Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Aerosol-cloud-precipitation interactions in marine stratocumulus clouds

Marine stratocumulus clouds are generally optically thick and shallow, exerting a net cooling influence on climate. Changes in atmospheric aerosol levels alter cloud microphysics (e.g., droplet size) and cloud macrophysics (e.g., liquid water path, cloud thickness), thereby affecting cloud albedo and Earth’s radiative balance. To understand the aerosol-cloud-precipitation interactions and to explore the dynamical effects, three-dimensional large-eddy simulations (LES) with detailed bin-resolved microphysics are performed to explore the diurnal variation of marine stratocumulus clouds under different aerosol levels and environmental conditions. It is shown that the marine stratocumulus cloud albedo is sensitive to aerosol perturbation under clean background conditions, and to environmental conditions such as large-scale divergence rate and free tropospheric humidity.

Based on the in-situ Eastern Pacific Emitted Aerosol Cloud Experiment (E-PEACE) during Jul. and Aug. 2011, and A-Train satellite observation of 589 individual ship tracks during Jun. 2006-Dec. 2009, an analysis of cloud albedo responses in ship tracks is presented. It is found that the albedo response in ship tracks depends on the mesoscale cloud structure, the free tropospheric humidity, and cloud top height. Under closed cell structure (i.e., cloud cells ringed by a perimeter of clear air), with sufficiently dry air above cloud tops and/or higher cloud top heights, the cloud albedo can become lower in ship tracks. Based on the satellite data, nearly 25% of ship tracks exhibited a decreased albedo. The cloud macrophysical responses are crucial in determining both the strength and the sign of the cloud albedo response to aerosols.

To understand the aerosol indirect effects on global marine warm clouds, multisensory satellite observations, including CloudSat, MODIS, CALIPSO, AMSR-E, ECMWF, CERES, and NCEP, have been applied to study the sensitivity of cloud properties to aerosol levels and to large scale environmental conditions. With an estimate of anthropogenic aerosol fraction, the global aerosol indirect radiative forcing has been assessed.

As the coupling among aerosol, cloud, precipitation, and meteorological conditions in the marine boundary layer is complex, the integration of LES modeling, in-situ aircraft measurements, and global multisensory satellite data analyses improves our understanding of this complex system.

Optimum hot electron production with low-density foams for laser fusion by fast ignition

We propose a foam cone-in-shell target design aiming at optimum hot electron production for the fast ignition. A thin low-density foam is proposed to cover the inner tip of a gold cone inserted in a fuel shell. An intense laser is then focused on the foam to generate hot electrons for the fast ignition. Element experiments demonstrate increased laser energy coupling efficiency into hot electrons without increasing the electron temperature and beam divergence with foam coated targets in comparison with solid targets. This may enhance the laser energy deposition in the compressed fuel plasma.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Control of filamentation induced by femtosecond laser pulses propagating in air

Filamentation formed by self-focusing of intense laser pulses propagating in air is investigated. It is found that the position of filamentation can be controlled continuously by changing the laser power and divergence angle of the laser beam. An analytical model for the process is given.

Long Run Trends in the Price of Energy and Energy Services

4 p.

The variations and climatic significance of D/H ratios in the carbon-bound hydrogen of cellulose in trees

The σD values of nitrated cellulose from a variety of trees covering a wide geographic range have been measured. These measurements have been used to ascertain which factors are likely to cause σD variations in cellulose C-H hydrogen.

It is found that a primary source of tree σD variation is the σD variation of the environmental precipitation. Superimposed on this are isotopic variations caused by the transpiration of the leaf water incorporated by the tree. The magnitude of this transpiration effect appears to be related to relative humidity.

Within a single tree, it is found that the hydrogen isotope variations which occur for a ring sequence in one radial direction may not be exactly the same as those which occur in a different direction. Such heterogeneities appear most likely to occur in trees with asymmetric ring patterns that contain reaction wood. In the absence of reaction wood such heterogeneities do not seem to occur. Thus, hydrogen isotope analyses of tree ring sequences should be performed on trees which do not contain reaction wood.

Comparisons of tree σD variations with variations in local climate are performed on two levels: spatial and temporal. It is found that the σD values of 20 North American trees from a wide geographic range are reasonably well-correlated with the corresponding average annual temperature. The correlation is similar to that observed for a comparison of the σD values of annual precipitation of 11 North American sites with annual temperature. However, it appears that this correlation is significantly disrupted by trees which grew on poorly drained sites such as those in stagnant marshes. Therefore, site selection may be important in choosing trees for climatic interpretation of σD values, although proper sites do not seem to be uncommon.

