scholarly journals An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Kernel Method ◽  
A Priori ◽  
Diffusion Problem ◽  
Two Dimensional ◽  
A Posteriori ◽  
Numerical Examples ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Convergence Estimates ◽  
Inverse Diffusion

In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data. A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization parameters. The numerical examples illustrate the behavior of the proposed method.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals Robust Bayesian Joint Inversion of Gravimetric and Muographic Data for the Density Imaging of the Puy de Dôme Volcano (France)

2021 ◽  
Vol 8 ◽  
Author(s):  
Anne Barnoud ◽  
Valérie Cayol ◽  
Peter G. Lelièvre ◽  
Angélie Portal ◽  
Philippe Labazuy ◽  
...  
Keyword(s):  
Joint Inversion ◽  
A Priori ◽  
Constant Density ◽  
Acquisition Duration ◽  
Regularization Parameters ◽  
Absolute Density ◽  
Gravimetric Data ◽  
Ill Posed ◽  
Leave One Out

Imaging the internal structure of volcanoes helps highlighting magma pathways and monitoring potential structural weaknesses. We jointly invert gravimetric and muographic data to determine the most precise image of the 3D density structure of the Puy de Dôme volcano (Chaîne des Puys, France) ever obtained. With rock thickness of up to 1,600 m along the muon lines of sight, it is, to our knowledge, the largest volcano ever imaged by combining muography and gravimetry. The inversion of gravimetric data is an ill-posed problem with a non-unique solution and a sensitivity rapidly decreasing with depth. Muography has the potential to constrain the absolute density of the studied structures but the use of the method is limited by the possible number of acquisition view points, by the long acquisition duration and by the noise contained in the data. To take advantage of both types of data in a joint inversion scheme, we develop a robust method adapted to the specificities of both the gravimetric and muographic data. Our method is based on a Bayesian formalism. It includes a smoothing relying on two regularization parameters (an a priori density standard deviation and an isotropic correlation length) which are automatically determined using a leave one out criterion. This smoothing overcomes artifacts linked to the data acquisition geometry of each dataset. A possible constant density offset between both datasets is also determined by least-squares. The potential of the method is shown using the Puy de Dôme volcano as case study as high quality gravimetric and muographic data are both available. Our results show that the dome is dry and permeable. Thanks to the muographic data, we better delineate a trachytic dense core surrounded by a less dense talus.

scholarly journals Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations

2004 ◽  
Vol 2004 (37) ◽  
pp. 1973-1996 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Wide Class ◽  
Noise Level ◽  
A Priori ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.

scholarly journals An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hao Cheng ◽  
Ping Zhu ◽  
Jie Gao
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Regularization Method ◽  
A Priori ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Regularization Parameters ◽  
The Cauchy Problem

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. Thea priorianda posteriorirules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.

Quasi-solution approach for a two dimensional nonlinear inverse diffusion problem

2013 ◽  
Vol 219 (23) ◽  
pp. 10956-10960 ◽  
Author(s):  
Yiliang Liu ◽  
Salih Tatar ◽  
Suleyman Ulusoy
Keyword(s):  
Diffusion Problem ◽  
Two Dimensional ◽  
Solution Approach ◽  
Inverse Diffusion

scholarly journals Naming and necessity in a two-dimensional semantic framework

KÜLÖNBSÉG ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
János Kovács
Keyword(s):  
Philosophy Of Language ◽  
Causal Model ◽  
A Priori ◽  
Proper Names ◽  
Two Dimensional ◽  
A Posteriori ◽  
Semantic Framework ◽  
Contingent A Priori ◽  
Necessary A Posteriori ◽  
Naming And Necessity

This paper surveys the relevance of Kripke’s semantics of proper names. In his Naming and Necessity Kripke takes issue with Frege’s and Russell’s descriptive semantics of proper names. He proposes a new model called the causal model of proper names. Kripke’s model of the philosophy of language have challenged the relation of the metaphysical concepts necessity/contingency and the epistemological concepts apriority/a posteriority, respectively. Since Kant it has been accepted that all a priori truth is necessary, while all a posteriori truth is contingent. Kripke’s book has changed these tenets and nowadays it is accepted that the four concepts are independent of each other and that the complex concepts generated with them have instance.   This paper investigates Kripke’s arguments on necessity and apriority in a two-dimensional semantic framework. The paper argues that the two-dimensional model is in harmony with Kripke’s model although Soames has been claiming the opposite in several publications. The paper claims that Soames’ theory of direct reference is unable to account for necessary a posteriori and contingent a priori statements.

On the Recovery of Continuous Functions from Noisy Fourier Coefficients

2011 ◽  
Vol 11 (1) ◽  
pp. 75-82 ◽  
Author(s):  
Kosnazar Sharipov
Keyword(s):  
Error Bound ◽  
Fourier Coefficients ◽  
Regularization Parameter ◽  
A Priori ◽  
Continuous Functions ◽  
A Posteriori ◽  
Parameter Choice ◽  
Optimal Error ◽  
Ill Posed ◽  
Optimal Error Bound

AbstractWe consider the classical ill-posed problem of the recovery of continuous functions from noisy Fourier coefficients. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.

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