Interval-Based Approach for Uncertainty Propagation in Inverse Problems
Publication: Journal of Engineering Mechanics
Volume 141, Issue 1
Abstract
A new class of interval-based computational algorithms for parameter identification under uncertainty in structural engineering problems is presented. The iterative method allows passing directly from uncertain raw measurements to sharp (tight) bound estimates of the unknown parameters by exploiting interval FEMs and adjoint-based optimization techniques. Overestimation in interval width is handled successfully using the inclusion isotonicity property of interval arithmetic. First, an update of the iterative solution proceeds in a degenerated interval form until it becomes insignificant, and then the update is switched to full interval form, allowing uncertainty propagation and sensitivity analysis. A new containment-stopping criterion, which is intrinsic to intervals, is used. Example problems are then presented and discussed to show the effectiveness of the proposed inverse method in estimating the range of Young’s moduli from given ranges in displacements.
Get full access to this article
View all available purchase options and get full access to this article.
References
Alefeld, G., and Herzberger, J. (1983). Introduction to interval computations, Academic, New York.
Brown, R. G., and Hwang, P. Y. C. (1992). Introduction to random signals and applied Kalman filtering, 2nd Ed., Wiley, New York.
Chen, S. H., Lian, H. D., and Yang, X. W. (2002). “Interval static displacement analysis for structures with interval parameters.” Int. J. Numer. Methods Eng., 53(2), 393–407.
Cook, D., Fedele, F., and Yezzi, A. (2010). “Detection of spherical inclusions using active surfaces.” Proc., Int. Conf. on Synthetic Aperture Sonar and Synthetic Aperture Radar, Vol. 32, Institute of Acoustics, St. Albans, Hertfordshire, U.K., 43–50.
Corliss, G., Foley, G., and Kearfott, R. B. (2007). “Formulation for reliable analysis of structural frames.” Reliab. Comput., 13(2), 125–147.
De Munck, M., Moens, D., Desmet, W., and Vandepitte, D. (2008). “A response surface based optimisation algorithm for the calculation of fuzzy envelope FRFs of models with uncertain properties.” Comput. Struct., 86(10), 1080–1092.
Engl, H. W., Hanke, M., and Neubauer, A. (2000). Regularization of inverse problems, Kluwer Academic, Dordrecht, Netherlands.
Eppstein, M., Fedele, F., Laible, J. P., Zhang, C., Godavarty, A., and Sevick-Muraca, E. M. (2003). “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography.” IEEE Trans. Med. Imaging, 22(10), 1215–1223.
Fedele, F., Laible, J. P., and Eppstein, M. (2003). “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: Theory and vectorized implementation.” J. Comput. Phys., 187(2), 597–619.
Fedele, F., Muhanna, R. L., and Xiao, N. (2012). “Interval-based inverse problems with uncertainties.” Proc., 16th Int. Federation of Automatic Control (IFAC) Symp. on System Identification, Vol. 16, IFAC, New York, 1079–1084.
Hansen, E. (1965). “Interval arithmetic in matrix computation, part I.” SIAM J. Numer. Anal., 2(2), 308–320.
Hansen, E. (1992). Global optimization using interval analysis, Marcel Dekker, New York.
Hansen, E., and Walster, G. W. (2004). Global optimization using interval analysis, Marcel Dekker, New York.
Kalman, R. E. (1960). “A new approach to linear filtering and prediction problems.” J. Fluids Eng., 82(1), 35–45.
Köylüoğlu, H. U., Çakmak, A. Ş., and Nielsen, S. R. K. (1995). “Interval algebra to deal with pattern loading and structural uncertainties.” J. Eng. Mech., 1149–1157.
Marchuk, G. I. (1995). Adjoint equations and analysis of complex systems, Kluwer Academic, Dordrecht, Netherlands.
McWilliam, S. (2001). “Anti-optimisation of uncertain structures using interval analysis.” Comput. Struct, 79(4), 421–430.
