Browsing by Subject "Multiscale method"
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Item Multiscale Simulation and Uncertainty Quantification Techniques for Richards' Equation in Heterogeneous Media(2012-10-19) Kang, Seul KiIn this dissertation, we develop multiscale finite element methods and uncertainty quantification technique for Richards' equation, a mathematical model to describe fluid flow in unsaturated porous media. Both coarse-level and fine-level numerical computation techniques are presented. To develop an accurate coarse-scale numerical method, we need to construct an effective multiscale map that is able to capture the multiscale features of the large-scale solution without resolving the small scale details. With a careful choice of the coarse spaces for multiscale finite element methods, we can significantly reduce errors. We introduce several methods to construct coarse spaces for multiscale finite element methods. A coarse space based on local spectral problems is also presented. The construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions. These basis functions are complemented using weighted local spectral eigenfunctions. These newly constructed basis functions can capture the small scale features of the solution within a coarse-grid block and give us an accurate coarse-scale solution. However, it is expensive to compute the local basis functions for each parameter value for a nonlinear equation. To overcome this difficulty, local reduced basis method is discussed, which provides smaller dimension spaces with which to compute the basis functions. Robust solution techniques for Richards' equation at a fine scale are discussed. We construct iterative solvers for Richards' equation, whose number of iterations is independent of the contrast. We employ two-level domain decomposition pre-conditioners to solve linear systems arising in approximation of problems with high contrast. We show that, by using the local spectral coarse space for the preconditioners, the number of iterations for these solvers is independent of the physical properties of the media. Several numerical experiments are given to support the theoretical results. Last, we present numerical methods for uncertainty quantification applications for Richards' equation. Numerical methods combined with stochastic solution techniques are proposed to sample conductivities of porous media given in integrated data. Our proposed algorithm is based on upscaling techniques and the Markov chain Monte Carlo method. Sampling results are presented to prove the efficiency and accuracy of our algorithm.Item Numerical methods for highly oscillatory dynamical systems using multiscale structure(2013-08) Kim, Seong Jun; Tsai, Yen-Hsi R.The main aim of this thesis is to design efficient and novel numerical algorithms for a class of deterministic and stochastic dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become, therefore, inefficient when the much longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three different papers. The framework of the heterogeneous multiscale method (HMM) is considered as a general strategy both for the design and the analysis of multiscale methods. In Chapter 2, we consider a new class of multiscale methods that use a technique related to the construction of a Poincaré map. The main idea is to construct effective paths in the state space whose projection onto the slow subspace shows the correct dynamics. More precisely, we trace the evolution of the invariant manifold M(t), identified by the level sets of slow variables, by introducing a slowly evolving effective path which crosses M(t). The path is locally constructed through interpolation of neighboring points generated from our developed map. This map is qualitatively similar to a Poincaré map, but its construction is based on the procedure which solves two split equations successively backward and forward in time only over a short period. This algorithm does not require an explicit form of any slow variables. In Chapter 3, we present efficient techniques for numerical averaging over the invariant torus defined by ergodic dynamical systems which may not be mixing. These techniques are necessary, for example, in the numerical approximation of the effective slow behavior of highly oscillatory ordinary differential equations in weak near-resonance. In this case, the torus is embedded in a higher dimensional space and is given implicitly as the intersection of level sets of some slow variables, e.g. action variables. In particular, a parametrization of the torus may not be available. Our method constructs an appropriate coordinate system on lifted copies of the torus and uses an iterated convolution with respect to one-dimensional averaging kernels. Non-uniform invariant measures are approximated using a discretization of the Frobenius-Perron operator. These two numerical averaging strategies play a central role in designing multiscale algorithms for dynamical systems, whose fast dynamics is restricted not to a circle, but to the tori. The efficiency of these methods is illustrated by numerical examples. In Chapter 4, we generalize the classical two-scale averaging theory to multiple time scale problems. When more than two time scales are considered, the effective behavior may be described by the new type of slow variables which do not have formally bounded derivatives. Therefore, it is necessary to develop a theory to understand them. Such theory should be applied in the design of multiscale algorithms. In this context, we develop an iterated averaging theory for highly oscillatory dynamical systems involving three separated time scales. The relevant multiscale algorithm is constructed as a family of multilevel solvers which resolve the different time scales and efficiently computes the effective behavior of the slowest time scale.