Geometric quantization and foliation reduction

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kähler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kähler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.

In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kähler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or “admissible”, values of momentum.

We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.

We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.

Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.

A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.

Full and Model-Reduced Structure-Preserving Simulation of Incompressible Fluids

This thesis outlines the construction of several types of structured integrators for incompressible fluids. We first present a vorticity integrator, which is the Hamiltonian counterpart of the existing Lagrangian-based fluid integrator. We next present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness to coarse spatial and temporal resolutions of geometric integrators, and the simplicity of homogenized boundary conditions on regular grids to deal with arbitrarily-shaped domains with sub-grid accuracy.

Both these numerical methods involve approximating the Lie group of volume-preserving diffeomorphisms by a finite-dimensional Lie-group and then restricting the resulting variational principle by means of a non-holonomic constraint. Advantages and limitations of this discretization method will be outlined. It will be seen that these derivation techniques are unable to yield symplectic integrators, but that energy conservation is easily obtained, as is a discretized version of Kelvin's circulation theorem.

Finally, we outline the basis of a spectral discrete exterior calculus, which may be a useful element in producing structured numerical methods for fluids in the future.

Stabilization of collective motion in three dimensions: A consensus approach

This paper proposes a methodology to stabilize relative equilibria in a model of identical, steered particles moving in three-dimensional Euclidean space. Exploiting the Lie group structure of the resulting dynamical system, the stabilization problem is reduced to a consensus problem. We first derive the stabilizing control laws in the presence of all-to-all communication. Providing each agent with a consensus estimator, we then extend the results to a general setting that allows for unidirectional and time-varying communication topologies. © 2007 IEEE.

积分偏移的计算几何与并行实现

The Second Round of Oil & Gas Exploration needs more precision imaging method, velocity vs. depth model and geometry description on Complicated Geological Mass. Prestack time migration on inhomogeneous media was the technical basic of velocity analysis, prestack time migration on Rugged surface, angle gather and multi-domain noise suppression. In order to realize this technique, several critical technical problems need to be solved, such as parallel computation, velocity algorithm on ununiform grid and visualization. The key problem is organic combination theories of migration and computational geometry. Based on technical problems of 3-D prestack time migration existing in inhomogeneous media and requirements from nonuniform grid, parallel process and visualization, the thesis was studied systematically on three aspects: Infrastructure of velocity varies laterally Green function traveltime computation on ununiform grid, parallel computational of kirchhoff integral migration and 3D visualization, by combining integral migration theory and Computational Geometry. The results will provide powerful technical support to the implement of prestack time migration and convenient compute infrastructure of wave number domain simulation in inhomogeneous media. The main results were obtained as follows: 1. Symbol of one way wave Lie algebra integral, phase and green function traveltime expressions were analyzed, and simple 2-D expression of Lie algebra integral symbol phase and green function traveltime in time domain were given in inhomogeneous media by using pseudo-differential operators’ exponential map and Lie group algorithm preserving geometry structure. Infrastructure calculation of five parts, including derivative, commutating operator, Lie algebra root tree, exponential map root tree and traveltime coefficients , was brought forward when calculating asymmetry traveltime equation containing lateral differential in 3-D by this method. 2. By studying the infrastructure calculation of asymmetry traveltime in 3-D based on lateral velocity differential and combining computational geometry, a method to build velocity library and interpolate on velocity library using triangulate was obtained, which fit traveltime calculate requirements of parallel time migration and velocity estimate. 3. Combining velocity library triangulate and computational geometry, a structure which was convenient to calculate differential in horizontal, commutating operator and integral in vertical was built. Furthermore, recursive algorithm, for calculating architecture on lie algebra integral and exponential map root tree (Magnus in Math), was build and asymmetry traveltime based on lateral differential algorithm was also realized. 4. Based on graph theory and computational geometry, a minimum cycle method to decompose area into polygon blocks, which can be used as topological representation of migration result was proposed, which provided a practical method to block representation and research to migration interpretation results. 5. Based on MPI library, a process of bringing parallel migration algorithm at arbitrary sequence traces into practical was realized by using asymmetry traveltime based on lateral differential calculation and Kirchhoff integral method. 6. Visualization of geological data and seismic data were studied by the tools of OpenGL and Open Inventor, based on computational geometry theory, and a 3D visualize system on seismic imaging data was designed.