Moens, D., and Vandepitte, D. (2005). “A survey of non-probabilistic uncertainty treatment in finite element analysis.” Comput. Meth. Appl. Mech. Eng., 194(12–16), 1527–1555.
Möller, B., Graf, W., and Beer, M. (2000). “Fuzzy structural analysis using alpha-level optimization.” Comput. Mech., 26(6), 547–565.
Moore, R. E. (1966). Interval analysis, Prentice Hall, Englewood Cliffs, NJ.
Moore, R. E. (1979). Methods and applications of interval analysis, Society for Industrial and Applied Mechanics (SIAM), Philadelphia.
Moore, R. E., Kearfott, R. B., and Cloud, M. J. (2008). Introduction to interval analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
Muhanna, R. L., and Mullen, R. L. (1995). “Development of interval based methods for fuzziness in continuum mechanics.” Proc., Int. Symp. of Uncertainty Modelling and Analysis–North American Fuzzy Information Processing Society (ISUMA–NAFIPS), IEEE Computer Society, Los Alamos, CA, 705–710.
Muhanna, R. L., and Mullen, R. L. (2001). “Uncertainty in mechanics problems—Interval–based approach.” J. Eng. Mech., 557–566.
Muhanna, R. L., Mullen, R. L., and Zhang, H. (2005). “Penalty-based solution for the interval finite element methods.” J. Eng. Mech., 1102–1111.
Muhanna, R. L., Zhang, H., and Mullen, R. L. (2007). “Interval finite element as a basis for generalized models of uncertainty in engineering mechanics.” Reliab. Comput., 13(2), 173–194.
Mullen, R. L., and Muhanna, R. L. (1999). “Bounds of structural response for all possible loading combinations.” J. Struct. Eng., 98–106.
Neumaier, A. (1990). Interval methods for systems of equations, Cambridge University Press, Cambridge, U.K.
Neumaier, A., and Pownuk, A. (2007). “Linear systems with large uncertainties, with applications to truss structures.” Reliab. Comput., 13(2), 149–172.
Popova, E., Iankov, R., and Bonev, Z. (2006). “Bounding the response of mechanical structures with uncertainties in all the parameters.” Proc., National Science Foundation (NSF) Workshop on Reliable Engineering Computing (REC), R. L. Muhannah and R. L. Mullen, eds., Georgia Institute of Technology, Atlanta, 245–265.
Pownuk, A. (2004). “Efficient method of solution of large scale engineering problems with interval parameters.” Proc., National Science Foundation (NSF) Workshop on Reliable Engineering Computing (REC), R. L. Muhannah and R. L. Mullen, eds., Georgia Institute of Technology, Atlanta, 305–316.
Qiu, Z., and Elishakoff, I. (1998). “Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis.” Comput. Meth. Appl. Mech. Eng., 152(3–4), 361–372.
Rama Rao, M. V., Mullen, R. L., and Muhanna, R. L. (2011). “A new interval finite element formulation with the same accuracy in primary and derived variables.” Int. J. Reliab. Saf., 5(3–4), 336–357.
Rump, S. M. (1983). “Solving algebraic problems with high accuracy.” A new approach to scientific computation, U. Kulisch and W. Miranker, eds., Academic, New York.
Rump, S. M. (1992). “On the solution of interval linear systems.” Computing, 47(3–4), 337–353.
Rump, S. M. (2001). “Self-validating methods.” Linear Algebra Appl., 324(1–3), 3–13.
Sun Microsystems. (2002). “Interval arithmetic in high performance technical computing.” White Paper, Sun Microsystems, Santa Clara, CA.
Tarantola, A. (1987). Inverse problem theory: Methods for data fitting and model parameter estimation, Elsevier, Amsterdam, Netherlands.
Tikhonov, A., and Arsenin, V. (1977). Solution of ill-posed problems, Wiley, New York.
Zhang, H. (2005). “Nondeterministic linear static finite element analysis: An interval approach.” Ph.D. dissertation, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 141 • Issue 1 • January 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Dec 30, 2012
Accepted: Apr 29, 2014
Published online: May 29, 2014
Published in print: Jan 1, 2015
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.