南海张裂大陆边缘数值模拟研究

The South China Sea (SCS) is one of the largest marginal seas in the western Pacific, which is located at the junction of Eurasian plate, Pacific plate and Indian-Australian plate. It was formed by continent breakup and sea-floor spreading in Cenozoic. The complicated interaction among the three major plates made tectonic movement complex and geological phenomena very rich in this area. The SCS is an ideal place to study the formation and evolution of rifted continental margin and sea-floor spreading since it is old enough to have experienced the major stages of the basin evolution but still young enough to have preserved its original nature. As the demand for energy grows day by day in our country, the deep water region of the northern continental margin in the SCS has become a focus of oil and gas exploration because of its huge hydrocarbon potential. Therefore, to study the rifted continental margin of the SCS not only can improve our understanding of the formation and evolution processes of rifted continental margin, but also can provide theoretical support for hydrocarbon exploration in rifted continental margin. This dissertation mainly includes five topics as follows: (1) Various classic lithosphere stretching models are reviewed, and the continuous non-uniform stretching model is modified to make it suitable for the case where the extension of lithopheric mantle exceeds that of the crust. Then simple/pure shear flexural cantilever model is applied to model the basement geometries of SO49-18 profile in the northern continental margin of the SCS. By fitting the basements obtained by using 2DMove software with modeling results, it is found that the reasonable effective elastic thickness is less than 5km in this region. According to this result, it is assumed that there is weak lower crust in the northern continental margin in the SCS. (2) We research on the methods for stretching factor estimation based on various lithosphere stretching models, and apply the method based on multiple finite rifting model to estimate the stretching factors of several wells and profiles in the northern continental margin of the SCS. (3) We improve one-dimension strain rate inversion method with conjugate gradient method, and apply it to invert the strain rate of several wells in the northern continental margin of the SCS. Two-dimension strain rate forward modeling is carried out, and the modeling results show that effective elastic thickness is a key parameter to control basin’s geometry. (4) We simulate divergent upwelling mantle flow model using finite difference method, and apply this newly developed model to examine the formation mechanism of the northwest and central sub-basin in the SCS. (5) We inverse plate thickness and basal temperature of oceanic lithosphere using sea-floor ages and bathymetries of the North Pacific and the North Atlantic based on varied-parameters plate model, in which the heat conductivity, heat capacity and coefficient of thermal expansion depend on temperature or depth. A new empirical formula is put forward based the inversed parameters, which depicts the relation among sea-floor age, bathymetry and heat flow. Then various similar empirical formulae, including the newly developed one, are applied to examine the sea-floor spread issue in the SCS based on the heat flow and bathymetry data of the abyssal sub-basin.

地质规律约束下的波阻抗反演

Impedance inversion is very important in seismic technology. It is based on seismic profile. Good inversion result is derived from high quality seismic profile, which is formed using high resolution imaging resolution. High-resolution process demands that signal/noise ratio is high. It is very important for seismic inversion to improve signal/noise ratio. the main idea is that the physical parameter (wave impedance), which describes the stratigraphy directly, is achieved from seismic data expressing structural style indirectly. The solution of impedance inversion technology, which is based on convolution model, is arbitrary. It is a good way to apply the priori information as the restricted condition in inversion. An updated impedance inversion technology is presented which overcome the flaw of traditional model and highlight the influence of structure. Considering impedance inversion restricted by sedimentary model, layer filling style and congruence relation, the impedance model is built. So the impedance inversion restricted by geological rule could be realized. there are some innovations in this dissertation: 1. The best migration aperture is achieved from the included angle of time surface of diffracted wave and reflected wave. Restricted by structural model, the dip of time surface of reflected wave and diffracted wave is given. 2. The conventional method of FXY forcasting noise is updated, and the signal/noise ratio is improved. 3. Considering the characteristic of probability distribution of seismic data and geological events fully, an object function is constructed using the theory of Bayes estimation as the criterion. The mathematics is used here to describe the content of practice theory. 4. Considering the influence of structure, the seismic profile is interpreted to build the model of structure. A series of structure model is built. So as the impedance model. The high frequency of inversion is controlled by the geological rule. 5. Conjugate gradient method is selected to improve resolving process for it fit the demands of geophysics, and the efficiency of algorithm is enhanced. As the geological information is used fully, the result of impedance inversion is reasonable and complex reservoir could be forecasted further perfectly.

