Mathematics
http://hdl.handle.net/1903/2261
2020-02-26T23:38:05ZWeyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces
http://hdl.handle.net/1903/25268
Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces
Walnut, David Francis
The theory of wavelets can be used to obtain expansions of
vectors in certain spaces. These expansions are like Fourier
series in that each vector can be written in terms of a fixed
collection of vectors in the Banach space and the coefficients
satisfy a "Plancherel Theorem" with respect to some sequence
space. In Weyl-Heisenberg expansions, the expansion vectors
(wavelets) are translates and modulates of a single vector (the
analyzing vector) .
The thesis addresses the problem of the existence and
stability of Weyl-Heisenberg expansions in the space of functions
square-integrable with respect to the measure w(x) dx for a
certain class of weights w. While the question of the existence
of such expansions is contained in more general theories, the
techniques used here enable one to obtain more general and
explicit results. In Chapter 1, the class of weights of interest is defined and
properties of these weights proven.
In Chapter 2, it is shown that Weyl-Heisenberg expansions
exist if the analyzing vector is locally bounded and satisfies a
certain global decay condition.
In Chapter 3, it is shown that these expansions persist if
the translations and modulations are not taken at regular
intervals but are perturbed by a small amount. Also, the
expansions are stable if the analyzing vector is perturbed. It is
also shown here that under more general assumptions, expansions
exist if translations and modulations are taken at any
sufficiently dense lattice of points.
Like orthonormal bases, the coefficients in Weyl-Heisenberg
expansions can be formed by the inner product of the vector being
expanded with a collection of wavelets generated by a transformed
version of the analyzing vector. In Chapter 4, it is shown that
this transformation preserves certain decay and smoothness
conditions and a formula for this transformation is given.
In Chapter 5, results on Weyl-Heisenberg expansions in the
space of square-integrable functions are presented.
1989-01-01T00:00:00ZSonic Limit Singularities in the Hodograph Method
http://hdl.handle.net/1903/25234
Sonic Limit Singularities in the Hodograph Method
Schot, Steven H.
In the hodograph transformation, introduced to linerize the equations governing the two-dimensional inviscid potential flow of a compressible fluid, there may appear so-called limit-points and limit-lines at which the Jacobian J = ∂(x,y)/ ∂(q,θ) of the transformation vanishes. This thesis investigate these singularities when they occur at points or segments of arc of the sonic line (Mach number unity).
Assuming the streamfunction to be regular in the hodograph variables, it is show that sonic limit points cannot be isolated but must lie on a supersonic limit line or form a sonic limit line [cf. H. Geiringer, Math. Zeitschr., 63, (1956), 514-524]. Using this dichotomy a classification of sonic limit points is set up and certain geometrical properties of the mapping in the neighborhood of the singularity are discussed. In particular the general sonic limit line is shown to be an equipotential and an isovel; an envelope of both families of characteristics; and the locus of cusps of the streamlines and the isoclines. Flows containing sonic limit lines may be constructed by forming suitable linear combinations of the Chaplygin product solutions for any value of the separation constant n ≥ 0. For n less than a certain value n0 and greater than zero (n = 0 corresponds to the well-known radial flow), these flows represent a compressible analogue of the incompressible corner flows and may be envisaged as taking place on a quadruply-sheeted surface. The sheets are joined at a super-sonic limit line and at the sonic limit line which has the shape of a hypocycloid (n >1), cycloid (n = 1), or epicycloid (n <1). To exemplify the general behavior, the flows are constructed explicitly for n = 1/2, 1, and 2. The shape of the sonic limit line is also discussed when solutions corresponding to different n are superposed, and it is shown how then the supersonic limit line can be eliminated so that an isolated sonic limit line is obtained. A flow containing such an isolated sonic limit line is presented. An appendix derives the asymptotic solution for large values of n which corresponds to the sonic limit solution.
The above results have been published in part in Math. Zeitschr., 67, (1957), 229-237. Other portions of this thesis will appear in two papers in Archive Rational Mech. and Anal., 2, (1958).
1958-01-01T00:00:00ZDevelopments in Lagrangian Data Assimilation and Coupled Data Assimilation to Support Earth System Model Initialization
http://hdl.handle.net/1903/25059
Developments in Lagrangian Data Assimilation and Coupled Data Assimilation to Support Earth System Model Initialization
Sun, Luyu
The air-sea interface is one of the most physically active interfaces of the Earth's environments and significantly impacts the dynamics in both the atmosphere and ocean. In this doctoral dissertation, developments are made to two types of Data Assimilation (DA) applied to this interface: Lagrangian Data Assimilation (LaDA) and Coupled Data Assimilation (CDA).
LaDA is a DA method that specifically assimilates position information measured from Lagrangian instruments such as Argo floats and surface drifters. To make a better use of this Lagrangian information, an augmented-state LaDA method is proposed using Local Ensemble Transform Kalman Filter (LETKF), which is intended to update the ocean state (T/S/U/V) at both the surface and at depth by directly assimilating the drifter locations. The algorithm is first tested using "identical twin" Observing System Simulation Experiments (OSSEs) in a simple double gyre configuration with the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model version 4.1 (MOM4p1). Results from these experiments show that with a proper choice of localization radius, the estimation of the state is improved not only at the surface, but throughout the upper 1000m. The impact of localization radius and model error in estimating accuracy of both fluid and drifter states are investigated.
