Conjugate Gradient

We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically faithful: the lines exhibit no 'wraparound effects'. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n 2 is the number of pixels. This relies on a discrete projection-slice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2-D Fourier transform on a non-Cartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT-the process of rapid exact evaluation of the 2-D Fourier transform at these non-Cartesian grid points-is possible using chirp-Z transforms. This Radon transform is one-to-one and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3-D version of the transform. We review closely related algorithmic work of Mersereau and Oppenheim (1974), Pasciak (1980), Edholm and Herman (1988), Lawton (1988) and Bailey and Swartztrauber (1991) which are precursors of these ideas, but miss the mathematical framework, the geometric faithfulness, and the invertibility.

With respect to the microwave imaging of the dielectric properties in an imaging region, the full derivation of a new inversion algorithm based on the contrast source inversion (CSI) algorithm and a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The unknown dielectric properties are represented as nodal values on a two-dimensional (2D) arbitrary triangular mesh using linear basis functions. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The resulting sparse and symmetric FEM matrix equation can be solved efficiently, and it is shown how its solution can be used to calculate the gradient operators required in the conjugate-gradient CSI update without storing the inverse of the FEM matrix. The inversion algorithm is applied to conductive-enclosures of various shapes and unbounded-region microwave tomography configurations where the 2D transverse magnetic (TM) approximation can be applied.

An efficient algorithm combining the adaptive integral method and the discrete complex-image method (DCIM) is presented in this paper for analyzing large-scale microstrip structures. The arbitrarily shaped microstrips are discretized using triangular elements with Rao-Wilton-Glisson basis functions. These basis functions are then projected onto a rectangular grid, which enables the calculation of the resultant matrix-vector product using the fast Fourier transform. The method retains the advantages of the well-known conjugate-gradient fast-Fourier-transform method, as well as the excellent modeling capability offered by triangular elements. The resulting algorithm has the memory requirement proportional to ( ) and the operation count for the matrix-vector multiplication proportional to ( log ), where denotes the number of unknowns. The required spatial Green's functions are computed efficiently using the DCIM, which further speeds up the algorithm. Numerical results for some microstrip circuits and a microstrip antenna array are presented to demonstrate the efficiency and accuracy of this method.

This paper describes our experience with developing a parallel weighted- least-square (WLS) state estimation (SE) program for shared-memory parallel computers. Since the key computational kernel of the WLS algorithm based on the Newton-Raphson approach is a solver of sparse linear equations, a significant part of our effort was focused on selecting, implementing and evaluating this algorithm. An optimized shared memory version of the conjugate gradient (CG) algorithm was found to be competitive to state-of-the-art implementation of LU solvers for the SE problem on the SGI Altix, a shared memory architecture. We also ported the full SE algorithm including CG to the Cray MTA-2 shared memory multithreaded architecture and investigated its performance.

We develop algorithms to forward model and invert magnetometric resistivity (MMR) responses over an arbitrary 3-D conductivity structure. The observed data can be at the surface or in the borehole. In the forward modelling algorithm, the secondorder partial differential equations for the scalar and vector potentials are discretized on a staggered-grid using the finitevolume technique. In the inversion method, we discretize the 3-D model into a large number of rectangular cells of constant conductivity, and the final solution is obtained by minimizing a global objective function composed of the model objective function and data misfit. All minimizations are carried out with the Gauss-Newton algorithm and model perturbations at each iteration are obtained by a conjugate gradient least squares method (CGLS), in which only the sensitivity matrix and its transpose multiplying a vector are required. A depth weighting and a sensitivity-based weighting are respectively applied to inversion of surface and down-hole MMR data. The inversion method has been tested with a field data set from Western Australia.

In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable quadrature formulas that provide (second order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology.

ABSTRACT We consider the interior Dirichlet problem for Laplace’s equation on a non-simply connected two-dimensional regions with smooth boundaries. The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral. The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind. Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “modified Dirichlet problem”. The Nyström’s method coupled with the trapezoidal rule is used as numerical integration scheme. The linear system derived from the integral equation is solved using the conjugate gradient method applied to the normqal equation. Theoretical and computational details of the method are presented.