三维油藏数值模拟正反演算法研究

The dynamic prediction of complex reservoir development is one of the important research contents of dynamic analysis of oil and gas development. With the increase development of time, the permeabilities and porosities of reservoirs and the permeability of block reservoir at its boundaries are dynamically changing. How to track the dynamic change of permeability and porosity and make certain the permeability of block reservoir at its boundary is an important practical problem. To study developing dynamic prediction of complex reservoir, the key problem of research of dynamic prediction of complex reservoir development is realizing inversion of permeability and porosity. To realize the inversion, first of all, the fast forward and inverse method of 3-dimension reservoir simulation must be studied. Although the inversion has been widely applied to exploration and logging, it has not been applied to3-dimension reservoir simulation. Therefore, the study of fast forward and inverse method of 3-dimension reservoir simulation is a cutting-edge problem, takes on important realistic signification and application value. In this dissertation, 2-dimension and 3-dimension fluid equations in porous media are discretized by finite difference, obtaining finite difference equations to meet the inner boundary conditions by Peaceman's equations, giving successive over relaxation iteration of 3-dimension fluid equations in porous media and the dimensional analysis. Several equation-solving methods are compared in common use, analyzing its convergence and convergence rate. The alternating direction implicit procedure of 2-dimension has been turned into successive over relaxation iteration of alternating direction implicit procedure of 3-dimension fluid equations in porous media, which possesses the virtues of fast computing speed, needing small memory of computer, good adaptability for heterogeneous media and fast convergence rate. The geological model of channel-sandy reservoir has been generated with the help of stochastic simulation technique, whose cross sections of channel-sandy reservoir are parabolic shapes. This method makes the hard data commendably meet, very suit for geological modeling of containing complex boundary surface reservoir. To verify reliability of the method, theoretical solution and numerical solution are compared by simplifying model of 3-dimension fluid equations in porous media, whose results show that the only difference of the two pressure curves is that the numerical solution is lower than theoretical at the wellbore in the same space. It proves that using finite difference to solve fluid equations in porous media is reliable. As numerical examples of 3-dimension heterogeneous reservoir of the single-well and multi-well, the pressure distributions have been computed respectively, which show the pressure distributions there are clearly difference as difference of the permeabilities is greater than one order of magnitude, otherwise there are no clearly difference. As application, the pressure distribution of the channel-sandy reservoir have been computed, which indicates that the space distribution of pressure strongly relies on the direction of permeability, and is sensitive for space distributions of permeability. In this dissertation, the Peaceman's equations have been modified into solving vertical well problem and horizontal well problem simultaneously. In porous media, a 3D layer reservoir in which contain vertical wells and horizontal wells has been calculated with iteration. For channel-sandy reservoir in which there are also vertical wells and horizontal wells, a 3D transient heterogeneous fluid equation has been discretized. As an example, the space distribution of pressure has been calculated with iteration. The results of examples are accord with the fact, which shows the modification of Peaceman's equation is correct. The problem has been solved in the space where there are vertical and horizontal wells. In the dissertation, the nonuniform grid permeability integration equation upscaling method, the nonuniform grid 2D flow rate upscaling method and the nonuniform grid 3D flow rate upscaling method have been studied respectively. In those methods, they enhance computing speed greatly, but the computing speed of 3D flow rate upscaling method is faster than that of 2D flow rate upscaling method, and the precision of 3D flow rate upscaling method is better than that of 2D flow rate upscaling method. The results also show that the solutions of upscaling method are very approximating to that of fine grid blocks. In this paper, 4 methods of fast adaptive nonuniform grid upscaling method of 3D fluid equations in porous media have been put forward, and applied to calculate 3D heterogeneous reservoir and channel-sandy reservoir, whose computing results show that the solutions of nonuniform adaptive upscaling method of 3D heterogeneous fluid equations in porous media are very approximating to that of fine grid blocks in the regions the permeability or porosity being abnormity and very approximating to that of coarsen grid blocks in the other region, however, the computing speed of adaptive upscaling method is 100 times faster than that of fine grid block method. The formula of sensitivity coefficients are derived from initial boundary value problems of fluid equations in porous media by Green's reciprocity principle. The sensitivity coefficients of wellbore pressure to permeability parameters are given by Peaceman's equation and calculated by means of numerical calculation method of 3D transient anisotropic fluid equation in porous media and verified by direct method. The computing results are in excellent agreement with those obtained by the direct method, which shows feasibility of the method. In the dissertation, the calculating examples are also given for 3D reservoir, channel-sandy reservoir and 3D multi-well reservoir, whose numerical results indicate: around the well hole, the value of the sensitivity coefficients of permeability is very large, the value of the sensitivity coefficients of porosity is very large too, but the sensitivity coefficients of porosity is much less than the sensitivity coefficients of permeability, so that the effect of the sensitivity coefficients of permeability for inversion of reservoir parameters is much greater than that of the sensitivity coefficients of porosity. Because computing the sensitivity coefficients needs to call twice the program of reservoir simulation in one iteration, realizing inversion of reservoir parameters must be sustained by the fast forward method. Using the sensitivity coefficients of permeability and porosity, conditioned on observed valley erosion thickness in wells (hard data), the inversion of the permeabilities and porosities in the homogeneous reservoir, homogeneous reservoir only along the certain direction and block reservoir are implemented by Gauss-Newton method or conjugate gradient method respectively. The results of our examples are very approximating to the real data of permeability and porosity, but the convergence rate of conjugate gradient method is much faster than that of Gauss-Newton method.

