Numerical Methods For The PDE-based Inverse Problems And Applications

Study on inverse problems originated in mathematical and physical equations, which isalso known as a mathematical and physical inverse problems or inverse problem inmathematical physics. In the past decades, with the development of computational tools andmethods, the research on inverse problem and its numerical methods research has become avery active research direction in science and engineering technology field and demonstratedthe important theoretical significance and wide application prospect. Especially in theGeophysical Sciences, life sciences and medicine, financial engineering, materials science,geological and environmental science, information and control field, inverse problem researchhas achieved great success.Inverse problems involve the determination of one or more unknown quantities usuallyappearing in the mathematical formulation of a physical problem. These unknown quantitiesmay be boundary heat flux, various source terms, thermal and material properties, boundaryshape and size. Solving inverse problems requires additional information through in-situ datameasurements of the field variables of the physical problems. These problems are alsoill-posed because the solution itself is sensitive to random errors in the measured input data.Regularisation techniques are usually used in order to deal with the instability of the solution.In the past decades, many methods based on the nonlinear least squares model, bothdeterministic (CGM) and stochastic (GA, PSO), have been investigated for numerical inverseproblems.The goal of this thesis is to examine and explore new techniques for numerical inverseproblems. The background theory of population-based heuristic algorithm known asquantum-behaved particle swarm optimisation (QPSO) is re-visited and examined. Toenhance the global search ability of QPSO for complex multi-modal problems, severalmodifications to QPSO are proposed. These include perturbation operation, Gaussianmutation and ring topology model. Several parameter selection methods for these algorithmsare proposed. Benchmark functions were used to test the performance of the modifiedalgorithms. To address the high computational cost of complex engineering optimisationproblems, two parallel models of the QPSO (master-slave, static subpopulation) weredeveloped for different distributed systems. A hybrid method, which makes use ofdeterministic (CGM) and stochastic (QPSO) methods, is proposed to improve the estimatedsolution and the performance of the algorithms for solving the inverse problems.Finally, the proposed methods are used to solve typical problems as appeared in manyresearch papers. The numerical results demonstrate the feasibility and efficiency of QPSO andthe global search ability and stability of the modified versions of QPSO. Two novel methodsof providing initial guess to CGM with approximated data from QPSO are also proposed foruse in the hybrid method and were applied to estimate heat fluxes and boundary shapes. Thesimultaneous estimation of temperature dependent thermal conductivity and heat capacity wasaddressed by using QPSO with Gaussian mutation. This combination provides a stablealgorithm even with noisy measurements. Comparison of the performance between different methods for solving inverse problems is also presented in this thesis.