Organizational Unit:
School of Mathematics
School of Mathematics
Permanent Link
Research Organization Registry ID
Description
Previous Names
Parent Organization
Parent Organization
Organizational Unit
Includes Organization(s)
ArchiveSpace Name Record
Publication Search Results
Now showing
1 - 10 of 156
-
ItemAsymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time Dependence(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
-
ItemDynamics of Almost Periodic Scalar Parabolic Equations(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
-
ItemRandom Restarts in Global Optimization(Georgia Institute of Technology, 2009-12-07) Hu, X. ; Shonkwiler, Ronald W. ; Spruill, Marcus C.In this article we study stochastic multistart methods for global optimization, which combine local search with random initialization, and their parallel implementations. It is shown that in a minimax sense the optimal restart distribution is uniform. We further establish the rate of decrease of the ensemble probability that the global minimum has not been found by the nth iteration. Turning to parallelization issues, we show that under independent identical processing (iip), exponential speedup in the time to hit the goal bin normally results. Our numerical studies are in close agreement with these finndings.
-
ItemA Curious Binomial Identity(Georgia Institute of Technology, 2009-12-07) Calkin, Neil J.In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.
-
ItemConverse Poincaré Type Inequalities for Convex Functions(Georgia Institute of Technology, 2009-12-07) Bobkov, S. G. ; Houdré, ChristianConverse Poincaré type inequalities are obtained within the class of smooth convex functions. This is, in particular, applied to the double exponential distribution.
-
ItemSome results on linear discrepancy for partially ordered sets(Georgia Institute of Technology, 2009-11-24) Keller, Mitchel ToddTanenbaum, Trenk, and Fishburn introduced the concept of linear discrepancy in 2001, proposing it as a way to measure a partially ordered set's distance from being a linear order. In addition to proving a number of results about linear discrepancy, they posed eight challenges and questions for future work. This dissertation completely resolves one of those challenges and makes contributions on two others. This dissertation has three principal components: 3-discrepancy irreducible posets of width 3, degree bounds, and online algorithms for linear discrepancy. The first principal component of this dissertation provides a forbidden subposet characterization of the posets with linear discrepancy equal to 2 by completing the determination of the posets that are 3-irreducible with respect to linear discrepancy. The second principal component concerns degree bounds for linear discrepancy and weak discrepancy, a parameter similar to linear discrepancy. Specifically, if every point of a poset is incomparable to at most D other points of the poset, we prove three bounds: the linear discrepancy of an interval order is at most D, with equality if and only if it contains an antichain of size D; the linear discrepancy of a disconnected poset is at most the greatest integer less than or equal to (3D-1)/2; and the weak discrepancy of a poset is at most D. The third principal component of this dissertation incorporates another large area of research, that of online algorithms. We show that no online algorithm for linear discrepancy can be better than 3-competitive, even for the class of interval orders. We also give a 2-competitive online algorithm for linear discrepancy on semiorders and show that this algorithm is optimal.
-
ItemEigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian(Georgia Institute of Technology, 2009-11-11) Yildirim Yolcu, SelmaSome eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.
-
ItemMultiscale Modeling and Simulation: The Interplay Beween Mathematics and Engineering Applications(Georgia Institute of Technology, 2009-10-26) Hou, Thomas Y.Many problems of fundamental and practical importance contain multiple scale solutions. Composite and nano materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Direct numerical simulations using a fine grid are very expensive. Developing effective multiscale methods that can capture accurately the large scale solution on a coarse grid has become essential in many engineering applications. In this talk, I will use two examples to illustrate how multiscale mathematics analysis can impact engineering applications. The first example is to develop multiscale computational methods to upscale multi-phase flows in strongly heterogeneous porous media. Multi-phase flows arise in many applications, ranging from petroleum engineering, contaminant transport, and fluid dynamics applications. Multiscale computational methods guided by multiscale analysis have already been adopted by the industry in their flow simulators. In the second example, we will show how to develop a systematic multiscale analysis for incompressible flows in three space dimensions. Deriving a reliable turbulent model has a significant impact in many engineering applications, including the aircraft design. This is known to be an extremely challenging problem. So far, most of the existing turbulent models are based on heuristic closure assumption and involve unknown parameters which need to be fitted by experimental data. We will show that how multiscale analysis can be used to develop a systematic multiscale method that does not involve any closure assumption and there are no adjustable parameters.
-
ItemMathematical approaches to digital color image denoising(Georgia Institute of Technology, 2009-09-14) Deng, HaoMany mathematical models have been designed to remove noise from images. Most of them focus on grey value images with additive artificial noise. Only very few specifically target natural color photos taken by a digital camera with real noise. Noise in natural color photos have special characteristics that are substantially different from those that have been added artificially. In this thesis previous denoising models are reviewed. We analyze the strengths and weakness of existing denoising models by showing where they perform well and where they don't. We put special focus on two models: The steering kernel regression model and the non-local model. For Kernel Regression model, an adaptive bilateral filter is introduced as complementary to enhance it. Also a non-local bilateral filter is proposed as an application of the idea of non-local means filter. Then the idea of cross-channel denoising is proposed in this thesis. It is effective in denoising monochromatic images by understanding the characteristics of digital noise in natural color images. A non-traditional color space is also introduced specifically for this purpose. The cross-channel paradigm can be applied to most of the exisiting models to greatly improve their performance for denoising natural color images.
-
ItemInitial-boundary value problems in fluid dynamics modeling(Georgia Institute of Technology, 2009-08-31) Zhao, KunThis thesis is devoted to studies of initial-boundary value problems (IBVPs) for systems of partial differential equations (PDEs) arising from fluid mechanics modeling, especially for the compressible Euler equations with frictional damping, the Boussinesq equations, the Cahn-Hilliard equations and the incompressible density-dependent Navier-Stokes equations. The emphasis of this thesis is to understand the influences to the qualitative behavior of solutions caused by boundary effects and various dissipative mechanisms including damping, viscosity and heat diffusion.