Systematic model reduction for complex systems through data mining and dimensionality reduction

This thesis consists of six chapters. The first chapter presents an overview of the thesis. The next three chapters pertain to effective descriptions of complex, dynamic phenomena through the use of diffusion maps (DMAPs), while Chapters 5 and 6 describe techniques relating to initialization on a slow manifold and the detection of the coarse-scale steady states of the dynamics which give rise to this slow manifold.;In Chapter 2, we study a variant of the Ising model which emulates the movement of a driven interface in the presence of mobile impurities. Through the use of diffusion maps, we are able to provide an effective coarse description of the dynamics of the system based on two DMAP-generated coarse variables. We construct a lifting operator which allows us to translate between DMAP variables and snapshots of the original system, and we use this lifting operator to validate these two coarse variables.;The focus in Chapter 3 is the application of diffusion maps to dissipative partial differential equations (PDEs). Model reduction techniques such as Approximate Inertial Manifolds and the POD-Galerkin approach are often able to provide reasonably accurate reduced dynamic models of the long-term dynamics of dissipative partial differential equations. Nevertheless, the methods are not well-suited to high-dimensional global attractors of a low intrinsic dimension whose geometry markedly "nonflat"; such global attractors, although theoretically embeddable in a low-dimensional space, require a large linear basis to be represented effectively. In this chapter, as a conceptual extension of the POD-Galerkin approach, we apply non-linear manifold learning techniques to evolution equations with time scale separation.;Despite the success of nonlinear dimensionality reduction techniques in finding meaningful reduced descriptions for complex systems, they still suffer from the curse of dimensionality; the amount of data required to successfully recover d "instrinsic" dimensions grows exponentially with d. When the dynamics of the system are equivariant with respect to a k-dimensional symmetry group, however, one may use vector diffusion maps to reduce the data required to an amount which is only exponential in d - k. In Chapter 4, we describe both vector diffusion maps and the so-called eigenvector method upon which this technique is based. We also demonstrate the ability of both vector diffusion maps and this eigenvector method to parameterize, denoise, and align data sets of interest through the use of two prototypical examples.;In Chapter 5, we study a technique which may be used for both the computation of coarse-scale steady states and initialization on the slow manifold for dynamical systems with time scale separation. The technique alleviates certain problems encountered by similar approaches, and is also simple enough to be implemented as a "wrapper" around an existing "black box" or legacy code. We demonstrate the technique on a lattice Boltzmann fine-scale code.;By coupling the techniques of Chapter 5 with routines in a software suite created by Mike Henderson, "Multifario", we are able to construct an algorithm capable of covering a d-dimensional slow manifold in n-dimensional space by d-dimensional spheres. This algorithm, discussed in Chapter 6, bears similarity to the computations that arise in both the computation of intrinsic low-dimensional manifolds and other model reduction techniques. However, it requires only a forward ("black box") integrator, and it is capable of sweeping out slow manifolds of arbitrarily high dimension (given enough CPU time) with folds, wrinkles, and geometries known to be difficult for many related methods.