The measurement of σD values in 5-year samples from the tree ring sequences of 13 trees from 11 North American sites reveals a variety of relationships with local climate. As it was for the spatial σD vs climate comparison, site selection is also apparently important for temporal tree σD vs climate comparisons. Again, it seems that poorly-drained sites are to be avoided. For nine trees from different "well-behaved" sites, it was found that the local climatic variable best related to the σD variations was not the same for all sites.

Two of these trees showed a strong negative correlation with the amount of local summer precipitation. Consideration of factors likely to influence the isotopic composition of summer rain suggests that rainfall intensity may be important. The higher the intensity, the lower the σD value. Such an effect might explain the negative correlation of σD vs summer precipitation amount for these two trees. A third tree also exhibited a strong correlation with summer climate, but in this instance it was a positive correlation of σD with summer temperature.

The remaining six trees exhibited the best correlation between σD values and local annual climate. However, in none of these six cases was it annual temperature that was the most important variable. In fact annual temperature commonly showed no relationship at all with tree σD values. Instead, it was found that a simple mass balance model incorporating two basic assumptions yielded parameters which produced the best relationships with tree σD values. First, it was assumed that the σD values of these six trees reflected the σD values of annual precipitation incorporated by these trees. Second, it was assumed that the σD value of the annual precipitation was a weighted average of two seasonal isotopic components: summer and winter. Mass balance equations derived from these assumptions yielded combinations of variables that commonly showed a relationship with tree σD values where none had previously been discerned.

It was found for these "well-behaved" trees that not all sample intervals in a σD vs local climate plot fell along a well-defined trend. These departures from the local σD VS climate norm were defined as "anomalous". Some of these anomalous intervals were common to trees from different locales. When such widespread commonalty of an anomalous interval occurred, it was observed that the interval corresponded to an interval in which drought had existed in the North American Great Plains.

Consequently, there appears to be a combination of both local and large scale climatic information in the σD variations of tree cellulose C-H hydrogen.

Effects of self energy of the ions on the double layer structure and properties at the dielectric interface

Although numerous theoretical efforts have been put forth, a systematic, unified and predictive theoretical framework that is able to capture all the essential physics of the interfacial behaviors of ions, such as the Hofmeister series effect, Jones-Ray effect and the salt effect on the bubble coalescence remain an outstanding challenge. The most common approach to treating electrostatic interactions in the presence of salt ions is the Poisson-Boltzmann (PB) theory. However, there are many systems for which the PB theory fails to offer even a qualitative explanation of the behavior, especially for ions distributed in the vicinity of an interface with dielectric contrast between the two media (like the water-vapor/oil interface). A key factor missing in the PB theory is the self energy of the ion.

In this thesis, we develop a self-consistent theory that treats the electrostatic self energy (including both the short-range Born solvation energy and the long-range image charge interactions), the nonelectrostatic contribution of the self energy, the ion-ion correlation and the screening effect systematically in a single framework. By assuming a finite charge spread of the ion instead of using the point-charge model, the self energy obtained by our theory is free of the divergence problems and gives a continuous self energy across the interface. This continuous feature allows ions on the water side and the vapor/oil side of the interface to be treated in a unified framework. The theory involves a minimum set of parameters of the ion, such as the valency, radius, polarizability of the ions, and the dielectric constants of the medium, that are both intrinsic and readily available. The general theory is first applied to study the thermodynamic property of the bulk electrolyte solution, which shows good agreement with the experiment result for predicting the activity coefficient and osmotic coefficient.

Next, we address the effect of local Born solvation energy on the bulk thermodynamics and interfacial properties of electrolyte solution mixtures. We show that difference in the solvation energy between the cations and anions naturally gives rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The miscibility of the mixture can either increases or decreases depending on the competition between the solvation energy and translation entropy of the ions. The interfacial tension shows a non-monotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations, and decreases approximately as the square root of the salt concentration for dilute solutions, which is in agreement with the Jones-Ray effect observed in experiment.

Next, we investigate the image effects on the double layer structure and interfacial properties near a single charged plate. We show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. The image charge effect is then studied for electrolyte solutions between two plates. For two neutral plates, we show that depletion of the salt ions by the image charge repulsion results in short-range attractive and long-range repulsive forces. If cations and anions are of different valency, the asymmetric depletion leads to the formation of an induced electrical double layer. For two charged plates, the competition between the surface charge and the image charge effect can give rise to like- charge attraction.

Then, we study the inhomogeneous screening effect near the dielectric interface due to the anisotropic and nonuniform ion distribution. We show that the double layer structure and interfacial properties is drastically affected by the inhomogeneous screening if the bulk Debye screening length is comparable or smaller than the Bjerrum length. The width of the depletion layer is characterized by the Bjerrum length, independent of the salt concentration. We predict that the negative adsorption of ions at the interface increases linearly with the salt concentration, which cannot be captured by either the bulk screening approximation or the WKB approximation. For asymmetric salt, the inhomogeneous screening enhances the charge separation in the induced double layer and significantly increases the value of the surface potential.

