基于概率和区间的工程不确定性反问题研究
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摘要
工程反问题的研究对促进现代工业技术发展具有重要意义。然而，由于实验条件的局限性、测量数据的随机性、结构模型的复杂性以及认知能力的差异性等众多不确定性因素的影响，导致工程反问题的研究面临着新的挑战。这主要表现为，在输入、输出和模型中存在不确定性因素条件下，确定性反求方法很难定量度量反求结果的可信性，因而需要发展不确定性反求方法对反求结果进行有效评价。目前，在不确定性反问题的求解效率、不确定性反问题向确定性反问题的转化、多源不确定性的混合度量和融合等诸多方面尚存在一系列的技术难点有待解决。
为此，本文围绕工程不确定性反问题中输出不确定性、模型不确定性和多源不确定性三方面问题，力求在工程不确定性反问题的实用性计算反求方法方面做出一系列的探索与尝试。借鉴目前不确定性研究领域中概率、区间和证据理论等求解策略，构造出具有工程实用价值的不确定性反求方法。从而，一定程度上能够针对固体力学、车辆工程和冲击工程中典型不确定性反问题的反求结果进行有效评价。鉴于此思路，本文开展并完成了以下几方面的工作：
（1）针对输入和（或）输出中只有部分参数已知且这些已知参数服从随机分布的工程反问题，提出了基于敏感性矩阵法和最大似然法相结合的反求方法。首先，利用敏感性矩阵方法将输入和（或）输出中具有不充足参数的欠定方程转化为适定方程。然后，基于这个适定方程，利用最大似然方法反求出未知参数的均值和置信区间。该方法能够结合敏感性矩阵法和最大似然法的各自优点，有效地量化部分已知参数的随机不确定性对反求结果的影响。
（2）针对贝叶斯识别理论在实际工程应用中的计算效率问题，提出了基于自适应近似加密技术的抽样方法。首先，采用径向基函数近似代替真实的未知参数联合后验概率密度分布函数。然后，利用自适应加密技术确保近似模型的精度。最后，基于这个高精度近似模型，采用马尔科夫链蒙特卡罗法对未知参数的边缘后验概率密度分布进行求解。该方法不需大量调用原耗时的仿真模型，而是每次利用自适应近似模型进行准确而快速地抽样，从而大幅提高贝叶斯识别的计算效率。
（3）针对输出和模型中不确定性参数难以采用精确概率描述的复杂工程反问题，发展了基于区间分析的反求方法。以区间数学理论为基础采用区间来定量描述不确定性。利用区间分析方法将不确定性反问题转化为两类确定性反问题，即未知参数中点与半径的反求。利用基于近似模型信赖域管理策略的反求方法和隔代遗传算法稳定实现这两类确定性反问题的求解，进而通过区间运算实现未知参数上下界的确定。在不确定性参数获取样本信息较少的情况下，该方法能够高效客观地评价输出和模型中同时存在的区间不确定性对反求结果的影响。
（4）针对处理模型中非物理不确定性因素的影响，提出了基于模型确认的反求方法。首先，采用模型确认方法获得最佳的数值模型，并对影响数值模型计算结果的不确定性因素进行评价。然后，利用合适的反求方法求解基于此模型构造出的确定性反问题。该方法将不确定性因素在模型建立时进行有效的处理和评价，进而在反求过程中避免构造复杂繁琐的不确定性算法。另外，该方法概念明确、操作简单，计算效率高，在考虑模型非物理因素不确定性对反求结果影响的意义上，是基于区间分析的反求方法的重要补充。
（5）针对输出和模型中同时存在不同类型不确定性参数的工程反问题，提出了基于概率与区间混合度量的反求方法。该方法在贝叶斯理论框架下，采用概率描述测量数据中的随机不确定性，采用区间描述结构模型中的认知不确定性。由于不确定性区间参数的存在，导致未知参数边缘后验概率密度分布并不是单一形式，而是带状形式。分析区间不确定性参数对此带状分布空间的影响，然后利用马尔科夫链蒙特卡罗法求解未知参数的均值区间和置信区间。该方法能够结合不同类型的不确定性处理方法的各自优点对反求结果进行有效评价。
（6）针对工程反问题中相同类型不确定性参数的多源数据融合，发展了基于证据理论的反求方法，提出了一种基于高冲突证据加权和低冲突证据聚焦相结合的多源证据融合方法。基于证据距离的可信程度对高冲突证据进行加权平均，进而克服冲突性证据融合问题。同时，继承与运算的优点并引入聚焦系数，进而反映低冲突证据间的交叉融合程度。该方法能够同时考虑证据冲突和聚焦问题，有效地放宽了多源证据融合的条件，同时拓宽了证据理论在评价多源不确定性对反求结果影响中的应用范围。
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.