Next, the algorithm is applied to a realistic eddy-resolving model of the Gulf of Mexico (GoM) using Modular Ocean Model version 6 (MOM6) numerics, which is related to the 1/4-degree resolution configuration in transition to operational use at NOAA/NCEP. Atmospheric forcing is first used to produce the nature run simulation with forcing ensembles created using the spread provided by the 20 Century Reanalysis version 3 (20CRv3). In order to assist the examination on the proposed LaDA algorithm, an updated online drifter module adapted to MOM6 is developed, which resolves software issues present in the older MOM4p1 and MOM5 versions of MOM. In addition, new attributions are added, such as: the output of the intermediate trajectories and the interpolated variables: temperature, salinity, and velocity. The twin experiments with the GoM also show that the proposed algorithm provides positive impacts on estimating the ocean state variables when assimilating the drifter position together with surface temperature and salinity.
Lastly, an investigation of CDA explores the influence of different couplings on improving the simultaneous estimation of atmosphere and ocean state variables. Synchronization theory of the drive-response system is applied together with the determination of Lyapunov Exponents (LEs) as an indication of the error convergence within the system. A demonstration is presented using the Ensemble Transform Kalman Filter on the simplified Modular Arbitrary-Order Ocean-Atmosphere Model, a three-layer truncated quasi-geostrophic model. Results show that strongly coupled data assimilation is robust in producing more accurate state estimates and forecasts than traditional approaches of data assimilation.
2019-01-01T00:00:00ZTopics in Stochastic Optimization
http://hdl.handle.net/1903/25056
Topics in Stochastic Optimization
Sun, Guowei N/A
In this thesis, we work with three topics in stochastic optimization: ranking and selection (R&S), multi-armed bandits (MAB) and stochastic kriging (SK). For R&S, we first consider the problem of making inferences about all candidates based on samples drawn from one. Then we study the problem of designing efficient allocation algorithms for problems where the selection objective is more complex than the simple expectation of a random output. In MAB, we use the autoregressive process to capture possible temporal correlations in the unknown reward processes and study the effect of such correlations on the regret bounds of various bandit algorithms. Lastly, for SK, we design a procedure for dynamic experimental design for establishing a good global fit by efficiently allocating simulation budgets in the design space.
The first two Chapters of the thesis work with variations of the R&S problem. In Chapter 1, we consider the problem of choosing the best design alternative under a small simulation budget, where making inferences about all alternatives from a single observation could enhance the probability of correct selection. We propose a new selection rule exploiting the relative similarity between pairs of alternatives and show its improvement on selection performance, evaluated by the Probability of Correct Selection, compared to selection based on collected sample averages. We illustrate the effectiveness by applying our selection index on simulated R\&S problems using two well-known budget allocation policies. In Chapter 2, we present two sequential allocation frameworks for selecting from a set of competing alternatives when the decision maker cares about more than just the simple expected rewards. The frameworks are built on general parametric reward distributions and assume the objective of selection, which we refer to as utility, can be expressed as a function of the governing reward distributional parameters. The first algorithm, which we call utility-based OCBA (UOCBA), uses the Delta-technique to find the asymptotic distribution of a utility estimator to establish the asymptotically optimal allocation by solving the corresponding constrained optimization problem. The second, which we refer to as utility-based value of information (UVoI) approach, is a variation of the Bayesian value of information (VoI) techniques for efficient learning of the utility. We establish the asymptotic optimality of both allocation policies and illustrate the performance of the two algorithms through numerical experiments.
Chapter 3 considers the restless bandit problem where the rewards on the arms are stochastic processes with strong temporal correlations that can be characterized by the
well-known stationary autoregressive-moving-average time series models. We argue that despite the statistical stationarity of the reward processes, a linear improvement in cumulative
reward can be obtained by exploiting the temporal correlation, compared to
policies that work under the independent reward assumption. We introduce the
notion of temporal exploration-exploitation trade-off, where a policy has to balance
between learning more recent information to track the evolution of all reward processes and utilizing currently available predictions to gain better immediate reward.
We prove a regret lower bound characterized by the bandit problem complexity
and correlation strength along the time index and propose policies that achieve a
matching upper bound.
Lastly, Chapter 4 proposes a fully sequential experimental design procedure for the stochastic kriging (SK) methodology of fitting unknown response surfaces from simulation experiments. The procedure first estimates the current SK model performance by jackknifing the existing data points. Then, an additional SK model is fitted on the jackknife error estimates to capture the landscape of the current SK model performance. Methodologies for balancing exploration and exploitation trade-off in Bayesian optimization are employed to select the next simulation point. Compared to existing experimental design procedures relying on the posterior uncertainty estimates from the fitted SK model for evaluating model performance, our method is robust to the SK model specifications. We design a dynamic allocation algorithm, which we call kriging-based dynamic stochastic kriging (KDSK), and illustrate its performance through two numerical experiments.
2019-01-01T00:00:00Z