The thesis aims at showing some important methods, theories and applications about non-parametric density estimation via regularization in univariate setting. It gives a brief introduction to non-parametric density estimation, and discuss several well-known methods, for example, histogram and kernel methods. Regularized methods with penalization and shape constraints are the focus of the thesis. Maximum entropy density estimation is introduced and the relationship between taut string and maximum entropy density estimation is explored. Furthermore, the dual and primal theories are discussed and some theoretical proofs corresponding to quasi-concave density estimation are presented. Different the numerical methods of non-parametric density estimation with regularization are classified and compared. Finally, a real data experiment will also be discussed in the last part of the thesis. / Statistics

We consider the convex parametric quadratic programming problem when the end of the parametric interval is caused by a multiplicity of possibilities ("ties"). In such cases, there is no clear way for the proper active set to be determined for the parametric analysis to continue. In this thesis, we show that the proper active set may be determined in general by solving a certain non-parametric quadratic programming problem. We simplify the parametric quadratic programming problem with a parameter both in the linear part of the objective function and in the right-hand side of the constraints to a quadratic programming without a parameter. We break the analysis into three parts. We first study the parametric quadratic programming problem with a parameter only in the linear part of the objective function, and then a parameter only in the right-hand side of the constraints. Each of these special cases is transformed into a quadratic programming problem having no parameters. A similar approach is then applied to the parametric quadratic programming problem having a parameter both in the linear part of the objective function and in the right-hand side of the constraints.

Nonlinear acoustic media for implementations of parametric generation of low frequencies has so far been restricted to small values of the parameter B/A, typically between 3 and 13. Parametric amplification, defined as the generation of a low difference frequency signal resulting from the nonlinear interactions of two higher frequency fundamentals is enhanced by medium with a large coefficient of nonlinearity and low sound speed. The acoustic properties of a highly nonlinear medium were estimated and introduced in a numerical model, to evaluate the parametric amplification induced by a thin layer of such material in contact with a spherical transducer. The numerical model predicted a significant enhancement of the sound pressure level for the difference frequency component relative to that obtained when the transducer is driven linearly at the difference frequency. A source was then constructed to compare the theoretical predictions with experimental values and an enhancement of 17dB compared to the linear operation of the transducer was measured. The difference between the parametric amplification achieved with the nonlinear medium and the parametric amplification that would be obtained in water is 73dB.

Optical source generation has attracted significant attention recently, especially in fiber optical communications. Today there is a growing a demand for optical source generation beyond conventional telecommunication wavelength bands. However, high quality and versatile optical source is generally not available over those wavelength bands due to the lack of efficient gain medium. Thanks to fiber optical parametric amplifier (FOPA), which is based on the third order nonlinear susceptibility of optical fibers, offers ultrafast response, wide-gain bandwidth, high gain and large frequency detune from the pump, serves as a promising candidate for signal amplification over those wavelength bands. By using the corresponding fiber optical parametric oscillator (FOPO) configuration, widely tunable source from continuous-wave (CW) to sub-picosecond pulses can be potentially generated to serve different applications from communication to biomedical imaging. In this thesis, we first demonstrate an all-fiber widely-tunable picosecond FOPO using highly-nonlinear fiber (HNLF). The tuning range is as wide as 250 nm, which is higher than previous picosecond FOPOs reported in the 1550-nm region. Second, time-dispersion tuning of the FOPO is investigated with fixed pump wavelength. It is a relatively simple and economic approach, and there will be no filter induced cavity loss. We then describe using FOPO to generated nearly-transform limited sub-picosecond pulses with a 60-nm tuning range. Another FOPO with a tuning range of 440-nm with dispersion-shifted fiber (DSF) as the gain medium is proposed and demonstrated. Compared with FOPOs demonstrated using HNLF as the gain medium, the use of DSF offers two key advantages: a wider tuning range and a narrower linewidth. In addition to picosecond FOPO, CW FOPO is also of great interest in fiber optical communications and biomedical imaging. We also demonstrate an all-fiber CW single-longitudinal-mode (SLM) FOPO with tuning range covers the S and L bands. SLM oscillation with a side-mode suppression ratio greater than 43 dB is achieved, which has been extended to 1-μm region under stable operation. Apart from static tuning, dynamic wavelength tuning of the FOPO is also discussed in this thesis with a cumulative speed exceeds 4,000,000 nm/s, which is higher than previous work reported in wavelength-swept FOPOs. The high-speed swept source would be useful in biomedical imaging and sensing applications. The amplification of the sub-picosecond pulses of the FOPO output is also investigated, for the first time to our knowledge, by using a fiber optical parametric chirped pulse amplifier(FOPCPA).The totally fiber-integrated nature of the whole system allows complete self-alignment and further integration to other fiber-based systems. All these research effort will show the versatility of FOPO techniques for generating wide range of optical sources for varies applications. These schemes may be useful in generating CW and short pulse for potential optical communication and biomedical imaging in non-conventional wavelength bands. / published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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Thesis (M. Phil.)--University of Hong Kong, 2008. / Also available in print.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 92-95).

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 84-88). Also available in electronic version.

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