Research On Calibration Of Mathematical Models

With the rapid development of science and engineering technology, humans arebeginning to explore more and more physical phenomenons, such as cosmic activitiesand nuclear fusion. However, there are often many problems occuring in these physicalprocesses such as high security risk, high costs or even impossibilities to implementthese processes. Thus, people build different mathematical models and make computersimulations to study these physical processes. Input parameters of mathematical modelsoften contain some fixed but unknown parameters, which can be confirmed by outputsof computer simulations and a small amount of physical observations and the process isoften reffered as calibration. The purpose of calibration is to determine values ofcalibration parameters and makes outputs from mathematical models and physicalobservations into good agreement, so mathematical models can better simulate andpredict real physical processes.Based on existing calibration methods, the dissertation divids them into Bayesiancalibration and Frequentist calibration. The former is applied to the case of largedeviations between mathematical modeles and physical processes, while the latter isapplied to the case of small deviations between mathematical models and physicalprocesses. Based on the deep investigation on the two types of methods and relatedrequirements of calibration, the dissertation tries to improve traditional calibrationmethods and get better calibration results. What’s more, time-varying calibration issuesare also analyzed. The main achievements of the dissertation are as follows:(1) Improved Bayesian calibration method. A detailed description of Bayesiancalibration is present first. And then we advance the idea of Modularization and Plug-inin Bayesian calibration from the point of engneering to deal with the problem ofcomplicated calculating and time-consuming. Simulation examples are compared toverify the effectiveness of the method.(2) Improved Frequentist calibration method. ANLS(Approximate Non-linearLeast Square) calibration method is described first and we propose a new method offrequentist calibration—IANLS(Improved Approximate Non-linear Least Square)calibration method to overcome the drawbacks of ANLS. Through three differentsimulation examples from low-dimension to high-dimension, we verify the validity ofIANLS method in solving calibration problems, especially high-dimensional ones.(3) Time-varying calibration problem research. Based on the problem of howmathematical models track physical processes and achieve high consistency whenphysical processes are changing slowly, we propose two predictive models of LinearRegression and Gaussian Random Process. Through the simulation examples, weanalyze characteristics and application occasions of the two models and give some advices for further studies.