地震波场模拟及叠前深度偏移的李群方法

The processes of seismic wave propagation in phase space and one way wave extrapolation in frequency-space domain, if without dissipation, are essentially transformation under the action of one parameter Lie groups. Consequently, the numerical calculation methods of the propagation ought to be Lie group transformation too, which is known as Lie group method. After a fruitful study on the fast methods in matrix inversion, some of the Lie group methods in seismic numerical modeling and depth migration are presented here. Firstly the Lie group description and method of seismic wave propagation in phase space is proposed, which is, in other words, symplectic group description and method for seismic wave propagation, since symplectic group is a Lie subgroup and symplectic method is a special Lie group method. Under the frame of Hamiltonian, the propagation of seismic wave is a symplectic group transformation with one parameter and consequently, the numerical calculation methods of the propagation ought to be symplectic method. After discrete the wave field in time and phase space, many explicit, implicit and leap-frog symplectic schemes are deduced for numerical modeling. Compared to symplectic schemes, Finite difference (FD) method is an approximate of symplectic method. Consequently, explicit, implicit and leap-frog symplectic schemes and FD method are applied in the same conditions to get a wave field in constant velocity model, a synthetic model and Marmousi model. The result illustrates the potential power of the symplectic methods. As an application, symplectic method is employed to give synthetic seismic record of Qinghai foothills model. Another application is the development of Ray+symplectic reverse-time migration method. To make a reasonable balance between the computational efficiency and accuracy, we combine the multi-valued wave field & Green function algorithm with symplectic reverse time migration and thus develop a new ray+wave equation prestack depth migration method. Marmousi model data and Qinghai foothills model data are processed here. The result shows that our method is a better alternative to ray migration for complex structure imaging. Similarly, the extrapolation of one way wave in frequency-space domain is a Lie group transformation with one parameter Z and consequently, the numerical calculation methods of the extrapolation ought to be Lie group methods. After discrete the wave field in depth and space, the Lie group transformation has the form of matrix exponential and each approximation of it gives a Lie group algorithm. Though Pade symmetrical series approximation of matrix exponential gives a extrapolation method which is traditionally regarded as implicit FD migration, it benefits the theoretic and applying study of seismic imaging for it represent the depth extrapolation and migration method in a entirely different way. While, the technique of coordinates of second kind for the approximation of the matrix exponential begins a new way to develop migration operator. The inversion of matrix plays a vital role in the numerical migration method given by Pade symmetrical series approximation. The matrix has a Toepelitz structure with a helical boundary condition and is easy to inverse with LU decomposition. A efficient LU decomposition method is spectral factorization. That is, after the minimum phase correlative function of each array of matrix had be given by a spectral factorization method, all of the functions are arranged in a position according to its former location to get a lower triangular matrix. The major merit of LU decomposition with spectral factorization (SF Decomposition) is its efficiency in dealing with a large number of matrixes. After the setup of a table of the spectral factorization results of each array of matrix, the SF decomposition can give the lower triangular matrix by reading the table. However, the relationship among arrays is ignored in this method, which brings errors in decomposition method. Especially for numerical calculation in complex model, the errors is fatal. Direct elimination method can give the exact LU decomposition But even it is simplified in our case, the large number of decomposition cost unendurable computer time. A hybrid method is proposed here, which combines spectral factorization with direct elimination. Its decomposition errors is 10 times little than that of spectral factorization, and its decomposition speed is quite faster than that of direct elimination, especially in dealing with a large number of matrix. With the hybrid method, the 3D implicit migration can be expected to apply on real seismic data. Finally, the impulse response of 3D implicit migration operator is presented.