ω循环型预条件矩阵及其应用

In this paper, we apply the preconditioned conjugate gradient method to the solution of positive-definite Toeplitz systems, especially we introduce a new kind of co-circulant preconditioners Pn[ca] by the use of embedding method. We have also discussed the properties of these new preconditioners and proved that many of former preconditioners can be considered as some special cases of Pn[co\. Because of the introduction of co-circulant preconditioners pn[a>], we can greatly overcome the singularity caused by circulant preconditioners. We have discussed the oo-circulant series and functions. We compare the ordinary circularity with the co-circularity, showing that the latter one can be considered as the extended form of the former one; correspondingly, many methods and theorems of the ordinary circularity can be extended. Furthermore, we present the co-circulant decompositional method. By the use of this method, we can divide any co-circulant signal into a summation of many sub-signals; especially among those sub-signals, there are many subseries of which their period is just equal to 1, which are actually the frequency elements of the original co-circulant signal. In this way, we can establish the relationship between the signal and its frequency elements, that is, the frequency elements hi the frequency domain are actually signals with the period of 1 in the spatial domain. We have also proved that the co-circulant has already existed in the traditional Fourier theory. By the use of different criteria for constructing preconditioners, we can get many different preconditioned systems. From the preconditioned systems PN[

井间地震技术研究和应用

Cross well seismic technique is a new type of geophysical method, which observes the seismic wave of the geologic body by placing both the source and receiver in the wells. By applying this method, it averted the absorption to high-frequency component of seismic signal caused by low weathering layers, thus, an extremely high-resolution seismic signal can be acquired. And extremely fine image of cross well formations, structure, and reservoir can be achieved as well. An integrated research is conducted to the high-frequency S-wave and P-wave data and some other data to determine the small faults, small structure and resolving the issues concerning the thin bed and reservoir's connectivity, fluid distribution, steam injection and fracture. This method connects the high-resolution surface seismic, logging and reservoir engineering. In this paper, based on the E & P situation in the oilfield and the theory of geophysical exploration, a research is conducted on cross well seismic technology in general and its important issues in cross well seismic technology in particular. A technological series of integrated field acquisition, data processing and interpretation and its integrated application research were developed and this new method can be applied to oilfield development and optimizing oilfield development scheme. The contents and results in this paper are as listed follows: An overview was given on the status quo and development of the cross well seismic method and problems concerning the cross well seismic technology and the difference in cross well seismic technology between China and international levels; And an analysis and comparison are given on foreign-made field data acquisition systems for cross-well seismic and pointed out the pros and cons of the field systems manufactured by these two foreign companies and this is highly valuable to import foreign-made cross well seismic field acquisition system for China. After analyses were conducted to the geometry design and field data for the cross well seismic method, a common wave field time-depth curve equation was derived and three types of pipe waves were discovered for the first time. Then, a research was conducted on the mechanism for its generation. Based on the wave field separation theory for cross well seismic method, we believe that different type of wave fields in different gather domain has different attributes characteristics, multiple methods (for instance, F-K filtering and median filtering) were applied in eliminating and suppressing the cross well disturbances and successfully separated the upgoing and downgoing waves and a satisfactory result has been achieved. In the area of wave field numerical simulation for cross well seismic method, a analysis was conducted on conventional ray tracing method and its shortcomings and proposed a minimum travel time ray tracing method based on Feraiat theory in this paper. This method is not only has high-speed calculation, but also with no rays enter into "dead end" or "blinded spot" after numerous iterations and it is become more adequate for complex velocity model. This is first time that the travel time interpolation has been brought into consideration, a dynamic ray tracing method with shortest possible path has been developed for the first arrivals of any complex mediums, such as transmission, diffraction and refraction, etc and eliminated the limitation for only traveling from one node to another node and increases the calculation accuracy for minimum travel time and ray tracing path and derives solution and corresponding edge conditions to the fourth-order differential sonic wave equation. The final step is to calculate cross well seismic synthetics for given source and receivers from multiple geological bodies. Thus, real cross-well seismic wave field can be recognized through scientific means and provides important foundation to guide the cross well seismic field geometry designing. A velocity tomographic inversion of the least square conjugated gradient method was developed for cross well seismic velocity tomopgraphic inversion and a modification has been made to object function of the old high frequency ray tracing method and put forward a thin bed oriented model for finite frequency velocity tomographic inversion method. As the theory model and results demonstrates that the method is simple and effective and is very important in seismic ray tomographic imaging for the complex geological body. Based on the characteristics of the cross well seismic algorithm, a processing flow for cross well seismic data processing has been built and optimized and applied to the production, a good section of velocity tomopgrphic inversion and cross well reflection imaging has been acquired. The cross well seismic data is acquired from the depth domain and how to interprets the depth domain data and retrieve the attributes is a brand new subject. After research was conducted on synthetics and trace integration from depth domain for the cross well seismic data interpretation, first of all, a research was conducted on logging constraint wave impedance of cross well seismic data and initially set up cross well seismic data interpretation flows. After it applied and interpreted to the cross well seismic data and a good geological results has been achieved in velocity tomographic inversion and reflection depth imaging and a lot of difficult problems for oilfield development has been resolved. This powerful, new method is good for oilfield development scheme optimization and increasing EOR. Based on conventional reservoir geological model building from logging data, a new method is also discussed on constraining the accuracy of reservoir geological model by applying the high resolution cross well seismic data and it has applied to Fan 124 project and a good results has been achieved which it presents a bight future for the cross well seismic technology.

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.

