A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces

Eldén, Lars; Simoncini, Valeria

doi:10.1088/0266-5611/25/6/065002

Cited by 24 publications

(25 citation statements)

References 36 publications

(62 reference statements)

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“…It is also well known that this problem can be stabilized by adding the condition requiring that u ≤ M [9,7,8].…”

Section: The Solution Depends Continuously On the Data (Stability)mentioning

confidence: 99%

Non-linear inverse geothermal problems

Wokiyi¹

2017

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The inverse geothermal problem consist of estimating the temperature distribution below the earth's surface using temperature and heat-flux measurements on the earth's surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard's definition of well-posedness.We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces.Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a wellposed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.i

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“…It is also well known that this problem can be stabilized by adding the condition requiring that u ≤ M [9,7,8].…”

Section: The Solution Depends Continuously On the Data (Stability)mentioning

confidence: 99%

Non-linear inverse geothermal problems

Wokiyi¹

2017

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“…This trick has been widely used to deal with the nonhomogeneous ill-posed problems, see, e.g., the appendix in [25].…”

Section: Regularization Parameter Choicementioning

confidence: 99%

“…The Cauchy problem for an elliptic equation is a classical ill-posed problem and occurs in several important applications, such as inverse scattering [17,37], electrical impedance tomography [10], optical tomography [7], and thermal engineering [23]. The topic is treated in several monographs [16,36,37,41,42,45], and in numerous papers, see [1,3,6,8,9,18,24,25,34,47,48,51,57,61] and the references therein. Even if some of the theoretical investigations are quite general, the numerical procedures proposed are typically for the two dimensional case and often only valid for the problem with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Feng¹,

Eldén²

2013

Inverse Problems

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An ill-posed Cauchy problem for a 3D elliptic PDE with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability error estimates are given. For the numerical implementation, a large sparse system is obtained from the discretizing the QBV problem using finite difference method (FDM). A LeftPreconditioner Generalized Minimum Residual (GMRES) method is used to solve the large system effectively. For the preconditioned system, a fast solver using the Fast Fourier Transform (FFT). Numerical results show that the method works well.

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“…where f (x) ∈ L 2 (R) is unknown and need to be determined, g(x) is the data given approximately by g δ (x) satisfying (9).…”

Section: Applicationsmentioning

confidence: 99%

Numerical pseudodifferential operator and Fourier regularization

Qian

2009

Adv Comput Math

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The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.

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A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces

Cited by 24 publications

References 36 publications

Non-linear inverse geothermal problems

Non-linear inverse geothermal problems

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Numerical pseudodifferential operator and Fourier regularization

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