An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

He, Dongdong; Pan, Kejia

doi:10.1016/j.apm.2014.11.027

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…Formally, we have established a so-called Cauchy problem on an annulus (He and Pan, 2015;Jaoua et al, 2008;Wei and Zhou, 2009). The approach can be extended to any topologically equivalent two-dimensional domain by applying, for example, conformal mapping (Liu, 2015;Pommerenke, 1992).…”

Section: Formulation Of Air Gap Field As An Initial Value Problemmentioning

confidence: 99%

Towards design of electrical machines from given air gap field

Poutala

Suuriniemi

Tarhasaari

et al. 2017

COMPEL

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Purpose The purpose of this paper is to introduce a reverted way to design electrical machines. The authors present a work flow that systematically yields electrical machine geometries from given air gap fields. Design/methodology/approach The solution process exploits the inverse Cauchy problem. The desired air gap field is inserted to this as the Cauchy data, and the solution process is stabilized with the aid of linear algebra. Findings The results are verified by solving backwards the air gap fields in the standard way. They match well with the air gap fields inserted as an input to the system. Originality/value The paper reverts the standard design work flow of electrical motor by solving directly for a geometry that yields the desired air gap field. In addition, a stabilization strategy for the underlying Cauchy problem is introduced.

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Section: Formulation Of Air Gap Field As An Initial Value Problemmentioning

confidence: 99%

Towards design of electrical machines from given air gap field

Poutala

Suuriniemi

Tarhasaari

et al. 2017

COMPEL

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show abstract

“…Due to its importance the literature on this problem is vast. Recent results on this particular problem includes meshless methods [12,13], an optimal regularization method [14], a fitting algorithm [15], singular boundary method [16], discrete Fourier transform method [17], a method based on proper solution space [18], and an energy regularization method [19].…”

Section: Introductionmentioning

confidence: 99%

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

Tadi

Radenković

2021

Mathematics

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This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise.

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“…Stable method for solving three-dimensional inverse Cauchy-type problem of heat transfer based on the BEM, regularization technique called truncated SVD method (TSVD) and Tikhonov regularization technique with a selection of the regularization parameter using the L-curve was discussed in the reference paper (Wang et al , 2016). Currently, modified methods of Tikhonov regularization are also used (He and Pan, 2015;Ma and Fu, 2012; Yang et al , 2015). The reference paper (Yeih et al , 2014) includes the solution of the non-linear stationary Cauchy problem with the use of the modified Tikhonov regularization technique.…”

Section: Introductionmentioning

confidence: 99%

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

Joachimiak

Ciałkowski

Frąckowiak

2019

HFF

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Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Findings Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Practical implications Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. Originality/value In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

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An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

Cited by 6 publications

References 31 publications

Towards design of electrical machines from given air gap field

Towards design of electrical machines from given air gap field

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

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