地震偏移宽角吸收边界条件和Toeplitz矩阵反演中的边界效应研究

The content of this paper is based on the research work while the author took part in the key project of NSFC and the key project of Knowledge Innovation of CAS. The whole paper is expanded by introduction of the inevitable boundary problem during seismic migration and inversion. Boundary problem is a popular issue in seismic data processing. At the presence of artificial boundary, reflected wave which does not exist in reality comes to presence when the incident seismic wave arrives at the artificial boundary. That will interfere the propagation of seismic wave and cause alias information on the processed profile. Furthermore, the quality of the whole seismic profile will decrease and the subsequent work will fail.This paper has also made a review on the development of seismic migration, expatiated temporary seismic migration status and predicted the possible break through. Aiming at the absorbing boundary problem in migration, we have deduced the wide angle absorbing boundary condition and made a compare with the boundary effect of Toepiitz matrix fast approximate computation.During the process of fast approximate inversion computation of Toepiitz system, we have introduced the pre-conditioned conjugate gradient method employing co circulant extension to construct pre-conditioned matrix. Especially, employment of combined preconditioner will reduce the boundary effect during computation.Comparing the boundary problem in seismic migration with that in Toepiitz matrix inversion we find that the change of boundary condition will lead to the change of coefficient matrix eigenvalues and the change of coefficient matrix eigenvalues will cause boundary effect. In this paper, the author has made an qualitative analysis of the relationship between the coefficient matrix eigenvalues and the boundary effect. Quantitative analysis is worthy of further research.

Fast Convergence of Regularised Region-based Mixture of Gaussians for Dynamic Background Modelling

The momentum term has long been used in machine learning algorithms, especially back-propagation, to improve their speed of convergence. In this paper, we derive an expression to prove the O(1/k2) convergence rate of the online gradient method, with momentum type updates, when the individual gradients are constrained by a growth condition. We then apply these type of updates to video background modelling by using it in the update equations of the Region-based Mixture of Gaussians algorithm. Extensive evaluations are performed on both simulated data, as well as challenging real world scenarios with dynamic backgrounds, to show that these regularised updates help the mixtures converge faster than the conventional approach and consequently improve the algorithm’s performance.

Performance and Fault Tolerance of Preconditioned Iterative Solvers on Low-Power ARM Architectures