A model for an infrared (IR) flame detection system using multiple artificial neural networks (ANN) is presented. The present work offers significant improvements over our previous design (Huseynov et al., 2005). Feature extraction only in the relevant frequency band using joint time-frequency analysis yields an input to a series of conjugate-gradient (CG) method-based ANNs. Each ANN is trained to distinguish all hydrocarbon flames from a particular type of environmental nuisance and ambient noise. Signal saturation caused by the increased intensity of IR sources at closer distances is resolved by adjustable gain control.

... Naser El-Sheimy, Kai-Wei Chiang, and Aboelmagd Noureldin Abstract—Inertial-navigation system (INS) and global position system (GPS) technologies have been widely applied in many po-sitioning and navigation applications. ... [23] and Noureldin et al. ...

The Radon transform (RT) suffers from the typical problems of loss of resolution and aliasing that arise as a consequence of incomplete information, including limited aperture and discretization. Sparseness in the Radon domain is a valid and useful criterion for supplying this missing information, equivalent somehow to assuming smooth amplitude variation in the transition between known and unknown (missing) data. Applying this constraint while honoring the data can become a serious challenge for routine seismic processing because of the very limited processing time available, in general, per common midpoint. To develop methods that are robust, easy to use and flexible to adapt to different problems we have to pay attention to a variety of algorithms, operator design, and estimation of the hyperparame-ters that are responsible for the regularization of the solution. In this paper, we discuss fast implementations for several varieties of RT in the time and frequency domains. An iterative conjugate gradient algorithm with fast Fourier transform multiplication is used in all cases. To preserve the important property of iterative subspace methods of regularizing the solution by the number of iterations, the model weights are incorporated into the operators. This turns out to be of particular importance, and it can be understood in terms of the singular vectors of the weighted transform. The iterative algorithm is stopped according to a general cross validation criterion for subspaces. We apply this idea to several known implementations and compare results in order to better understand differences between, and merits of, these algorithms.

A spectral stochastic finite element formulation with consideration of multi-layer effect and spatial variability of material properties is developed for probabilistic analysis of laminated composite plates. The material properties of each lamina are modeled as a set of random fields and represented by the Karhunen-Loève expansion. The expansion is incorporated into a spatial discretization in accordance with a standard finite element procedure based on the first-order shear deformation theory. A spectral expansion with use of polynomial chaos is employed to represent the stochastic nodal displacements in terms of standard normal random variables. A preconditioning matrix is then proposed for the solution of spectral stochastic finite element equations with use of a preconditioning conjugate gradient technique. The various statistics of interest of the system responses are finally obtained by means of the coefficients of the spectral expansion. The numerical accuracy and the computational efficiency of the method are demonstrated by comparison with Monte-Carlo simulation.

An algorithm based on the finite element modified method of characteristics (FEMMC) is presented to solve convection-diffusion, Burgers and unsteady incompressible Navier-Stokes equations for laminar flow. Solutions for these progressively more involved problems are presented so as to give numerical evidence for the robustness, good error characteristics and accuracy of our method. To solve the Navier-Stokes equations, an approach that

... Con-sidering that the Superior Province was constructed largely from the addition of mantle-derived material between 3100 and 2600 Ma (Card 1990) and that the Grenville Province comprises tectonically stacked slices of reworked Archean and Proterozoic rock assemblages ...

The aim of this study is to make a contribution towards the efforts of reducing the use of CFCs by finding a drop-in replacement for pure refrigerants used in domestic and industrial appliances. The suggested solution is the use of HFC and HC based refrigerant mixtures. In this study, we investigate different possible ratios of these mixtures and their corresponding performances by using Artificial Neural-Networks (ANNs). We believe this dramatically reduces the times and efforts required to achieve these targets. Coefficients of Performances (COPs) and Total Irreversibilities (TIs) of refrigerants and their mixtures have been calculated for a vapor-compression refrigeration system with a liquid/suction line heat-exchanger. The constant cooling-load method is taken as a reference. The thermodynamic properties of refrigerants have been taken from REFPROP 6.01. To train the network, based on Scaled Conjugate Gradient (SCG), Pola-Ribiere Conjugate Gradient (CGP), and Levenberg-Marquardt (LM) learning algorithms and a logistic sigmoid transfer function, we have used various ratios of 7 refrigerant mixtures of HFCs and HCs along with three CFCs (R12, R22, and R502). They were used as inputs while the COP and TI values, calculated as above, were the outputs. The network has yielded R 2 values of 0.9999 and maximum errors for training and test data were found to be 2 and 3%, respectively.

