# Browsing by Subject "PDE"

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Item The dynamics of bose gases(2015-05) Taliaferro, Kenneth William; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Maggi, Francesco; Pavlovic, Natasa; Tzirakis, Nikolaos; Vasseur, AlexisShow more We study the Gross-Pitaevskii (GP) hierarchy, which is an infinite sequence of coupled partial differential equations that models the dynamics of Bose gases and arises in the derivation of the cubic and quintic nonlinear Schrödinger equations from an N-body linear Schrödinger equation. In Chapter 2, we consider the cubic case in R³ and derive the GP hierarchy in the strong topology corresponding to the spaces used by Klainerman and Machedon in (82). We also prove that positive semidefiniteness of solutions is preserved over time and use this result to prove global well-posedness of solutions to the GP hierarchy. This is based on a joint work with Thomas Chen (24). In Chapters 3 and 4, we prove uniqueness of solutions to the GP hierarchy in R[superscript d] in a low regularity Sobolev type space in the cubic and quintic cases, respectively. These chapters are an extension of the work of Chen-Hainzl-Pavlović-Seiringer (17) and are based on joint works with Younghun Hong and Zhihui Xie (70,71).Show more Item Kalman filtering for state estimation of advection diffusion PDE from sparse observation(2019-09-17) An, Haocheng; Ghattas, Omar N.Show more Kalman filter is an important algorithm used in control theory. It takes an initial state as input and a series of observations over time and output the hidden state. The advection-diffusion equation is a PDE that characterizes the combination effect of advection and diffusion of a given object in the solvent. Such a problem is within the domain that the Kalman filter can solve. In this report, I will first derive the Kalman filter algorithm, then examine its application to an advection-diffusion equation. I will use different metrics to quantify the numerical performance of the algorithm. The contribution of this report lies in the combination of the Kalman filter algorithm with the advection equation. Also, an ample amount of graphs that can visually tell us the evolving trend of the stateShow more Item Numerical methods for averaging and homogenization(2020-05-08) Dussinger, Milica; Engquist, Björn, 1945-; Gamba, Irene M; Arbogast, Todd; Elber, RonShow more Science and engineering are full of examples of multiscale problems, which pose severe challenges to numerical simulations. In multiscale problems, processes interact on different scales in space and time. Numerical methods, which by direct simulation fully resolves this interaction demands a tremendous amount of computational time as well as memory resources. The smallest scale should be well approximated over the full computational domain. This thesis is concerned with developing and studying numerical algorithms following the framework of the heterogeneous multiscale methods (HMM). We will focus on two numerical methods that mimic the analytical techniques of averaging and homogenization respectively. The goal is to approximate the effective or averaged solution even when the explicit analytic form may not be available. The computational challenge is to include the effects of the small scales without the cost of resolving them over the full domain. In the first part of the thesis, we focus on a class of methods for the numerical averaging of highly oscillatory ordinary differential equations. The algorithms will represent an extension to the previous work done by Tao, Owhadi and Marsden. We present analysis and apply the technique to model equations. In the second part of the thesis, we focus on methods for numerical computing the effective or homogenized form of multiscale elliptic equations. We present a procedure that reduces the effect from boundary conditions, or the so-called cell resonance error. This has been an active field of research during the last few years. We use averaging kernels that have special regularity and vanishing "negative" moment properties in order to average and thereby reduce the boundary error.Show more Item The De Giorgi method : applications to degenerate PDE(2020-05) Stokols, Logan Frank; Vasseur, Alexis F.; Caffarelli, Luis; Pavlovic, Natasa; Silvestre, LuisShow more The De Giorgi method was developed in 1957 for showing continuity of non-linear elliptic problems. In this work we will apply generalizations of that method to a variety of degenerate problems. Such problems include first-order equations with negative viscosity, hypoelliptic equations including the nonlocal Focker-Planck equation, and transport-diffusion equations with boundary, for which the diffusion is of critical order and degenerates near the boundary. We will also consider a separate problem in which energy techniques can be brought to bear on a hyperbolic problem, namely the stability of shocks to conservation laws.Show more