Estimation of a Regularisation Parameter for a Robin Inverse Problem

AbstractWe consider the nonlinear and ill-posed inverse problem where the Robin coefficient in the Laplace equation is to be estimated using the measured data from the accessible part of the boundary. Two regularisation methods are considered — viz. L2 and H1 regularisation. The regularised problem is transformed to a nonlinear least squares problem; and a suitable regularisation parameter is chosen via the normalised cumulative periodogram (NCP) curve of the residual vector under the assumption of white noise, where information on the noise level is not required. Numerical results show that the proposed method is efficient and competitive.

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Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

Complexity ◽

10.1155/2020/9619427 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

Ke Wang ◽

Guolin Liu ◽

Qiuxiang Tao ◽

Min Zhai

Keyword(s):

Least Squares ◽

Least Squares Method ◽

Nonlinear Least Squares ◽

Coefficient Matrix ◽

Nonlinear Parameters ◽

Nonlinear Least Squares Problem ◽

Ill Posed ◽

Characteristic Values ◽

Parameter Separation ◽

Fitting Problem

In this work, we combine the special structure of the separable nonlinear least squares problem with a variable projection algorithm based on singular value decomposition to separate linear and nonlinear parameters. Then, we propose finding the nonlinear parameters using the Levenberg–Marquart (LM) algorithm and either solve the linear parameters using the least squares method directly or by using an iteration method that corrects the characteristic values based on the L-curve, according to whether or not the nonlinear function coefficient matrix is ill posed. To prove the feasibility of the proposed method, we compared its performance on three examples with that of the LM method without parameter separation. The results show that (1) the parameter separation method reduces the number of iterations and improves computational efficiency by reducing the parameter dimensions and (2) when the coefficient matrix of the linear parameters is well-posed, using the least squares method to solve the fitting problem provides the highest fitting accuracy. When the coefficient matrix is ill posed, the method of correcting characteristic values based on the L-curve provides the most accurate solution to the fitting problem.

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A Regularization Homotopy Iterative Method for Ill-Posed Nonlinear Least Squares Problem and its Application

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.90-93.3268 ◽

2011 ◽

Vol 90-93 ◽

pp. 3268-3273 ◽

Cited By ~ 3

Author(s):

Li Min Tang

Keyword(s):

Iterative Method ◽

Least Squares ◽

Regularization Parameter ◽

Nonlinear Least Squares ◽

Least Squares Problems ◽

Least Squares Problem ◽

Nonlinear Least Squares Problem ◽

Calculation Results ◽

Ill Posed ◽

Defor Mation

A regularization homotopy iterative method established for ill-posed nonlinear least squares problem. Two new regularization parameter selecting strategies are proposed, which are called direct search method and interval division method. The calculation results of nonlinear least squares problems show that the regularization homotopy iterative method and parameter selecting strategies proposed in this paper are correctly and applicable. And also calculation results of nonlinear adjustment of free networks with rank deficiency of Jianglong bridge pier displacement defor-mation monitoring control network show that the method not only decrease the iterative matrix con-dition number, but also make the condition number small fluctuation in full iterative process.

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Conjugate Gradient Method for Estimation of Robin Coefficients

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.101213.260314a ◽

2014 ◽

Vol 4 (2) ◽

pp. 189-204 ◽

Cited By ~ 2

Author(s):

Yan-Bo Ma ◽

Fu-Rong Lin

Keyword(s):

Inverse Problem ◽

Integral Equation ◽

Conjugate Gradient Method ◽

Conjugate Gradient ◽

Gradient Method ◽

Nonlinear Least Squares ◽

Boundary Integral ◽

Numerical Examples ◽

Nonlinear Least Squares Problem ◽

Ill Posed

AbstractWe consider a Robin inverse problem associated with the Laplace equation, which is a severely ill-posed and nonlinear. We formulate the problem as a boundary integral equation, and introduce a functional of the Robin coefficient as a regularisation term. A conjugate gradient method is proposed for solving the consequent regularised nonlinear least squares problem. Numerical examples are presented to illustrate the effectiveness of the proposed method.

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