Abstract: Determining the rigid transformation relating 2D images to known 3D geometry is a classical problem in photogrammetry and computer vision. Heretofore, the best methods for solving the problem have relied on iterative optimization methods which cannot be proven to converge and/or which do not effectively account for the orthonormal structure of rotation matrices. We show that the pose estimation problem can be formulated as that of minimizing an error metric based on collinearity in object (as opposed to image) space. Using object space collinearity error, we derive an iterative algorithm which directly computes orthogonal rotation matrices and which is globally convergent. Experimentally, we show that the method is computationally efficient, that it is no less accurate than the best currently employed optimization methods, and that it outperforms all tested methods in robustness to outliers.

Abstract: This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance.

Abstract: An image-processing technique is proposed, which performs iterative interrogation of particle image velocimetry (PIV) recordings. The method is based on cross-correlation, enhancing the matching performances by means of a relative transformation between the interrogation areas. On the basis of an iterative prediction of the tracers motion, window offset and deformation are applied, accounting for the local deformation of the fluid continuum. In addition, progressive grid refinement is applied in order to maximise the spatial resolution. The performances of the method are analysed and compared with the conventional cross correlation with and without the effect of a window discrete offset. The assessment of performance through synthetic PIV images shows that a remarkable improvement can be obtained in terms of precision and dynamic range. Moreover, peak-locking effects do not affect the method in practice. The velocity gradient range accessed with the application of a relative window deformation (linear approximation) is significantly enlarged, as confirmed in the experimental results.

Abstract: In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well.

Abstract: Notation and Convention.- Variational Inequalities with Continuous Mappings.- Problem Formulation and Basic Facts Main Idea of CR Methods Implementable CR Methods Modified Rules for Computing Iteration Parameters CR Method Based on a Frank-Wolfe Type Auxiliary Procedure CR Method for Variational Inequalities with Nonlinear Constraints Variational Inequalities with Multivalued Mappings.- Problem Formulation and Basic Facts CR Method for the Mixed Variational Inequality Problem CR Method for the Generalized Variational Inequality Problem CR Method for Multivalued Inclusions Decomposable CR Method Applications and Numerical Experiments.- Iterative Methods for Variational Inequalities with non Strictly Monotone Mappings Economic Equilibrium Problems Numerical Experiments with Test Problems Auxiliary Results.- Feasible Quasi-Nonexpansive Mappings Error Bounds for Linearly Constrained Problems A Relaxation Subgradient Method Without Linesearch

Abstract: The paper deals with topology optimization of structures undergoing large deformations. The geometrically nonlinear behaviour of the structures are modelled using a total Lagrangian finite element formulation and the equilibrium is found using a Newton-Raphson iterative scheme. The sensitivities of the objective functions are found with the adjoint method and the optimization problem is solved using the Method of Moving Asymptotes. A filtering scheme is used to obtain checkerboard-free and mesh-independent designs and a continuation approach improves convergence to efficient designs. Different objective functions are tested. Minimizing compliance for a fixed load results in degenerated topologies which are very inefficient for smaller or larger loads. The problem of obtaining degenerated "optimal" topologies which only can support the design load is even more pronounced than for structures with linear response. The problem is circumvented by optimizing the structures for multiple loading conditions or by minimizing the complementary elastic work. Examples show that differences in stiffnesses of structures optimized using linear and nonlinear modelling are generally small but they can be large in certain cases involving buckling or snap-through effects.

Abstract: We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies.

Abstract: Numerical solution of ill-posed problems is often accomplished by discretization (projection onto a finite dimensional subspace) followed by regularization. If the discrete problem has high dimension, though, typically we compute an approximate solution by projecting the discrete problem onto an even smaller dimensional space, via iterative methods based on Krylov subspaces. In this work we present a common framework for efficient algorithms that regularize after this second projection rather than before it. We show that determining regularization parameters based on the final projected problem rather than on the original discretization has firmer justification and often involves less computational expense. We prove some results on the approximate equivalence of this approach to other forms of regularization, and we present numerical examples.