Finally, to account for the ion specificity, we study the self energy of a single ion across the dielectric interface. The ion is considered to be polarizable: its charge distribution can be self-adjusted to the local dielectric environment to minimize the self energy. Using intrinsic parameters of the ions, such as the valency, radius, and polarizability, we predict the specific ion effect on the interfacial affinity of halogen anions at the water/air interface, and the strong adsorption of hydrophobic ions at the water/oil interface, in agreement with experiments and atomistic simulations.

The theory developed in this work represents the most systematic theoretical technique for weak-coupling electrolytes. We expect the theory to be more useful for studying a wide range of structural and dynamic properties in physicochemical, colloidal, soft-matter and biophysical systems.

A influência da feminilização da Aids sobre a prática da enfermeira em relação ao aleitamento materno

Trata-se de uma pesquisa qualitativa cujo objeto é a percepção da enfermeira sobre a prática do aleitamento materno no contexto da feminilização da Aids. Tem por objetivos: analisar a percepção das enfermeiras de maternidade sobre a prática do aleitamento materno e a feminilização da Aids e discutir a prática da enfermeira em relação ao aleitamento materno a partir da feminilização da Aids. Os sujeitos do estudo foram nove enfermeiras de três maternidades municipais do Rio de Janeiro que possuem título de Hospitais Amigo da Criança. A coleta de dados foi realizada através de entrevistas semi-estruturadas. A técnica de análise do conteúdo foi baseada em Bardin. Emergiram três categorias: a) A percepção da enfermeira sobre sua prática quanto ao aleitamento materno; b) As percepções da enfermeira sobre a feminilização da Aids; c) A prática da enfermeira em relação ao aleitamento materno a partir da feminilização da Aids. Constatamos que a enfermeira percebe sua prática em relação ao aleitamento materno sob influência das Políticas Públicas voltadas para a promoção e proteção do mesmo, como a Iniciativa Hospital Amigo da Criança e o Alojamento Conjunto. Em relação à Aids, o advento da feminilização surpreende as enfermeiras que reagem com indignação, tristeza, medo e angústia. Estes sentimentos justificam-se, pois, para elas, pensar soropositividade em mulheres significa privá-las de exercer sua saúde reprodutiva e sexual plenamente, ou seja, os papéis esperados socialmente de uma mulher, como ser mãe e amamentar. A condição social e sexual da mulher (gênero) também emergiu dos depoimentos como determinantes para soropositividade. Ao perceberem sua prática às mulheres soropositivas nas maternidades, as enfermeiras apontam dificuldades geradas pela dicotomia (incentivo ao aleitamento materno e inibição da lactação) tanto para elas profissionais quanto para as mulheres que não podem amamentar. O processo de feminilização e os investimentos e recursos voltados para este acarretaram mudanças na prática da enfermeira, que refere mais segurança pessoal após disponibilização de teste rápido para HIV e cursos de capacitação para os profissionais. Além da prática voltada para as questões técnicas, apontam uma nova abordagem às mulheres soropositivas, como o objetivo de não expô-las às outras puérperas nas enfermarias de alojamento conjunto. Desta maneira, constatamos que as mudanças ocorridas nas práticas das enfermeiras estão relacionadas com o estabelecimento de políticas públicas voltadas para a amamentação e o HIV/Aids. Os valores pessoais ainda interferem na prática das enfermeiras, e a Aids ainda é vista como uma doença possuidora de estigmas tanto sociais quanto culturais. Reforçamos a necessidade de estratégias que possam diminuir a divergência das Políticas de Incentivo ao Aleitamento Materno e as de Prevenção à Transmissão Vertical, a fim de qualificar a prática de enfermagem às mulheres soropositivas.

