track

Over-parameterization and over-correction are two of the major problems in the rational function model (RFM). A new approach of optimized RFM (ORFM) is proposed in this paper. By synthesizing stepwise selection, orthogonal distance regression, and residual systematic error correction model, the proposed ORFM can solve the ill-posed problem and over-correction problem caused by constant term. The least square, orthogonal distance, and the ORFM are evaluated with control and check grids generated from satellite observation Terre (SPOT-5) high-resolution satellite data. Experimental results show that the accuracy of the proposed ORFM, with 37 essential RFM parameters, is more accurate than the other two methods, which contain 78 parameters, in cross-track and along-track plane. Moreover, the over-parameterization and over-correction problems have been efficiently alleviated by the proposed ORFM, so the stability of the estimated RFM parameters and its accuracy have been significantly improved.

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MATHEMATICAL MODELLING OF WATER FLOW IN PAPER PRESS MACHINES

Mathematical Modelling and Analysis ◽

10.3846/13926292.2004.9637259 ◽

2004 ◽

Vol 9 (4) ◽

pp. 267-286

Author(s):

R. Čiegis ◽

M. Meilūnas ◽

A. Štikonas

Keyword(s):

Mathematical Models ◽

Numerical Algorithms ◽

Discrete Problem ◽

Time Step ◽

Water Losses ◽

Deformable Porous Media ◽

Ill Posed ◽

The Stability ◽

Liquid Movement ◽

The Mathematical Model

In this work, mathematical models of wet pressing of paper are studied. Our goal is to compare two mathematical models, which are developed for simulation of filtration processes in paper press machines. Both models were obtained from the same general model of the compressible porous medium, but different assumptions were used. Modified models are developed that describe water losses at the boundaries of the porous layer and the importance of this factor is investigated. Numerical algorithms are developed for simulation of the liquid movement in the deformable porous media. It is proved that the discrete problem is stable if the time step t satisfies the inequality τ ≤ Ch2 . It follows from the stability analysis that the mathematical model describes an ill-posed problem for some values of parameters used in simulations.

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TOWARDS THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS VIA REGULARIZATION AND EXTENSION THEORY

Reviews in Mathematical Physics ◽

10.1142/s0129055x96000251 ◽

1996 ◽

Vol 08 (05) ◽

pp. 715-740 ◽

Cited By ~ 6

Author(s):

HAGEN NEIDHARDT ◽

VALENTIN ZAGREBNOV

Keyword(s):

Operator Method ◽

Stability Domain ◽

Extension Theory ◽

Friedrichs Extension ◽

Natural Domain ◽

Abstract Operator ◽

Adjoint Extension ◽

Ill Posed ◽

The Stability ◽

The Right

For singular potentials in quantum mechanics it can happen that the Schrödinger operator is not esssentially self-adjoint on a natural domain, i.e., each self-adjoint extension is a candidate for the right physical Hamiltonian. Traditional way to single out this Hamiltonian is the removing cut-offs for regularizing potential. Connecting regularization and extension theory we develop an abstract operator method to treat the problem of the right Hamiltonian. We show that, using the notion of the maximal (with respect to the perturbation) Friedrichs extension of unperturbed operator, one can classify the above problem as wellposed or ill-posed depending on intersection of the quadratic form domain of perturbation and deficiency subspace corresponding to restriction of unperturbed operator to stability domain. If this intersection is trivial, then the right Hamiltonian is unique: it coincides with the form sum of perturbation and the Friedrich extension of the unperturbed operator restricted to the stability domain. Otherwise it is not unique: the family of “right Hamiltonians” can be described in terms of symmetric extensions reducing the ill-posed problem to the well-posed problem.

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An inverse problem in estimating the time dependent source term and initial temperature simultaneously by the polynomial regression and conjugate gradient method

Filomat ◽

10.2298/fil2010507c ◽

2020 ◽

Vol 34 (10) ◽

pp. 3507-3516

Author(s):

Arzu Coşkun

Keyword(s):

Source Term ◽

Initial Temperature ◽

Polynomial Regression ◽

Heat Conduction Problem ◽

Inverse Heat Conduction Problem ◽

Necessary Condition ◽

Inverse Heat Conduction ◽

Control Framework ◽

Ill Posed ◽

The Stability

From the final and interior temperature measurements identifying the source term with initial temperature simultaneously is an inverse heat conduction problem which is a kind of ill-posed. The optimal control framework has been found to be effective in dealing with these problems. However, they require to find the gradient information. This idea has been employed in this research. We derive the gradient of Tikhonov functional and establish the stability of the minimizer from the necessary condition. The stability and effectiveness of evolutionary algorithm are presented for various test examples.

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A NEW OPTIMIZED RFM OF HIGH-RESOLUTION SATELLITE IMAGERY

ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ◽

10.5194/isprs-archives-xli-b3-65-2016 ◽

2016 ◽

Vol XLI-B3 ◽

pp. 65-69

Author(s):

C. Li ◽

X. J. Liu ◽

T. Deng

Keyword(s):

High Resolution ◽

Constant Term ◽

Satellite Observation ◽

Least Square ◽

Orthogonal Distance Regression ◽

Ill Posed ◽

High Resolution Satellite Imagery ◽

The Stability ◽

Spot 5 ◽

Cross Track

Over-parameterization and over-correction are two of the major problems in the rational function model (RFM). A new approach of optimized RFM (ORFM) is proposed in this paper. By synthesizing stepwise selection, orthogonal distance regression, and residual systematic error correction model, the proposed ORFM can solve the ill-posed problem and over-correction problem caused by constant term. The least square, orthogonal distance, and the ORFM are evaluated with control and check grids generated from satellite observation Terre (SPOT-5) high-resolution satellite data. Experimental results show that the accuracy of the proposed ORFM, with 37 essential RFM parameters, is more accurate than the other two methods, which contain 78 parameters, in cross-track and along-track plane. Moreover, the over-parameterization and over-correction problems have been efficiently alleviated by the proposed ORFM, so the stability of the estimated RFM parameters and its accuracy have been significantly improved.

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