为此，本文围绕工程不确定性反问题中输出不确定性、模型不确定性和多源不确定性三方面问题，力求在工程不确定性反问题的实用性计算反求方法方面做出一系列的探索与尝试。借鉴目前不确定性研究领域中概率、区间和证据理论等求解策略，构造出具有工程实用价值的不确定性反求方法。从而，一定程度上能够针对固体力学、车辆工程和冲击工程中典型不确定性反问题的反求结果进行有效评价。鉴于此思路，本文开展并完成了以下几方面的工作：
（1）针对输入和（或）输出中只有部分参数已知且这些已知参数服从随机分布的工程反问题，提出了基于敏感性矩阵法和最大似然法相结合的反求方法。首先，利用敏感性矩阵方法将输入和（或）输出中具有不充足参数的欠定方程转化为适定方程。然后，基于这个适定方程，利用最大似然方法反求出未知参数的均值和置信区间。该方法能够结合敏感性矩阵法和最大似然法的各自优点，有效地量化部分已知参数的随机不确定性对反求结果的影响。
（2）针对贝叶斯识别理论在实际工程应用中的计算效率问题，提出了基于自适应近似加密技术的抽样方法。首先，采用径向基函数近似代替真实的未知参数联合后验概率密度分布函数。然后，利用自适应加密技术确保近似模型的精度。最后，基于这个高精度近似模型，采用马尔科夫链蒙特卡罗法对未知参数的边缘后验概率密度分布进行求解。该方法不需大量调用原耗时的仿真模型，而是每次利用自适应近似模型进行准确而快速地抽样，从而大幅提高贝叶斯识别的计算效率。
（3）针对输出和模型中不确定性参数难以采用精确概率描述的复杂工程反问题，发展了基于区间分析的反求方法。以区间数学理论为基础采用区间来定量描述不确定性。利用区间分析方法将不确定性反问题转化为两类确定性反问题，即未知参数中点与半径的反求。利用基于近似模型信赖域管理策略的反求方法和隔代遗传算法稳定实现这两类确定性反问题的求解，进而通过区间运算实现未知参数上下界的确定。在不确定性参数获取样本信息较少的情况下，该方法能够高效客观地评价输出和模型中同时存在的区间不确定性对反求结果的影响。
（4）针对处理模型中非物理不确定性因素的影响，提出了基于模型确认的反求方法。首先，采用模型确认方法获得最佳的数值模型，并对影响数值模型计算结果的不确定性因素进行评价。然后，利用合适的反求方法求解基于此模型构造出的确定性反问题。该方法将不确定性因素在模型建立时进行有效的处理和评价，进而在反求过程中避免构造复杂繁琐的不确定性算法。另外，该方法概念明确、操作简单，计算效率高，在考虑模型非物理因素不确定性对反求结果影响的意义上，是基于区间分析的反求方法的重要补充。
（5）针对输出和模型中同时存在不同类型不确定性参数的工程反问题，提出了基于概率与区间混合度量的反求方法。该方法在贝叶斯理论框架下，采用概率描述测量数据中的随机不确定性，采用区间描述结构模型中的认知不确定性。由于不确定性区间参数的存在，导致未知参数边缘后验概率密度分布并不是单一形式，而是带状形式。分析区间不确定性参数对此带状分布空间的影响，然后利用马尔科夫链蒙特卡罗法求解未知参数的均值区间和置信区间。该方法能够结合不同类型的不确定性处理方法的各自优点对反求结果进行有效评价。
（6）针对工程反问题中相同类型不确定性参数的多源数据融合，发展了基于证据理论的反求方法，提出了一种基于高冲突证据加权和低冲突证据聚焦相结合的多源证据融合方法。基于证据距离的可信程度对高冲突证据进行加权平均，进而克服冲突性证据融合问题。同时，继承与运算的优点并引入聚焦系数，进而反映低冲突证据间的交叉融合程度。该方法能够同时考虑证据冲突和聚焦问题，有效地放宽了多源证据融合的条件，同时拓宽了证据理论在评价多源不确定性对反求结果影响中的应用范围。
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.
引文
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