Abstract: We present an implicit solution method for the compressible Navier-Stokes equations based on a Discontinuous Galerkin space discretization and on the implicit backward Euler time integration scheme. The linear system arising from the implicit time stepping scheme are solved with the preconditioned GMRES iterative method. Several preconditioners have been considered. We describe the features of the method and investigate its accuracy and performance by computing several classical 2-dimensional test cases.

Abstract: This paper examines the problem of robust stabilization of a class of triangular structural time-delay nonlinear systems. Based on the constructive use of appropriate Lynapunov-Krasovskii functionals, an iterative procedure of constructing a stabilizing controller is developed. A practical industry process is provided to illustrate the application of the main result.

Abstract: Computational burden is a major concern when an iterative algorithm is used to reconstruct a three-dimensional (3-D) image with attenuation, detector response, and scatter corrections. Most of the computation time is spent executing the projector and backprojector of an iterative algorithm. Usually, the projector and the backprojector are transposed operators of each other. The projector should model the imaging geometry and physics as accurately as possible. Some researchers have used backprojectors that are computationally less expensive than the projectors to reduce computation time. This paper points out that valid backprojectors should satisfy a condition that the projector/backprojector matrix must not contain negative eigenvalues. This paper also investigates the effects when unmatched projector/backprojector pairs are used.

Abstract: We analyze the conjugate gradient (CG) method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as flexible CG, we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small. We further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments. Depending on the eigenvalue distribution corresponding to the fixed reference preconditioner, several situations have to be distinguished. In some cases, the convergence is as fast with truncated versions of the algorithm or even with the standard CG method, whereas quite large variations are allowed without too much penalty. In other cases, the flexible variant effectively outperforms the standard method, while the need for truncation limits the size of the variations that can be reasonably allowed.

Abstract: This paper proposes an iterative soft interference cancellation scheme for a multiple antenna system. The proposed scheme is derived from the maximum a posteriori (MAP) criterion. At each iteration, the a posteriori probability is obtained by cancelling out the previous soft symbols, new soft symbols are calculated based on the a posteriori probability, and the total interference-plus-noise power is dynamically updated to account for the uncertainty in the soft symbols. The computational cost of the proposed algorithm grows linearly with the number of antennas while that of the optimum MAP detector grows exponentially. Simulations results show that an SNR gain of about 3 to 5 dB can be obtained by the proposed soft cancellation scheme.

Abstract: Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi-definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time-stepping integration schemes based on Pade approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.

Abstract: Publisher Summary This chapter discusses the analysis of multigrid methods. It explains briefly the classical iterative methods for linear systems. This includes the estimate for the rate of convergence of linear iterative methods, the derivation of the preconditioned conjugate gradient (PCG) method, and an estimate for the rate of convergence of the PCG method as an iterative method. It considers a model finite element problem and discusses a two-level method. The model finite element equation serves as a background problem and the two-level method provides a motivation for the multigrid methods discussed briefly in the chapter. In addition, the chapter also details the construction of two classes of commonly used smoothing operators (smoothers): additive and multiplicative smoothers. It discusses the conditions under which the smoothers satisfy the smoothing conditions required in the abstract multilevel theory. Many examples of smoothers are provided and relevant smoothing conditions are verified in the chapter.

Abstract: In this paper, we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators. We consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism. The approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters. In particular, we propose an iterative algorithm for designing a controller which is based upon subgradient iterations. At each step of the sequence, we first generate a random sample and then we make a subgradient step for a convex constraint defined by the LQR problem. The main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps. In addition, at a fixed step, we compute a lower bound of the probability that a quadratically stabilizing controller is found.