Projetos de cuidado em Fisioterapia

As práticas de cuidado em fisioterapia, em muitas situações, resgatam a função do fisioterapeuta de executor de técnicas que lhe era atribuída nos primórdios da profissão. Ao exercer essa função meramente técnica, muitas vezes deixando-se substituir pelo equipamento nas suas ações, o profissional compromete o estabelecimento do vínculo terapeuta-paciente, contribuindo para o esvaziamento do encontro em saúde. Nessas situações, predomina o êxito técnico (a eficiência na realização do procedimento) sobre o sucesso prático (os benefícios trazidos para vida das pessoas). Para que o sucesso prático seja atingido, é fundamental que haja o questionamento sobre o que sonham as pessoas, profissionais e pacientes, para as suas vidas e para a saúde, quais são suas perspectivas e projetos de vida, seus projetos de felicidade. Nesse sentido, é imprescindível considerar, também, os projetos de felicidade dos profissionais da saúde enquanto sujeitos desse encontro. Afinal, é a partir deles que o profissional elabora o seu projeto de cuidado para cada paciente. Assim, esse trabalho buscou compreender os elementos que configuram a construção de projetos de cuidado em fisioterapia a partir da reflexão dos próprios fisioterapeutas. Para tanto, utilizou-se a metodologia qualitativa de pesquisa por meio de entrevistas semi-estruturadas para que fossem produzidas narrativas da história de vida do trabalho. Os discursos foram analisados integralmente e a categorização foi feita em três sub-temas: exercício profissional, relação com os pacientes e reflexões. Foi possível perceber que muitos dos arranjos de trabalho estabelecidos visam coibir o vínculo profissional-paciente, transformando-o em valor de troca e mercantilizando a relação terapêutica. Expropriada do vínculo, a prática se resume à realização de procedimentos independentes de sua finalidade, minando as possibilidades de sucesso prático. Nesse caso, não é a tecnologia que gera o afastamento e a mecanização, mas são as estratégias de mercantilização do cuidado fisioterápico. Essa situação só pode ocorrer por um processo de subordinação do profissional e seu saber, o que está fortemente associado a condições de trabalho exploratórias. Podemos dizer que a mercantilização do cuidado facilita a restrição sobre as condições de trabalho e também é fruto dela. O problema é que essa restrição é vista pelos profissionais como não definitiva, mas como um caminho para se alcançar reconhecimento na busca pelo exercício liberal da fisioterapia. A intenção desse estudo não foi traçar um plano normativo de conduta para a profissão, mas imaginamos que o equacionamento dessas questões passa necessariamente pelo reconhecimento, pelas inquietações e indignações com o problema. O que se espera, portanto, é facilitar esse processo de desconforto por meio da proximidade dessas questões trazidas pelas narrativas.

近衍射极限半导体激光束波面检测

在星间半导体激光通信系统中，如何检测发射光束波面的质量是个较难处理的问题，为了较好地解决这一问题，在简单介绍白光横向双剪切干涉仪的基础上，报道了用此干涉仪对近衍射极限半导体激光光束波面的检测，在此基础上推导出计算远场发散度的公式。实验测得近场光束的波高差为0．2A，通过夫朗和费衍射求得光束的发散度仅为64．8μrad，这表明光束接近光学衍射极限。同时，表明双剪切干涉仪灵敏度高、实用性好。

Wave-front analysis method of circular aperture sampling for collimation testing

A new type of wave-front analysis method for the collimation testing of laser beams is proposed. A concept of wave-front height is defined, and, on this basis, the wave-front analysis method of circular aperture sampling is introduced. The wave-front height of the tested noncollimated wave can be estimated from the distance between two identical fiducial diffraction planes of the sampled wave, and then the divergence is determined. The design is detailed, and the experiment is demonstrated. The principle and experiment results of the method are presented. Owing to the simplicity of the method and its low cost, it is a promising method for checking the collimation of a laser beam with a large divergence. © 2005 Optical Society of America.

提高能量密度的超衍射极限激光光束相位补偿技术

定义了激光光束衍射远场光斑压缩前后的能量比以及能量密度比来衡量超衍射极限激光光束的质量。通过利用反向传递算法设计了合适的补偿相位板，不但对准直放大的单一横模激光光束进行小于光学衍射极限的发散度的压缩，同时又保证光束能量集中于压缩后的远场衍射主瓣中，使压缩后的远场衍射光斑的能量密度增加。给出了相应的实例。这一结论不但解决了光学超分辨中光束压缩与能量损失不可避免这一矛盾，而且为发散度小且能量密度高的超衍射极限激光光束的实验工作以及该类光束的实际应用提供了理论基础。

Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

A Direct Approach to Robustness Optimization

This dissertation reformulates and streamlines the core tools of robustness analysis for linear time invariant systems using now-standard methods in convex optimization. In particular, robust performance analysis can be formulated as a primal convex optimization in the form of a semidefinite program using a semidefinite representation of a set of Gramians. The same approach with semidefinite programming duality is applied to develop a linear matrix inequality test for well-connectedness analysis, and many existing results such as the Kalman-Yakubovich--Popov lemma and various scaled small gain tests are derived in an elegant fashion. More importantly, unlike the classical approach, a decision variable in this novel optimization framework contains all inner products of signals in a system, and an algorithm for constructing an input and state pair of a system corresponding to the optimal solution of robustness optimization is presented based on this information. This insight may open up new research directions, and as one such example, this dissertation proposes a semidefinite programming relaxation of a cardinality constrained variant of the H ∞ norm, which we term sparse H ∞ analysis, where an adversarial disturbance can use only a limited number of channels. Finally, sparse H ∞ analysis is applied to the linearized swing dynamics in order to detect potential vulnerable spots in power networks.