Selection of regularization parameter for l1-regularized damage detection

Abstract The l1 regularization technique has been developed for structural health monitoring and damage detection through employing the sparsity condition of structural damage. The regularization parameter, which controls the trade-off between data fidelity and solution size of the regularization problem, exerts a crucial effect on the solution. However, the l1 regularization problem has no closed-form solution, and the regularization parameter is usually selected by experience. This study proposes two strategies of selecting the regularization parameter for the l1-regularized damage detection problem. The first method utilizes the residual and solution norms of the optimization problem and ensures that they are both small. The other method is based on the discrepancy principle, which requires that the variance of the discrepancy between the calculated and measured responses is close to the variance of the measurement noise. The two methods are applied to a cantilever beam and a three-story frame. A range of the regularization parameter, rather than one single value, can be determined. When the regularization parameter in this range is selected, the damage can be accurately identified even for multiple damage scenarios. This range also indicates the sensitivity degree of the damage identification problem to the regularization parameter.

[1]  V. A. Morozov,et al.  Methods for Solving Incorrectly Posed Problems , 1984 .

[2]  Richard G Baraniuk,et al.  More Is Less: Signal Processing and the Data Deluge , 2011, Science.

[3]  Demetri Terzopoulos,et al.  Regularization of Inverse Visual Problems Involving Discontinuities , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  C. Fritzen,et al.  DAMAGE DETECTION BASED ON MODEL UPDATING METHODS , 1998 .

[5]  U. Hämarik,et al.  On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data , 2006 .

[6]  Charles R. Farrar,et al.  Compressed sensing techniques for detecting damage in structures , 2013 .

[7]  P. Shearer,et al.  Compressive sensing of the Tohoku‐Oki Mw 9.0 earthquake: Frequency‐dependent rupture modes , 2011 .

[8]  Mark A. Lukas,et al.  On the discrepancy principle and generalised maximum likelihood for regularisation , 1995, Bulletin of the Australian Mathematical Society.

[9]  Stephen P. Boyd,et al.  An Interior-Point Method for Large-Scale l1-Regularized Logistic Regression , 2007, J. Mach. Learn. Res..

[10]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[11]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[12]  You-Lin Xu,et al.  Inverse substructure method for model updating of structures , 2012 .

[13]  Yong Xia,et al.  Experimental Investigation on Statistical Moment-based Structural Damage Detection Method , 2009 .

[14]  Yongchao Yang,et al.  Output-only modal identification by compressed sensing: Non-uniform low-rate random sampling , 2015 .

[15]  X. Y. Li,et al.  Adaptive Tikhonov regularization for damage detection based on nonlinear model updating , 2010 .

[16]  H. L. Taylor,et al.  Deconvolution with the l 1 norm , 1979 .

[17]  S. S. Law,et al.  DAMAGE LOCALIZATION BY DIRECTLY USING INCOMPLETE MODE SHAPES. TECHNICAL NOTE , 2000 .

[18]  Norris Stubbs,et al.  Crack detection in beam-type structures using frequency data , 2003 .

[19]  Gu-Quan Song,et al.  Structural damage identification by extended Kalman filter with L1-norm regularization scheme , 2017 .

[20]  Shun Weng,et al.  L1 regularization approach to structural damage detection using frequency data , 2015 .

[21]  Satish Nagarajaiah,et al.  Blind denoising of structural vibration responses with outliers via principal component pursuit , 2013, Structural Control and Health Monitoring.

[22]  Charles R. Farrar,et al.  Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review , 1996 .

[23]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[24]  Songye Zhu,et al.  Wavelet-based multi-scale finite element modeling and modal identification for structural damage detection , 2017 .

[25]  S. S. Law,et al.  Damage localization in plate structures from uniform load surface curvature , 2004 .

[26]  I. Johnstone.  On Minimax Estimation of a Sparse Normal Mean Vector , 1994 .

[27]  Satish Nagarajaiah,et al.  Structural damage identification via a combination of blind feature extraction and sparse representation classification , 2014 .

[28]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[29]  Yong Xia,et al.  Structural damage detection based on l1 regularization using natural frequencies and mode shapes , 2018 .

[30]  David L. Phillips,et al.  A Technique for the Numerical Solution of Certain Integral Equations of the First Kind , 1962, JACM.

[31]  Mark Richardson,et al.  PARAMETER ESTIMATION FROM FREQUENCY RESPONSE MEASUREMENTS USING RATIONAL FRACTION POLYNOMIALS (TWENTY YEARS OF PROGRESS) , 1982 .

[32]  Ying Wang,et al.  Damage Identification Scheme Based on Compressive Sensing , 2015, J. Comput. Civ. Eng..

[33]  Mark A. Lukas,et al.  Comparing parameter choice methods for regularization of ill-posed problems , 2011, Math. Comput. Simul..

[34]  Sergios Theodoridis,et al.  Sparsity-Aware Learning and Compressed Sensing: An Overview , 2012, ArXiv.

[35]  Edwin Reynders,et al.  Towards a more realistic modelling of the uncertainty on identified mode shapes due to measurement noise , 2011 .

[36]  You-Lin Xu,et al.  A new statistical moment-based structural damage detection method , 2008 .

[37]  Zhishen Wu,et al.  Assessment of Vibration-based Damage Identification Methods Using Displacement and Distributed Strain Measurements , 2009 .

[38]  B. Weber,et al.  Structural damage detection using nonlinear parameter identification with Tikhonov regularization , 2007 .

[39]  Yingsan Geng,et al.  Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields , 2018 .

[40]  C. D. Zhang,et al.  Comparative studies on damage identification with Tikhonov regularization and sparse regularization , 2016 .

[41]  John E. Mottershead,et al.  Model Updating In Structural Dynamics: A Survey , 1993 .

[42]  Nawawi Chouw,et al.  Vibration-based damage identification of an unreinforced masonry house model , 2017 .

[43]  Yuequan Bao,et al.  Emerging data technology in structural health monitoring: compressive sensing technology , 2014 .

[44]  S. Mallat.  A wavelet tour of signal processing , 1998 .