Raw materials play a vital role in the ferrochrome production using submerged arc furnace route. Optimized combination of different raw materials can improve the performance of furnace and minimize the power consumption. This process carries numerous process complexities as well as feed variation, which make it difficult to model mathematically. Artificial neural network known as a black box approach is attempted to predict the effect of various raw materials (pellets, briquettes, hard lumps, friable lumps, coke and quartzite) on the performance of submerged arc furnace by incorporating a production capability index (PCI). Production capability index is a ratio of the daily production and the maximum production achieved by the furnace in the ideal conditions. A detailed statistical analysis was carried on plant data to study relationship of raw material and furnace performance. In the first step of the study, the non-linear relationship between the raw material inputs and PCI is tried to predict by multivariable linear regression. Further feed forward back propagation neural network with three different learning algorithms were tried to improve the prediction accuracy (conjugant gradient decent, Levenberg-Marquardt optimization and resilient back propagation). Radial basis neural networks were also tried but no significant improvement was found in the performance prediction. The correlation coefficient is considered as a accuracy measure, and found that correlation between predicted and actual values were 0.64 for multilinear regression which was improved 0.70, 0.71 for radial basis neural network and feed forward neural network with resilient back propagation learning algorithm. Comparative analysis has been done among statistical analysis, neural network structures and the actual values of production capability index.

A systematic comparison of two basic types of neural network, static and dynamic, is presented in this study. Two backpropagation (BP) learning optimization algorithms, the standard BP and conjugate gradient (CG) method, are used for the static network, and the real-time recurrent learning (RTRL) algorithm is used for the dynamic-feedback network. Twenty-three storm-events, about 1632 rainfall and runoff data sets, of the Lan-Yang River in Taiwan are used to demonstrate the efficiency and practicability of the neural networks for one hour ahead streamflow forecasting. In a comparison of searching algorithms for a static network, the results show that the CG method is superior to the standard BP method in terms of the efficiency and effectiveness of the constructed network's performance. For a comparison of the static neural network using the CG algorithm with the dynamic neural network using RTRL, the results show that (1) the static-feedforward neural network could produce satisfactory results only when there is a sufficient and adequate training data set, (2) the dynamic neural network generally could produce better and more stable flow forecasting than the static network, and (3) the RTRL algorithm helps to continually update the dynamic network for learning-this feature is especially important for the extraordinary time-varying characteristics of rainfall -runoff processes. q

ABSTRAK : Telah dilakukan perancangan kinerja re-heater untuk mengurangi konsentrasi gas buang CO yang dihasikan dari knalpot tersebut dengan metode konjugate gradien. Dari metode tersebut diperoleh desain knalpot dengan panjang dan diameter re-heater sebesar masing-masing 0.2 m dan 0.12 m. Setelah diuji pada mobil Toyota Kijang produksi tahun 1990, knalpot tersebut mampu mengurangi emisi CO 0,35 %, CO2 14,6 % dan HC 76,29 %. Selain itu, desain knalpot tersebut juga mampu mengurangi kebisingan sekitar 3,79% pada saat putaran mesin stabil dan 3,00% pada saat putaran mesin ditingkatkan.

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an /2-orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders' MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.

Iterative solvers appear to be very promising in the development of efficient software, based on Interior Point methods, for large-scale nonlinear optimization problems. In this paper we focus on the use of preconditioned iterative techniques to solve the KKT system arising at each iteration of a Potential Reduction method for convex Quadratic Programming. We consider the augmented system approach and analyze the behaviour of the Constraint Preconditioner with the Conjugate Gradient algorithm. Comparisons with a direct solution of the augmented system and with MOSEK show the effectiveness of the iterative approach on large-scale sparse problems.