As the complexity of computing systems grows, reliability and energy are two crucial challenges asking for holistic solutions. In this paper, we investigate the interplay among concurrency, power dissipation, energy consumption and voltage-frequency scaling for a key numerical kernel for the solution of sparse linear systems. Concretely, we leverage a task-parallel implementation of the Conjugate Gradient method, equipped with an state-of-the-art pre-conditioner embedded in the ILUPACK software, and target a low-power multi core processor from ARM.In addition, we perform a theoretical analysis on the impact of a technique like Near Threshold Voltage Computing (NTVC) from the points of view of increased hardware concurrency and error rate.

Nichtüberlappende Gebietszerlegungsmethoden für lineare und quasilineare (monotone und nichtmonotone) Probleme

In dieser Arbeit werden nichtüberlappende Gebietszerlegungsmethoden einerseits hinsichtlich der zu lösenden Problemklassen verallgemeinert und andererseits in bisher nicht untersuchten Kontexten betrachtet. Dabei stehen funktionalanalytische Untersuchungen zur Wohldefiniertheit, eindeutigen Lösbarkeit und Konvergenz im Vordergrund. Im ersten Teil werden lineare elliptische Dirichlet-Randwertprobleme behandelt, wobei neben Problemen mit dominantem Hauptteil auch solche mit singulärer Störung desselben, wie konvektions- oder reaktionsdominante Probleme zugelassen sind. Der zweite Teil befasst sich mit (gleichmäßig) monotonen koerziven quasilinearen elliptischen Dirichlet-Randwertproblemen. In beiden Fällen wird das Lipschitz-Gebiet in endlich viele Lipschitz-Teilgebiete zerlegt, wobei insbesondere Kreuzungspunkte und Teilgebiete ohne Außenrand zugelassen sind. Anschließend werden Transmissionsprobleme mit frei wählbaren $L^{\infty}$-Parameterfunktionen hergeleitet, wobei die Konormalenableitungen als Funktionale auf geeigneten Funktionenräumen über den Teilrändern ($H_{00}^{1/2}(\Gamma)$) interpretiert werden. Die iterative Lösung dieser Transmissionsprobleme mit einem Ansatz von Deng führt auf eine Substrukturierungsmethode mit Robin-artigen Transmissionsbedingungen, bei der eine Auswertung der Konormalenableitungen aufgrund einer geschickten Aufdatierung der Robin-Daten nicht notwendig ist (insbesondere ist die bekannte Robin-Robin-Methode von Lions als Spezialfall enthalten). Die Konvergenz bezüglich einer partitionierten $H^1$-Norm wird für beide Problemklassen gezeigt. Dabei werden keine über $H^1$ hinausgehende Regularitätsforderungen an die Lösungen gestellt und die Gebiete müssen keine zusätzlichen Glattheitsvoraussetzungen erfüllen. Im letzten Kapitel werden nichtmonotone koerzive quasilineare Probleme untersucht, wobei das Zugrunde liegende Gebiet nur in zwei Lipschitz-Teilgebiete zerlegt sein soll. Das zugehörige nichtlineare Transmissionsproblem wird durch Kirchhoff-Transformation in lineare Teilprobleme mit nichtlinearen Kopplungsbedingungen überführt. Ein optimierungsbasierter Lösungsansatz, welcher einen geeigneten Abstand der rücktransformierten Dirichlet-Daten der linearen Teilprobleme auf den Teilrändern minimiert, führt auf ein optimales Kontrollproblem. Die dabei entstehenden regularisierten freien Minimierungsprobleme werden mit Hilfe eines Gradientenverfahrens unter minimalen Glattheitsforderungen an die Nichtlinearitäten gelöst. Unter zusätzlichen Glattheitsvoraussetzungen an die Nichtlinearitäten und weiteren technischen Voraussetzungen an die Lösung des quasilinearen Ausgangsproblems, kann zudem die quadratische Konvergenz des Newton-Verfahrens gesichert werden.

