The Research On Engineering Inverse Problems Under Uncertainty Based On Probability And Interval

The research on engineering inverse problems is very important for promoting thedevelop ment of modern industria l techno logy. However, due to the limitation ofexperime ntal conditions, the comp lexity of structure models, the randomness ofmeasured data and the diversity of episte mic ability etc., researches on engineeringinverse proble ms are face with new challenges. Under these uncertaint ies, the obtainedresults have been unable to meet the require ment of practical engineering. Thetraditiona l deterministic inverse methods are diffic ult to give us a clear ind icatio n ofthe degree to which we can trust estimates of the resulting parameters. Specifically,they are unable to answer questions that how many errors present in the knownimprecise parameters are transferred to the solution? And what are the confidenceintervals of the obtained solutions? Therefore, how to effective ly answer these twoquestions is the core of researches on engineering inverse problems under uncertainty.Developing effic ient computational inverse techniques for assessment of theobtained solutions has important value of engineering application. However, it is stillat its preliminary stage for the research on engineering inverse proble ms underuncertainty, especia lly for the nonprobability uncertainty inverse algorithms, stud iesfor which are just getting started. So me key technica l difficulties rema in, such aseffective solution for the probability inverse problem, solutio n for the comp lex inverseproblem under uncertaint ies, hybrid measure ment of uncertainties, fusion ofmulti-source uncertainties etc.This dissertation conducts a systematica l research for the engineering inverseproblems under uncertainty based on three key proble ms: na mely output uncertaint ies,model uncertaint ies and multi-source uncertainties, and aims at contributing someuseful inverse algorithms. Some comp utationa l methods in the present uncertaintyanalys is fie ld, such as probability, interva l and evidence, are extended into engineeringinverse proble ms under uncertainty fie ld, and whereby several effic ient inversealgorithms are constructed. As a result, the fo llowing studies are carried out in thisdissertatio n:(1) The uncertainty inverse proble ms with insufficie ncy and imprecis ion in theinput and/or output parameters are widely existing and unsolved in the practicalengineering. The ins uffic iency refers to the partly known parameters in the input and/or output, and the imprecis ion refers to the measurement errors of these ones. Acomb ined method is proposed to deal with such problems. In this method, theimprecis ion of these known parameters can be described by probability d istributionwith a certain mean va lue and varia nce. Sensitive matrix method is first used totransform the ins uffic ient formulation in the input and/or output to a resolvable one,and then the mean va lues of these unknown parameters can be identified bymaximizing the likelihood of the measurements. Finally, to quantify the uncertaintypropagatio n, confidence intervals of the obtained solutions are calc ulated based onlinearization and Monte Carlo methods. It is demonstrated that the proposed methodoffers a new viewpoint a nd strategy for effective ly quantifying the influence of therandom uncertaint ies to the obtained results.(2) A nove l algorithm is presented to promote the efficie ncy and accuracy ofBayesian approach for fast samp ling of posterior distributions of the unknownstructure para meters, whic h arise from a computatio nal cost proble m in Bayes ianidentifications. In this algorithm, the approximation model based on radius basisfunction is first used to replace the actua l joint posterior distribution o f the unknownparameters. The adaptive densifying technique is then suggested to guarantee theaccuracy of the approximation model by reconstructing the m with dens ified samp les.Fina lly, the marginal posterior distributions for each parameter with fine accuracy canbe efficie ntly achie ved by using the Markov Cha in Monte Carlo method based on thisdens ified approximation model. Two numerical examp les and two engineeringapplications are investigated, and the identified results show that the present methodcan achie ve significant computationa l ga ins without sacrific ing the accuracy.(3) An inverse method based on interva l ana lysis is presented for the comp lexengineering inverse proble m under uncertaint ies. In this method, interva l numbers areused to describe the uncertain parameters based on interva l mathematics theory. Byusing interva l analys is method based on the first-order Taylor expansion, theuncertainty inverse problem is transformed into two kinds of deterministic inverseproblems, i.e. determinatio n of a point a nd a radius of the unknown parameters. Usingthe deterministic inverse method based on trust region approximation manage mentstrategy and intergeneration projection genetic algorithm (IP-GA) to solving thesedeterministic inverse proble ms, respectively. Fina lly, the lower and upper bound of theunknown parameters can be determined by using interva l comp utation. It isdemonstrated that this method can give us a clear ind ication of the degree to which wecan trust estimates of the resulting parameters for the comple x engineering inverse problem under uncertaint ies.(4) An inverse method based on model validation is presented. This method firstuses the model va lidation techniques to obtain the best numerical model, throughwhich effects of the uncertainty factors on the numerical model can be well revealed.Then us ing the appropriate inverse method for solving deterministic inverse problem,which is constrted based on this model. It is demonstrated that this method can beassess the computationa l results under uncertaint ies in the model va lidation processe,which can avoid constructing the comp lex uncertainty algorithm in the inverse process.This method has a high computational e fficienc y, and provides a new idea foreffective ly quantifying the influe nce of the non-phys ical uncertainties to the obtainedresults.(5) A novel method based on Bayesian approach and interva l ana lys is is presentedfor uncertainty parameter identifications, which can deal with both me asurement no iseand model uncertainty. In this method, measurement no ises are described as randomvariables which are obeyed a certain probability distribution from the experiment. Anduncertain parameters of the forward model are treated as intervals, in which only the irbounds of the uncertainty are needed. For small uncertainty levels, model responsescan be approximated as a linear function of the uncertain parameters by using thefirst-order Taylor expansion. Because of the existence of the interval parameters, aposterior probability dens ity distribution strip enclosed by two bound ing distributionsis then resulted in the posterior space, instead of a single distribution as we usuallyobtain through Monte Carlo Markov Cha in method (MCMC) in traditio nal Bayes ianidentification. A monotonicity analys is is adopted for margina l posterior distributiontransformation, through whic h effects of the interval parameters on the posteriordistributio n strip can be well revealed. Based on the monotonicity analys is, fina lly, thepoint estimates and confidence interva ls of the unknown parameters are obtained fromtheir posterior distribution strip. Three numerical exa mples are investigated, and finenumerical results are obtained.(6) An inverse method based on evidence theory is developed for fusio n ofmulti-source uncertainties. A combined fus ion method is presented to deal with highlyconflicting evidences weighted and low ly conflicting evidences focused. Usingconfidence degree based on the evidence distance to weight the highly conflict ingevidences. A focus coeffic ient, whic h represents the comb ination degree of lowconflict evidence, is defined to focus the lowly conflicting evidences. This method canboth deal with e vidence conflict and focus proble ms, and relaxes effective ly the evidence fus ion conditions, so that extend the application fie ld of evidence theory inpractical engineering inverse proble ms.