We propose new conjugate gradient formulas for computing the search directions for unconstrained optimization problems. The new formulas turn out to be the conjugate descent formula if exact line searches are made. Some formulas possess the sufficient descent property without any line searches. General convergence results for the proposed formulas with the weak Wolfe-Powell conditions are studied. We prove that some of the formulas with the steplength technique which ensures the Zoutendijk condition to be held are globally convergent. In addition, the global convergence results for some other formulas with the standard Armijo line search are also given. Preliminary numerical results show that the proposed methods are very promising.

It is pointed out that the so called momentum method, much used in the neural network literature as an acceleration of the backpropagation method, is a stationary version of the conjugate gradient method. Connections with the continuous optimization method known as heavy ball with friction are also made. In both cases, adaptive (dynamic) choices of the so called learning rate and momentum parameters are obtained using a control Liapunov function analysis of the system.

We present a new variational-based method for automatic liver vessels segmentation from abdominal CTA images. The segmentation task is formulated as a functional minimization problem within a variational framework. We introduce a new functional that incorporates both geometrical vesselness measure and vessels surface properties. The functional describes the distance between the desired segmentation and the original image. To minimize the functional, we derive the Euler-Lagrange equation from it and solve it using the conjugate gradients algorithm. Our approach is automatic and improves upon other Hessian-based methods in the detection of bifurcations and complex vessels structures by incorporating a surface term into the functional. To assess our method, we conducted with an expert radiologist two comparative studies on 8 abdominal CTA clinical datasets. In the first study, the radiologist assessed the presence of 11 vascular bifurcations on each dataset, totaling of 73 bifurcations. The radiologist qualitatively compared the bifurcations segmentation of our method and that of a Hessian-based threshold method. Our method correctly segmented 88% of the bifurcations with a higher visibility score of 82%, as compared to only 55% in the Hessian-based method with a visibility score of 33%. In the second study, the radiologist assessed the individual vessels visibility on the 3D segmentation images and on the original CTA slices. Ten main liver vessels were examined in each dataset The overall visibility score was 93%. These results indicate that our method is suitable for the automatic segmentation and visualization of the liver vessels.

An iterative approach to full vector three-dimensional inverse scattering problems, where the unknown objects can have conductivity, permittivity and permeability different from the known background medium, is discussed. Since this problem involves a large number of unknowns, it has to be solved effectively and efficiently so that the results can be obtained in timely manner. The forward modeling is based on a domain integral equation approach formulated in terms of the electric and magnetic contrast sources normalized with the characteristic impedance of the background medium. Our numerical tests indicate that this formulation is prerequisite in order to arrive at a forward solution within an acceptable number of iterations, and hence it is also of significant importance in the optimization process of the inverse problem. The inverse scattering problem is attacked using the Multiplicative Regularized Contrast Source Inversion method as known in the literature. The complexity of this inverse method is approximately equal to the complexity of two equivalent forward algorithms of the conjugate gradient type. Furthermore, this inverse method has been armed with a weighted L 2 -norm regularizer which has been included as a multiplicative constraint. Some representative numerical testings will be presented to illustrate the ability of the our numerical algorithms.

The standard conjugate gradient (CG) method uses orthogonality of the residues to simplify the formulas for the parameters necessary for convergence. In adaptive filtering, the sample-by-sample update of the correlation matrix and the cross-correlation vector causes a loss of the residue orthogonality in a modified online algorithm, which, in turn, results in loss of convergence and an increase of the filter quadratic mean error. This letter extends a recently proposed control Liapunov function analysis of the CG method viewed as a dynamic system in the standard feedback configuration to the case of adaptive filtering.

A family of scaled conjugate gradient algorithms for large-scale unconstrained minimization is defined. The Perry, the Polak-Ribière and the Fletcher-Reeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization method. The best combination of formula, scaling and initial choice of step-length is compared against well known algorithms using a classical set of problems. An additional comparison involving an ill-conditioned estimation problem in Optics is presented.