Support Vector Machines: Training and Applications

The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Labs. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points. When the number of data points exceeds few thousands the problem is very challenging, because the quadratic form is completely dense, so the memory needed to store the problem grows with the square of the number of data points. Therefore, training problems arising in some real applications with large data sets are impossible to load into memory, and cannot be solved using standard non-linear constrained optimization algorithms. We present a decomposition algorithm that can be used to train SVM's over large data sets. The main idea behind the decomposition is the iterative solution of sub-problems and the evaluation of, and also establish the stopping criteria for the algorithm. We present previous approaches, as well as results and important details of our implementation of the algorithm using a second-order variant of the Reduced Gradient Method as the solver of the sub-problems. As an application of SVM's, we present preliminary results we obtained applying SVM to the problem of detecting frontal human faces in real images.

Approximate factorization constraint preconditioners for saddle-point matrices

We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.

Iterative Methods for Equality-Constrained Least Squares Problems

We consider the linear equality-constrained least squares problem (LSE) of minimizing ${\|c - Gx\|}_2 $, subject to the constraint $Ex = p$. A preconditioned conjugate gradient method is applied to the Kuhn–Tucker equations associated with the LSE problem. We show that our method is well suited for structural optimization problems in reliability analysis and optimal design. Numerical tests are performed on an Alliant FX/8 multiprocessor and a Cray-X-MP using some practical structural analysis data.

Least squares splines with free knots: global optimization approach

Splines with free knots have been extensively studied in regard to calculating the optimal knot positions. The dependence of the accuracy of approximation on the knot distribution is highly nonlinear, and optimisation techniques face a difficult problem of multiple local minima. The domain of the problem is a simplex, which adds to the complexity. We have applied a recently developed cutting angle method of deterministic global optimisation, which allows one to solve a wide class of optimisation problems on a simplex. The results of the cutting angle method are subsequently improved by local discrete gradient method. The resulting algorithm is sufficiently fast and guarantees that the global minimum has been reached. The results of numerical experiments are presented.

Mathematical modelling of the refractive index reconstruction through global optimization

We examine a mathematical model of non-destructive testing of planar waveguides, based on numerical solution of a nonlinear integral equation. Such problem is ill-posed, and the method of Tikhonov regularization is applied. To minimize Tikhonov functional, and find the parameters of the waveguide, we use two new optimization methods: the cutting angle method of global optimization, and the discrete gradient method of nonsmooth local optimization. We examine how the noise in the experimental data influences the solution, and how the regularization parameter has to be chosen. We show that even with significant noise in the data, the numerical solution is of high accuracy, and the method can be used to process real experimental da.ta..