指数矩阵近似计算在地震波场延拓中的应用

Seismic wave field numerical modeling and seismic migration imaging based on wave equation have become useful and absolutely necessarily tools for imaging of complex geological objects. An important task for numerical modeling is to deal with the matrix exponential approximation in wave field extrapolation. For small value size matrix exponential, we can approximate the square root operator in exponential using different splitting algorithms. Splitting algorithms are usually used on the order or the dimension of one-way wave equation to reduce the complexity of the question. In this paper, we achieve approximate equation of 2-D Helmholtz operator inversion using multi-way splitting operation. Analysis on Gauss integral and coefficient of optimized partial fraction show that dispersion may accumulate by splitting algorithms for steep dipping imaging. High-order symplectic Pade approximation may deal with this problem, However, approximation of square root operator in exponential using splitting algorithm cannot solve dispersion problem during one-way wave field migration imaging. We try to implement exact approximation through eigenfunction expansion in matrix. Fast Fourier Transformation (FFT) method is selected because of its lowest computation. An 8-order Laplace matrix splitting is performed to achieve a assemblage of small matrixes using FFT method. Along with the introduction of Lie group and symplectic method into seismic wave-field extrapolation, accurate approximation of matrix exponential based on Lie group and symplectic method becomes the hot research field. To solve matrix exponential approximation problem, the Second-kind Coordinates (SKC) method and Generalized Polar Decompositions (GPD) method of Lie group are of choice. SKC method utilizes generalized Strang-splitting algorithm. While GPD method utilizes polar-type splitting and symmetric polar-type splitting algorithm. Comparing to Pade approximation, these two methods are less in computation, but they can both assure the Lie group structure. We think SKC and GPD methods are prospective and attractive in research and practice.

基于信息重建技术的地震信号处理方法研究

The dissertation addressed the problems of signals reconstruction and data restoration in seismic data processing, which takes the representation methods of signal as the main clue, and take the seismic information reconstruction (signals separation and trace interpolation) as the core. On the natural bases signal representation, I present the ICA fundamentals, algorithms and its original applications to nature earth quake signals separation and survey seismic signals separation. On determinative bases signal representation, the paper proposed seismic dada reconstruction least square inversion regularization methods, sparseness constraints, pre-conditioned conjugate gradient methods, and their applications to seismic de-convolution, Radon transformation, et. al. The core contents are about de-alias uneven seismic data reconstruction algorithm and its application to seismic interpolation. Although the dissertation discussed two cases of signal representation, they can be integrated into one frame, because they both deal with the signals or information restoration, the former reconstructing original signals from mixed signals, the later reconstructing whole data from sparse or irregular data. The goal of them is same to provide pre-processing methods and post-processing method for seismic pre-stack depth migration. ICA can separate the original signals from mixed signals by them, or abstract the basic structure from analyzed data. I surveyed the fundamental, algorithms and applications of ICA. Compared with KL transformation, I proposed the independent components transformation concept (ICT). On basis of the ne-entropy measurement of independence, I implemented the FastICA and improved it by covariance matrix. By analyzing the characteristics of the seismic signals, I introduced ICA into seismic signal processing firstly in Geophysical community, and implemented the noise separation from seismic signal. Synthetic and real data examples show the usability of ICA to seismic signal processing and initial effects are achieved. The application of ICA to separation quake conversion wave from multiple in sedimentary area is made, which demonstrates good effects, so more reasonable interpretation of underground un-continuity is got. The results show the perspective of application of ICA to Geophysical signal processing. By virtue of the relationship between ICA and Blind Deconvolution , I surveyed the seismic blind deconvolution, and discussed the perspective of applying ICA to seismic blind deconvolution with two possible solutions. The relationship of PC A, ICA and wavelet transform is claimed. It is proved that reconstruction of wavelet prototype functions is Lie group representation. By the way, over-sampled wavelet transform is proposed to enhance the seismic data resolution, which is validated by numerical examples. The key of pre-stack depth migration is the regularization of pre-stack seismic data. As a main procedure, seismic interpolation and missing data reconstruction are necessary. Firstly, I review the seismic imaging methods in order to argue the critical effect of regularization. By review of the seismic interpolation algorithms, I acclaim that de-alias uneven data reconstruction is still a challenge. The fundamental of seismic reconstruction is discussed firstly. Then sparseness constraint on least square inversion and preconditioned conjugate gradient solver are studied and implemented. Choosing constraint item with Cauchy distribution, I programmed PCG algorithm and implement sparse seismic deconvolution, high resolution Radon Transformation by PCG, which is prepared for seismic data reconstruction. About seismic interpolation, dealias even data interpolation and uneven data reconstruction are very good respectively, however they can not be combined each other. In this paper, a novel Fourier transform based method and a algorithm have been proposed, which could reconstruct both uneven and alias seismic data. I formulated band-limited data reconstruction as minimum norm least squares inversion problem where an adaptive DFT-weighted norm regularization term is used. The inverse problem is solved by pre-conditional conjugate gradient method, which makes the solutions stable and convergent quickly. Based on the assumption that seismic data are consisted of finite linear events, from sampling theorem, alias events can be attenuated via LS weight predicted linearly from low frequency. Three application issues are discussed on even gap trace interpolation, uneven gap filling, high frequency trace reconstruction from low frequency data trace constrained by few high frequency traces. Both synthetic and real data numerical examples show the proposed method is valid, efficient and applicable. The research is valuable to seismic data regularization and cross well seismic. To meet 3D shot profile depth migration request for data, schemes must be taken to make the data even and fitting the velocity dataset. The methods of this paper are used to interpolate and extrapolate the shot gathers instead of simply embedding zero traces. So, the aperture of migration is enlarged and the migration effect is improved. The results show the effectiveness and the practicability.

