Theory And Algorithms Of Reconstructing Implicit Unknowns With Partial Differential Models And Random Big Data

The partial differential equations and their inverse problems are important branches in applied and computational mathematics.A large number of partial differential equation model data reconstruction problems appear in different forms in solving various physical background problems.In this thesis,the backward problem of parabolic equations and the contact inverse problem in the elasticity are studied.They are important as well as severely ill-posed in applications.The study is carried out from both the model itself and the observation data for reducing the error caused by ill-posedness in this thesis.In terms of model,uniqueness provides the possibility of reconstruction,and conditional stability estimation guarantees the stability of the global algorithm.In terms of data,due to the fact that it is difficult for the observation data to reach the theoretical accuracy in practice,a large number of data for reducing noise are considered.First,consider the model itself,it is transformed into an integral,which is expressed as a Green function.After that,raise the dimension of the funcion and extend it to the complex field.Then the conditional stability estimation could be obtained by analytic continuation.Fianlly,reduce the dimension to original form.In terms of numerical methods,previous studies on the backward problem of parabolic equations were mostly based on the complete and continuous values at the final time.However,the cost for obtaining complete data is high and the observation data are usually discrete in practice.In this thesis,finite discrete data are used for inversion,which makes the problem more practical.Numerically,the one-dimensional inverse problem is discretized into severely ill-posed linear equations by the collocation method.And then the Tikhonov regularization method is used to deal with the severely ill-posedness of the problem.The regularization parameter is determined by the generalized cross validation method,which expands the data set when the data is limited.And for the contact inverse problem in the elasticity,it is transformed into Cauchy problem.In order to reduce the number of conditions and make the model more stable,conditional stability is used in the thesis,and Tikhonov regularization is used to construct a stable algorithm.The unique continuation of the observation function is used to reduce the number of conditions of the problem.The unique continuation of the harmonic function on quadratic curves are studied firstly in this part.With the method of continuation,the data on the whole curve can be derived from the observations on a small section of curve,and the unique continuation of the problem of unknown boundary is studied.By unique continuation,more displacement information around the contact area can be obtained in the elastic contact inverse problem.It should be pointed out that Cauchy values are not needed in this problem.In numerical calculation,the thin plate spline is used to discretize the stress function,so that a continuous severely ill-posed problem can be transformed into a system of linear equations.After that,combined with conditional stability,a relatively stable numerical solution could be obtained by discrete Tikhonov regularization method.Small observation errors will have a great impact on the results for the model itself.Therefore,the observation data is needed to be preprocessed.First,a large amount of observation data is obtained through multiple measurements.Due to the statistical properties,the observation data can be reconstructed by the Tikhonov regularization method,which can reduce the noise of the observation data itself.The purpose of random big data reconstruction is to exchange the“quantity” of data for the accuracy of measurement data,so as to improve the model accuracy and realize the “credible” inversion of unknown quantities.Since the observation points are randomly,their distribution is uneven,thus the data results will be affected by the location of the observation points.To solve this problem and reduce the time and space cost at the same time,the finite dimensional spline function space is constructed by an appropriate spline and the regularization method of online updating.When the observation value of the model has a large error,doing data preprocessing in one-dimensional and twodimensional on the observation data with noise of two models respectively,the reconstructed observation values have higher accuracy than the original observation values.Then,the reconstructed data are substituted into the original model for calculation,so as to make up for some data errors caused by “inaccurate model”.