The Research On Computational Methods For Engineering Inverse Problems Under Interval Uncertainty

Inverse problems can be simply defined as assessing the causes from the results in contrast with the forward problems.Inverse problems widely exist in engineering,medical science,environment,meteorology,economy and other fields.The traditional inverse problem for the engineering problems are based on deterministic parameters and system models,and be solved by classical inverse methods.However,for the influences of the differences in cognitive styles,the limitations of experimental conditions and the randomness of measurement data,the traditional inverse methods are difficult to deal with the uncertain inverse problems.Therefore,developing the solving methods for the uncertain inverse problem is of great significance for the developments of modern industrial technology.The stochastic inverse method is a type of classical uncertain inverse problem,where a large number of strategy information is required to construct precise probability distribution functions of variables.However,for many practical engineering problems,obtaining sufficient uncertainty information is always expensive and sometimes very difficult,which may cause limitations in the applications of uncertainty inverse method.The interval inverse method is a relatively new uncertain inverse method,in which the interval model is used to describe the uncertainty of a variable.The interval model only needs the variation bounds of a variable,which can dramatically reduce the dependency of uncertain information and reflect better convenience and economy.Interval inverse method has been attracting more and more attention from researchers in recent years and shows a significant application potential.However,the research of interval inverse method still in its primary stage,especially for nonlinear interval inverse problem,which is studied in the last decades.There have some technical difficulties that need to be solved,such as comprehensive properties of interval analysis,low efficiency of double nested optimization etc.This dissertation carries out systematic research for the nonlinear interval inverse problem and aims at making some innovative researches and explorations on practical computational methods.The successful implementation of the inverse solution depends on repeated calculations of interval analysis.Therefore,an interval analysis method with higher computational efficiency and accuracy will be a better promotion to the solution of interval inverse problem.First,an efficient interval analysis method for the nonlinear forward problems is proposed,which realizes high-efficiency to calculate the interval outputs from interval inputs,and the idea of this part is used in the whole dissertation,which can be said to be the basis of the whole dissertation.On the other hand,in the aspect of the inverse method,three types of inverse problems are considered,namely output uncertainty inverse problems,model uncertainty inverse problems and multi-source uncertainty inverse problems,and corresponding effective inverse strategies are proposed to solve the low efficiency caused by double loop nested optimization problems.As a result,the following works are carried out and finished in this dissertation:(1)For a general uncertain forward problem,a nonlinear interval analysis method is proposed.First,the original function is transformed into a sum of several one-dimensional sub-functions based on the dimension-reduction method.Second,quadratic approximation models concerning the variables are created for the subfunctions through the second-order Taylor expansion method,and whereby complete quadratic form can be constructed.Third,the power function calculation of interval arithmetic is applied to calculate the corresponding intervals response.At last,combining the response intervals of sub-functions to obtain the response interval of the original function.The proposed method enables each variable appearing only once in a function,which can alleviate the overestimation of interval arithmetic to a great extent,and then it only acquires the first and second-order derivatives of onedimensional function which can be efficiently obtained.(2)For the output uncertainty inverse problem,a nonlinear interval inverse method is proposed.The interval inverse problem is transformed into an optimization problem that aims to minimize the 2-norm of the error function concerning computed responses and measured responses.Directly solving the optimization problem will encounter a double-loop optimization problem,which causes low efficiency of output uncertainty inverse calculation.A new sequential method is proposed to solve the transformed optimization problem based on the dimension-reduction interval method.At each iterative step,an approximate dimension-reduction model is first constructed at current interval midpoints to optimize interval radii;second,based on current interval radii,a new dimension-reduction model is constructed to optimize interval midpoints.The alternately solving of interval midpoints and radii dramatically increase the convergence efficiency and decrease the times of interval analyses.Moreover,the dimension-reduction interval analysis method does not need the timeconsuming original functions,but directly applies the interval arithmetic to calculate the function responding intervals.Therefore,the proposed method can improve the efficiency of solving output uncertainty inverse problems to a great extent.(3)For the output uncertainty inverse problem,a novel interval inverse framework is proposed.Under this framework,the interval inverse problem is transformed into an interval analysis problem and deterministic optimization problem.The position vector is created to establish the relationship of point vector in the interval vector and interval midpoint and radius vectors.At each iterative step,the corresponding position vectors of maximum and minimum points in every function at the current interval are firstly calculated by interval analysis method,based on which deterministic optimization can be constructed,and then interval solution can be updated by solving the established deterministic optimization.In the process of solving the position vector by interval analysis method,there is not any particular interval analysis method that is introduced.Therefore,theoretically,appropriate interval analysis methods can be selected in this framework according to practical engineering problems.Moreover,in an engineering problem,different interval analysis methods can be used to solve the corresponding parts.Therefore,the proposed method is called interval inverse framework,in which the double-loop nested optimization problem is transformed into interval analyses and deterministic optimizations that sequentially solved.(4)For the model uncertainty inverse problem,an efficient nonlinear interval inverse method is proposed based on the dimension-reduction method and adaptive strategy.Different from the output uncertainty problem,model uncertainty inverse problem can not be solved by directly applying the already proposed methods.Therefore,it is necessary to develop a corresponding model uncertainty inverse method.The interval inverse analysis method is used to transform model uncertainty inverse problem into an inverse-analysis problem with a double-loop nested problem,in which the out-loop is uncertainty problem and the inner-loop is deterministic inverse calculation.The double-loop problem will cause the low efficiency of solving the model uncertainty problem.An effective nonlinear model uncertainty inverse method is proposed.First,the inverse-analysis problem is transformed into several one-dimensional inverse anlysis problems by the dimension-reduction method.Second,for each one-dimensional inverse-analysis problem,the important points are selected to construct deterministic inverse problems based on adaptive strategy.Third,the interval of parameters can be obtained by combining the maximum and minimum value of the solutions of deterministic inverse problems.Moreover,according to the characteristics of the interval inverse problem,a termination criterion is provided to ensure the convergence of the proposed method.(5)For the multi-source uncertainty inverse problem with interval uncertainties in both outputs and system models,an effective nonlinear inverse method is proposed.At each iterative,multi-source uncertainty function is first transformed into a sum of output uncertainty function and model uncertainty function;second,the dimensionreduction is implemented both in output uncertainty function and model uncertainty function;third,the interval arithmetic is applied to calculate the interval responses in both output uncertainty and model uncertainty functions.Therefore,the nested problem is transformed into a traditional single-loop problem,and the computational efficiency can be improved.