Analyse de groupe d’un modèle de la plasticité idéale planaire et sur les solutions en termes d’invariants de Riemann pour les systèmes quasilinéaires du premier ordre

Les objets d’étude de cette thèse sont les systèmes d’équations quasilinéaires du premier ordre. Dans une première partie, on fait une analyse du point de vue du groupe de Lie classique des symétries ponctuelles d’un modèle de la plasticité idéale. Les écoulements planaires dans les cas stationnaire et non-stationnaire sont étudiés. Deux nouveaux champs de vecteurs ont été obtenus, complétant ainsi l’algèbre de Lie du cas stationnaire dont les sous-algèbres sont classifiées en classes de conjugaison sous l’action du groupe. Dans le cas non-stationnaire, une classification des algèbres de Lie admissibles selon la force choisie est effectuée. Pour chaque type de force, les champs de vecteurs sont présentés. L’algèbre ayant la dimension la plus élevée possible a été obtenues en considérant les forces monogéniques et elle a été classifiée en classes de conjugaison. La méthode de réduction par symétrie est appliquée pour obtenir des solutions explicites et implicites de plusieurs types parmi lesquelles certaines s’expriment en termes d’une ou deux fonctions arbitraires d’une variable et d’autres en termes de fonctions elliptiques de Jacobi. Plusieurs solutions sont interprétées physiquement pour en déduire la forme de filières d’extrusion réalisables. Dans la seconde partie, on s’intéresse aux solutions s’exprimant en fonction d’invariants de Riemann pour les systèmes quasilinéaires du premier ordre. La méthode des caractéristiques généralisées ainsi qu’une méthode basée sur les symétries conditionnelles pour les invariants de Riemann sont étendues pour être applicables à des systèmes dans leurs régions elliptiques. Leur applicabilité est démontrée par des exemples de la plasticité idéale non-stationnaire pour un flot irrotationnel ainsi que les équations de la mécanique des fluides. Une nouvelle approche basée sur l’introduction de matrices de rotation satisfaisant certaines conditions algébriques est développée. Elle est applicable directement à des systèmes non-homogènes et non-autonomes sans avoir besoin de transformations préalables. Son efficacité est illustrée par des exemples comprenant un système qui régit l’interaction non-linéaire d’ondes et de particules. La solution générale est construite de façon explicite.

