An Optimized Dynamic Mode Decomposition Model Robust to Multiplicative Noise

Dynamic mode decomposition (DMD) is an efficient tool for decomposing spatio-temporal data into a set of low-dimensional modes, yielding the oscillation frequencies and the growth rates of physically significant modes. In this paper, we propose a novel DMD model that can be used for dynamical systems affected by multiplicative noise. We first derive a maximum a posteriori (MAP) estimator for the data-based model decomposition of a linear dynamical system corrupted by certain multiplicative noise. Applying penalty relaxation to the MAP estimator, we obtain the proposed DMD model whose epigraphical limits are the MAP estimator and the conventional optimized DMD model. We also propose an efficient alternating gradient descent method for solving the proposed DMD model, and analyze its convergence behavior. The proposed model is demonstrated on both the synthetic data and the numerically generated one-dimensional combustor data, and is shown to have superior reconstruction properties compared to state-of-the-art DMD models. Considering that multiplicative noise is ubiquitous in numerous dynamical systems, the proposed DMD model opens up new possibilities for accurate data-based modal decomposition.Comment: 35 pages, 10 figure

Variational Domain Decomposition For Parallel Image Processing

Many important techniques in image processing rely on partial differential equation (PDE) problems, which exhibit spatial couplings between the unknowns throughout the whole image plane. Therefore, a straightforward spatial splitting into independent subproblems and subsequent parallel solving aimed at diminishing the total computation time does not lead to the solution of the original problem. Typically, significant errors at the local boundaries between the subproblems occur. For that reason, most of the PDE-based image processing algorithms are not directly amenable to coarse-grained parallel computing, but only to fine-grained parallelism, e.g. on the level of the particular arithmetic operations involved with the specific solving procedure. In contrast, Domain Decomposition (DD) methods provide several different approaches to decompose PDE problems spatially so that the merged local solutions converge to the original, global one. Thus, such methods distinguish between the two main classes of overlapping and non-overlapping methods, referring to the overlap between the adjacent subdomains on which the local problems are defined. Furthermore, the classical DD methods --- studied intensively in the past thirty years --- are primarily applied to linear PDE problems, whereas some of the current important image processing approaches involve solving of nonlinear problems, e.g. Total Variation (TV)-based approaches. Among the linear DD methods, non-overlapping methods are favored, since in general they require significanty fewer data exchanges between the particular processing nodes during the parallel computation and therefore reach a higher scalability. For that reason, the theoretical and empirical focus of this work lies primarily on non-overlapping methods, whereas for the overlapping methods we mainly stay with presenting the most important algorithms. With the linear non-overlapping DD methods, we first concentrate on the theoretical foundation, which serves as basis for gradually deriving the different algorithms thereafter. Although we make a connection between the very early methods on two subdomains and the current two-level methods on arbitrary numbers of subdomains, the experimental studies focus on two prototypical methods being applied to the model problem of estimating the optic flow, at which point different numerical aspects, such as the influence of the number of subdomains on the convergence rate, are explored. In particular, we present results of experiments conducted on a PC-cluster (a distributed memory parallel computer based on low-cost PC hardware for up to 144 processing nodes) which show a very good scalability of non-overlapping DD methods. With respect to nonlinear non-overlapping DD methods, we pursue two distinct approaches, both applied to nonlinear, PDE-based image denoising. The first approach draws upon the theory of optimal control, and has been successfully employed for the domain decomposition of Navier-Stokes equations. The second nonlinear DD approach, on the other hand, relies on convex programming and relies on the decomposition of the corresponding minimization problems. Besides the main subject of parallelization by DD methods, we also investigate the linear model problem of motion estimation itself, namely by proposing and empirically studying a new variational approach for the estimation of turbulent flows in the area of fluid mechanics

X-ray CT Image Reconstruction on Highly-Parallel Architectures.

Model-based image reconstruction (MBIR) methods for X-ray CT use accurate models of the CT acquisition process, the statistics of the noisy measurements, and noise-reducing regularization to produce potentially higher quality images than conventional methods even at reduced X-ray doses. They do this by minimizing a statistically motivated high-dimensional cost function; the high computational cost of numerically minimizing this function has prevented MBIR methods from reaching ubiquity in the clinic. Modern highly-parallel hardware like graphics processing units (GPUs) may offer the computational resources to solve these reconstruction problems quickly, but simply "translating" existing algorithms designed for conventional processors to the GPU may not fully exploit the hardware's capabilities. This thesis proposes GPU-specialized image denoising and image reconstruction algorithms. The proposed image denoising algorithm uses group coordinate descent with carefully structured groups. The algorithm converges very rapidly: in one experiment, it denoises a 65 megapixel image in about 1.5 seconds, while the popular Chambolle-Pock primal-dual algorithm running on the same hardware takes over a minute to reach the same level of accuracy. For X-ray CT reconstruction, this thesis uses duality and group coordinate ascent to propose an alternative to the popular ordered subsets (OS) method. Similar to OS, the proposed method can use a subset of the data to update the image. Unlike OS, the proposed method is convergent. In one helical CT reconstruction experiment, an implementation of the proposed algorithm using one GPU converges more quickly than a state-of-the-art algorithm converges using four GPUs. Using four GPUs, the proposed algorithm reaches near convergence of a wide-cone axial reconstruction problem with over 220 million voxels in only 11 minutes.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113551/1/mcgaffin_1.pd