Research On Accelerated Algorithms For Uncertainty Quantification Problems

With the development of science and technology,and the demand of practical application,computing science has become an important research tool for complex systems.For example,the dynamical system models with high cost and high risk in experimental research,such as aerospace and navigation simulation.However,it is often difficult to avoid the uncertainties caused by data missing and accuracy errors when modeling and simulating practical problems.Therefore,in order to obtain reliable prediction results,quantifying the uncertainties of the underlying models is one of the hot topics in the field of scientific computing and simulation.In recent decades,simulation models have gradually taken on the characteristics of high-dimensional,nonlinear,time-consuming and complex structure.Thus,effectively quantifying the uncertainties in complex systems is a huge challenge for current simulation modelling.Moreover,uncertainty affects the decision result of the system,so how to ensure calculation accuracy while improving efficiency is also one of the research objectives in this field.Based on the pursuit of accuracy and efficiency,this thesis carries out numerical algorithm research from three aspects:establishing surrogate model,improving reduced-order model(ROM),and rapidly calculating target statistics.Overall,the details are summarized as follows:To solve the problem of high simulation cost of complex system,polynomial chaos expansion(PCE)is adopted here.However,the scale of the expansion coefficient in the discrete case depends not only on the input space dimension and polynomial degree,but also on the degree of freedom of the discrete state.In order to build an efficient surrogate model at a low cost,proper orthogonal decomposition(POD)is used to reduce the original model,and then PCE is performed for the reduced states.The sparse coefficient matrix is determined by the compressive sensing method.The low-order moment approximations of the system state can be directly obtained by using the orthogonality of the polynomials.The error estimation and sparsity of the biorthogonal surrogate model are proved strictly,and its feasibility and effectiveness are verified numerically.In view of the dependence of POD modes on snapshots,a high-precision ROM is constructed for stochastic evolution equation by using statistical analysis techniques.Firstly,according to the similarity of the sample solutions of underlying system and the optimality of POD modes,a time-dependent clustering method is proposed based on the centroidal Voronoi tessellations.Each centroid corresponds to a set of projection optimal basis functions.Then the clustering results are used to train a classifier,which is utilized to select a group of suitable reduced-order modes from several sets of basis functions.This thesis first strictly proves the error estimation of the pre-classification based ROM,and then verifies numerically that the optimal modes generated after clustering can yield more accurate results.The research shows that although the classifier misjudgment will bring some accuracy loss,it is still more effective than the traditional ROM.Aiming at the huge cost of selecting a large number of high-precision samples in stochastic complex systems,a fast calculation method is proposed based on Bayesian framework and multilevel Monte Carlo.Firstly,the target is converted into an integral with respect to the output of interest by integral transformation.Then,take the output of interest as the research object,and its probability density function estimated by the coarse grid data is regarded as a prior.Based on the relationship between the density function and the distribution function,the likelihood operator is established by using the fine grid data.The posterior obtained by multi-resolution data will gradually approximate the real density function.Finally,an exact approximation of target statistics can be generated by using the quadrature rule.In order to further reduce the computational cost,the indication function is smoothed,and Latin hypercube sampling technique is used to collect nested data.The theoretical part strictly deduces the optimal number of levels and the required sample size of each level when the error threshold is given,and proves the error estimation of the approximate target statistics.Numerical results show that this method can obtain high-precision approximation of the objective at a low cost.