Introduction à quelques aspects de quantification géométrique.

On révise les prérequis de géométrie différentielle nécessaires à une première approche de la théorie de la quantification géométrique, c'est-à-dire des notions de base en géométrie symplectique, des notions de groupes et d'algèbres de Lie, d'action d'un groupe de Lie, de G-fibré principal, de connexion, de fibré associé et de structure presque-complexe. Ceci mène à une étude plus approfondie des fibrés en droites hermitiens, dont une condition d'existence de fibré préquantique sur une variété symplectique. Avec ces outils en main, nous commençons ensuite l'étude de la quantification géométrique, étape par étape. Nous introduisons la théorie de la préquantification, i.e. la construction des opérateurs associés à des observables classiques et la construction d'un espace de Hilbert. Des problèmes majeurs font surface lors de l'application concrète de la préquantification : les opérateurs ne sont pas ceux attendus par la première quantification et l'espace de Hilbert formé est trop gros. Une première correction, la polarisation, élimine quelques problèmes, mais limite grandement l'ensemble des observables classiques que l'on peut quantifier. Ce mémoire n'est pas un survol complet de la quantification géométrique, et cela n'est pas son but. Il ne couvre ni la correction métaplectique, ni le noyau BKS. Il est un à-côté de lecture pour ceux qui s'introduisent à la quantification géométrique. D'une part, il introduit des concepts de géométrie différentielle pris pour acquis dans (Woodhouse [21]) et (Sniatycki [18]), i.e. G-fibrés principaux et fibrés associés. Enfin, il rajoute des détails à quelques preuves rapides données dans ces deux dernières références.

Two-mode overconstrained three-DOFs rotational-translational linear-motor-based parallel-kinematics mechanism for machine tool applications

The paper introduces a family of three-DOFs translational-rotational Parallel-Kinematics Mechanisms (PKMs) as well as the mobility analysis of such family using Lie-group theory. Each member of this family has two-rotational one-translational DOFs. A novel mechanism is presented and analyzed as a representative of that family. The use and the practical value of that modular mechanism are emphasized.

Reversibility properties of semigroup actions on homogeneous spaces

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

We associate to an arbitrary Z-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer-Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.

Curves in homogeneous spaces and their contact with 1-dimensional orbits

Let alpha be a C(infinity) curve in a homogeneous space G/H. For each point x on the curve, we consider the subspace S(k)(alpha) of the Lie algebra G of G consisting of the vectors generating a one parameter subgroup whose orbit through x has contact of order k with alpha. In this paper, we give various important properties of the sequence of subspaces G superset of S(1)(alpha) superset of S(2)(alpha) superset of S(3)(alpha) superset of ... In particular, we give a stabilization property for certain well-behaved curves. We also describe its relationship to the isotropy subgroup with respect to the contact element of order k associated with alpha.

A homological version of the implicit function theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.

Symplectic actions on coadjoint orbits

We present a compact expression for the field theoretical actions based on the symplectic analysis of coadjoint orbits of Lie groups. The final formula for the action density α c becomes a bilinear form 〈(S, 1/λ), (y, m y)〉, where S is a 1-cocycle of the Lie group (a schwarzian type of derivative in conformai case), λ is a coefficient of the central element of the algebra and script Y sign ≡ (y, m y) is the generalized Maurer-Cartan form. In this way the action is fully determined in terms of the basic group theoretical objects. This result is illustrated on a number of examples, including the superconformal model with N = 2. In this case the method is applied to derive the N = 2 superspace generalization of the D=2 Polyakov (super-) gravity action in a manifest (2, 0) supersymmetric form. As a byproduct we also find a natural (2, 0) superspace generalization of the Beltrami equations for the (2, 0) supersymmetric world-sheet metric describing the transition from the conformal to the chiral gauge.