We present a robust iterative method to solve the inverse scattering problem in cases where the Born approximation is valid. We formulate this linearized inverse problem in terms of the unknown material contrast and the unknown contrast sources and we solve the problem by minimizing a cost functional consisting of two terms. The first term represents the differences between the actual data and the modeled data, while the second term represents the misfit in the constitutive relations between the contrast sources and the incident fields. In each iteration, the contrast sources and the contrast are reconstructed alternatingly, using subsequently a conjugate gradient step for the contrast source updates and a direct inversion of a diagonal matrix for the contrast. A further regularization with a multiplicative regularization factor is discussed. In this regularization procedure the relative variation of the contrast is minimized as well. As a test case we consider the two-dimensional (2-D) transverse magnetic polarization problem. Synthetic numerical examples are presented in order to compare the presented algorithm to the traditional Born algorithm. Results with respect to the inversion of experimental data are presented as well. In addition, some inversion results for the subsurface sensing problem, both in two and three dimensions, are presented.

In this paper, We propose a new nonlinear conjugate gradient method (FRA) that satisfies a sufficient descent condition and global convergence under the inexact line search of strong wolf powell. Our numerical experiment shaw the efficiency of the new method in solving a set of problems from the CUTEst package, the proposed new formula gives excellent numerical results at CPU time, number of iterations, number of gradient ratings when compared to WYL, DY, PRP, and FR methods.

The conjugate gradient is the most popular optimization method for solving large systems of linear equations. In a system identification problem, for example, where very large impulse response is involved, it is necessary to apply a particular strategy which diminishes the delay, while improving the convergence time. In this paper we propose a new scheme which combines frequency-domainadaptive filtering with a conjugate gradient technique in order to solve a high order multichannel daptive filter, while being delayless and guaranteeing a very short convergence time.

In four-dimensional variational data assimilation (4D-Var) an optimal estimate of the initial state of a dynamical system is obtained by solving a large-scale unconstrained minimization problem. The gradient of the cost functional may be efficiently computed using the adjoint modeling, at the expense equivalent to a few forward model integrations; for most practical applications, the evaluation of the Hessian matrix is not feasible due to the large dimension of the discrete state vector. Hybrid methods aim to provide an improved optimization algorithm by dynamically interlacing inexpensive L-BFGS iterations with fast convergent Hessian-free Newton (HFN) iterations. In this paper, a comparative analysis of the performance of a hybrid method vs. L-BFGS and HFN optimization methods is presented in the 4D-Var context. Numerical results presented for a two-dimensional shallow-water model show that the performance of the hybrid method is sensitive to the selection of the method parameters such as the length of the L-BFGS and HFN cycles and the number of inner conjugate gradient iterations during the HFN cycle. Superior performance may be obtained in the hybrid approach with a proper selection of the method parameters. The applicability of the new hybrid method in the framework of operational 4D-Var in terms of computational cost and performance is also discussed.

The ideal black body fully absorbs all incident rays, that is, all propagating waves created by arbitrary sources. A known idealized realization of the black body is the perfectly matched layer (PML), widely used in numerical electromagnetics. However, ideal black bodies and PMLs do not interact with evanescent fields that exists near any finite-size source, and the energy stored in these fields cannot be harvested. Here, we introduce the concept of the ideal conjugate matched layer (CML), which fully absorbs the energy of both propagating and evanescent fields of sources acting as an ideal sink for electromagnetic energy. Conjugate matched absorbers have exciting application potentials, as resonant attractors of electromagnetic energy into the absorber volume. We derive the conditions on the constitutive parameters of media which can serve as CML materials, numerically study the performance of planar and cylindrical CML and discuss possible realizations of such materials as metal-dielectric composites.

Three second-order, nonselfadjoint partial differential equations in two dimensions are discretized by the finite element method, leading to non-symmetric systems of linear equations. The systems are solved by several conjugate gradient-like methods, and compared for their computational cost. All methods are tested with and without preconditioning.

In this paper we present recent results for the solution of the three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the models considered in our previous work, we are considering a more complicated 3D model. Here we outline the basics of our solution, and show the results obtained for this model. The results are encouraging, and show that the software works. However, some unresolved problem remains, as usual. This problem is directly inherent in the physics of the 3D MT inverse problem, rather than related to some bugs of our solution. We strongly believe that it can be successfully resolved, if we apply more constraints on the model parameters, in addition to those imposed by traditional Tikhonov-type regularization.

