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# A Combined Sensitive Matrix Method and Maximum Likelihood Method for Uncertainty Inverse Problems

W. Zhang1, X. Han1,2, J. Liu1, Z. H. Tan1

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
Corresponding author. Tel: +86 731 88823993; fax: +86 731 88823945, E-mail address: hanxu@hnu.edu.cn (X. Han).

Computers, Materials & Continua 2011, 26(3), 201-226. https://doi.org/10.3970/cmc.2011.026.201

## Abstract

The uncertainty inverse problems with insufficiency and imprecision in the input and/or output parameters are widely existing and unsolved in the practical engineering. The insufficiency refers to the partly known parameters in the input and/or output, and the imprecision refers to the measurement errors of these ones. In this paper, a combined method is proposed to deal with such problems. In this method, the imprecision of these known parameters can be described by probability distribution with a certain mean value and variance. Sensitive matrix method is first used to transform the insufficient formulation in the input and/or output to a resolvable one, and then the mean values of these unknown parameters can be identified by maximizing the likelihood of the measurements. Finally, to quantify the uncertainty propagation, confidence intervals of the obtained solutions are calculated based on linearization and Monte Carlo methods. Two numerical examples are presented to demonstrate the effectiveness of the present method.

## Keywords

APA Style
Zhang, W., Han, X., Liu, J., Tan, Z.H. (2011). A combined sensitive matrix method and maximum likelihood method for uncertainty inverse problems. Computers, Materials & Continua, 26(3), 201-226. https://doi.org/10.3970/cmc.2011.026.201
Vancouver Style
Zhang W, Han X, Liu J, Tan ZH. A combined sensitive matrix method and maximum likelihood method for uncertainty inverse problems. Comput Mater Contin. 2011;26(3):201-226 https://doi.org/10.3970/cmc.2011.026.201
IEEE Style
W. Zhang, X. Han, J. Liu, and Z.H. Tan "A Combined Sensitive Matrix Method and Maximum Likelihood Method for Uncertainty Inverse Problems," Comput. Mater. Contin., vol. 26, no. 3, pp. 201-226. 2011. https://doi.org/10.3970/cmc.2011.026.201

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