Efficient Numerical Methods For Singular Perturbation Equation And Multi-Scale Data

Multi-scale models are widely used in various fields nowadays.Compared with traditional single-scale models,multi-scale models attempt to capture information from different levels when describing a system or a process.Therefore,by leveraging models at different scales(for example,atomic or molecular scale,macroscopic scale),multi-scale models are more accurate and can be solved more efficiently.Finding analytical solutions for modeling problems is usually difficult,especially for complicated multi-scale cases with information gathered at different scales.Designing accurate and efficient numerical schemes is very important in the study of multi-scale modeling.In this thesis,we propose numerical schemes for two different types of multiscale problems with both theoretical guarantees and empirical validation.The first type of problems we focus on is to solve singular perturbation equations with small parameters numerically,including the fourth-order linear singular perturbation equations and the second-order convection/reaction-diffusion singular perturbation equations.We study the asymptotic properties of their analytical solutions,and use them to construct our numerical scheme using the so-called Tailored Finite Point Method(TFPM).With the help of TFPM,we can obtain more accurate numerical solutions than those derived by traditional methods on a coarser grid.We analyze the error estimation of our numerical scheme and empirically validate its efficiency and accuracy.The second type of problems we are interested in is the design and analysis of sampling algorithms used in processing multi-scale data.Such algorithms play an important role in optimizing performance for different data-heavy tasks.For instance,using the Markov Chain Monte Carlo(MCMC)method to handle multi-scale data appearing in medical image processing is very common,which is shown to greatly speed up the imageprocessing procedure.In this thesis,we study the so-called Langevin Monte Carlo method(LMC),which is an indirect sampling algorithm belonging to the MCMC algorithms.We improve the original LMC algorithm and show the error analysis results for the improved algorithm.We also give an estimation for the convergence rate of the empirical measure of the sampling points,which greatly contributes to the theoretical foundation of sampling algorithms and multi-scale data processing.