A systematic comparison of two basic types of neural network, static and dynamic, is presented in this study. Two backpropagation (BP) learning optimization algorithms, the standard BP and conjugate gradient (CG) method, are used for the static network, and the real-time recurrent learning (RTRL) algorithm is used for the dynamic-feedback network. Twenty-three storm-events, about 1632 rainfall and runoff data sets, of the Lan-Yang River in Taiwan are used to demonstrate the efficiency and practicability of the neural networks for one hour ahead streamflow forecasting. In a comparison of searching algorithms for a static network, the results show that the CG method is superior to the standard BP method in terms of the efficiency and effectiveness of the constructed network's performance. For a comparison of the static neural network using the CG algorithm with the dynamic neural network using RTRL, the results show that (1) the static-feedforward neural network could produce satisfactory results only when there is a sufficient and adequate training data set, (2) the dynamic neural network generally could produce better and more stable flow forecasting than the static network, and (3) the RTRL algorithm helps to continually update the dynamic network for learning-this feature is especially important for the extraordinary time-varying characteristics of rainfall -runoff processes. q

ABSTRACT Explicit finite element approximate inverse preconditioning methods have been extensively used for solving efficiently sparse linear systems on multiprocessor and multicomputer systems. New parallel computational techniques are proposed for the parallelization of explicit preconditioned biconjugate conjugate gradient type methods, based on Portable Operating System Interface for UniX (POSIX) Threads, for multicore systems. Parallelization is achieved by assigning every loop of the parallel explicit preconditioned bi-conjugate conjugate gradient-STAB (PEPBiCG-STAB) to the desired number of threads, thus achieving for-loop parallelization. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PEPBiCG-STAB method for solving characteristic two dimensional boundary value problems on multicore computer systems are presented, which are favorably compared to corresponding results from multiprocessor systems. The implementation issues of the proposed method are also discussed using POSIX Threads on a multicore system.

In this paper a parallel shared memory Java multithreaded design and implementation of the explicit approximate inverse preconditioning is presented for solving efficiently arrow-type linear systems on symmetric multiprocessor systems. A new parallel algorithm for computing a class of optimized approximate inverse matrix is introduced. The performance on a symmetric multiprocessor system, using Java multithreading, is investigated by solving characteristic

A study of the fundamental obstacles to accelerate the preconditioned conjugate gradient (PCG) method on modern graphic processing units (GPUs) is presented and several techniques are proposed to enhance its performance over previous work independent of the GPU generation and the matrix sparsity pattern. The proposed enhancements increase the performance of PCG up to 23 times compared to vector optimized PCG results on modern CPUs and up to 3.4 times compared to previous GPU results.

We develop a non-linear conjugate gradient inversion for global long period electromagnetic induction studies. The scheme requires computation of derivatives of the regularized penalty functional. We derive analytical and numerical expressions for these derivatives, and the associated Jacobian, and show how these can be efficiently implemented by generalizing and extending an existing finite difference forward solver. Using layered spherical harmonics to parametrize the model space, we invert a range of synthetic data sets to test the inversion, and to study vertical and horizontal resolution of currently available data sets. We conclude that the currently available long-period global geomagnetic observatory data in the period range 5–107 d can resolve large scale (300–500 km vertically, thousands of km horizontally) heterogeneities in mantle electrical conductivity reliably at depths ∼ 670–1600 km. By extending induction response to 0.2–5 d (including daily variation periods), upper-mantle structure could also be resolved.

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries-stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB).

We present a very general geometrical correction method for enforcing collisions and other geometrical constraints between polygonal mesh surfaces. It is based on a global resolution scheme that takes advantage of an efficient use of the conjugate gradient algorithm to find the appropriate displacement of the mesh vertices that would satisfy all the constraints simultaneously, and according to momentum conservation laws. This method has been implemented in a cloth simulation system along with a collision response model that enforces a minimum "thickness" distance between cloth surfaces, which can be efficiently integrated in an simulation scheme based on implicit integration. Some provided examples illustrate the efficiency of the method.