scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

scholarly journals Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 705 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Three Dimensional ◽  
Wave Number ◽  
Modified Helmholtz Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

scholarly journals A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Convergence Estimate ◽  
Ill Posed

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

scholarly journals Smoothing and Regularization with Modified Sparse Approximate Inverses

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
T. Huckle ◽  
M. Sedlacek
Keyword(s):  
Laplace Operator ◽  
Constant Coefficient ◽  
Regularization Method ◽  
Low Frequency ◽  
Attractive Alternative ◽  
Iterative Regularization ◽  
Dynamic Computation ◽  
Approximate Inverses ◽  
Ill Posed ◽  
Original Information

Sparse approximate inverses which satisfy have shown to be an attractive alternative to classical smoothers like Jacobi or Gauss-Seidel (Tang and Wan; 2000). The static and dynamic computation of a SAI and a SPAI (Grote and Huckle; 1997), respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties (Bröker et al.; 2001). Here, we are interested in developing preconditioners that can incorporate probing conditions for improving the approximation relative to high- or low-frequency subspaces. We present analytically derived optimal smoothers for the discretization of the constant-coefficient Laplace operator. On this basis, we introduce probing conditions in the generalized Modified SPAI (MSPAI) approach (Huckle and Kallischko; 2007) which yields efficient smoothers for multigrid. In the second part, we transfer our approach to the domain of ill-posed problems to recover original information from blurred signals. Using the probing facility of MSPAI, we impose the preconditioner to act as approximately zero on the noise subspace. In combination with an iterative regularization method, it thus becomes possible to reconstruct the original information more accurately in many cases. A variety of numerical results demonstrate the usefulness of this approach.

scholarly journals Comparative study of conjugate gradient algorithms performance on the example of steady-state axisymmetric heat transfer problem

2013 ◽  
Vol 34 (3) ◽  
pp. 15-44 ◽  
Author(s):  
Paweł Ocłoń ◽  
Stanisław Łopata ◽  
Marzena Nowak
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Conjugate Gradient ◽  
Transfer Problem ◽  
Heat Conduction Problem ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Conjugate Gradient Algorithms ◽  
Gradient Algorithms ◽  
Biconjugate Gradient

Abstract The finite element method (FEM) is one of the most frequently used numerical methods for finding the approximate discrete point solution of partial differential equations (PDE). In this method, linear or nonlinear systems of equations, comprised after numerical discretization, are solved to obtain the numerical solution of PDE. The conjugate gradient algorithms are efficient iterative solvers for the large sparse linear systems. In this paper the performance of different conjugate gradient algorithms: conjugate gradient algorithm (CG), biconjugate gradient algorithm (BICG), biconjugate gradient stabilized algorithm (BICGSTAB), conjugate gradient squared algorithm (CGS) and biconjugate gradient stabilized algorithm with l GMRES restarts (BICGSTAB(l)) is compared when solving the steady-state axisymmetric heat conduction problem. Different values of l parameter are studied. The engineering problem for which this comparison is made is the two-dimensional, axisymmetric heat conduction in a finned circular tube.

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