Abstract: Investigated in this paper is the approximation in the ATC-40 nonlinear static procedure (NSP) that the earthquake-induced deformation of an inelastic single-degree-of-freedom (SDF) system can be estimated by an iterative method requiring analysis of a sequence of equivalent linear systems. Several deficiencies in the ATC-40 Procedure A are demonstrated. This iterative procedure did not converge for some of the systems analyzed. It converged in many cases, but to a deformation much different than dynamic (nonlinear response history or inelastic design spectrum) analysis of the inelastic system. The ATC-40 Procedure B always gives a unique value of deformation, same as that determined by Procedure A if it converged. These approximate procedures underestimate significantly the deformation for a wide range of periods and ductility factors with errors approaching 50%, implying that the estimated deformation is about half the “exact” value. Surprisingly, the ATC-40 procedures are deficient relative to even the...

Abstract: A fuzzy model predictive control (FMPC) approach is introduced to design a control system for a highly nonlinear process. In this approach, a process system is described by a fuzzy convolution model that consists of a number of quasi-linear fuzzy implications. In controller design, prediction errors and control energy are minimized through a two-layered iterative optimization process. At the lower layer, optimal local control policies are identified to minimize prediction errors in each subsystem. A near optimum is then identified through coordinating the subsystems to reach an overall minimum prediction error at the upper layer. The two-layered computing scheme avoids extensive online nonlinear optimization and permits the design of a controller based on linear control theory. The efficacy of the FMPC approach is demonstrated through three examples.

Abstract: Linearized methods are presented for appraising resolution and parameter accuracy in images generated with 2-D and 3-D nonlinear electromagnetic (EM) inversion schemes. When direct matrix inversion is used, the model resolution and a posteriori model covariance matrices can be calculated readily. By analyzing individual columns of the model resolution matrix, the spatial variation of the resolution in the horizontal and vertical directions can be estimated empirically. Plotting the diagonal of the model covariance matrix provides an estimate of how errors in the inversion process, such as data noise and incorrect a priori assumptions, map into parameter error and thus provides valuable information about the uniqueness of the resulting image. Methods are also derived for image appraisal when the iterative conjugate gradient technique is applied to solve the inverse. An iterative statistical method yields accurate estimates of the model covariance matrix as long as enough iterations are used. Although determining the entire model resolution matrix in a similar manner is computationally prohibitive, individual columns of this matrix can be determined. Thus, the spatial variation in image resolution can be determined by calculating the columns of this matrix for key points in the image domain and then interpolating between. Examples of the image analysis techniques are provided on 2-D and 3-D synthetic crosswell EM data sets as well as a field data set collected at Lost Hills oil field in central California.

Abstract: In image restoration and reconstruction applications, unconstrained Krylov subspace methods represent an attractive approach for computing approximate solutions. They are fast, but unfortunately they do not produce approximate solutions preserving nonnegativity. As a consequence the error of the computed approximate solution can be large. Enforcing a nonnegativity constraint can produce much more accurate approximate solutions, but can also be computationally expensive. This paper considers a nonnegativity constrained minimization algorithm which represents a variant of an algorithm proposed by Kaufman. Numerical experiments show that the algorithm can be more accurate and computationally competitive with unconstrained Krylov subspace methods.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: We propose optimal gradient operators based on a newly derived consistency criterion. This criterion is based on an orthogonal decomposition of the difference between a continuous gradient and discrete gradients into the intrinsic smoothing effect and the self-inconsistency involved in the operator. We show that consistency assures the exactness of gradient direction of a locally 1D pattern in spite of its orientation, spectral composition, and sub-pixel translation. Stressing that inconsistency reduction is of primary importance, we derive an iterative algorithm which leads to accurate gradient operators of arbitrary size. We compute the optimum 3/spl times/3, 4/spl times/4, and 5/spl times/5 operators, compare them with conventional operators and examine the performance for one synthetic and several real images. The results indicate that the proposed operators are superior with respect to accuracy, bandwidth and isotropy.

Abstract: An overview of the polarization method is presented. The method can by applied for different regimes of the electromagnetic field as well as for electric circuits. Criteria for the choice of the permeability are proposed, so that the iterative scheme leads to a Picard-Banach fixed point procedure. The errors are evaluated. An efficient overrelaxation method is presented. The modality of using FEM numerical method is analyzed in order to ensure the convergence of the method.

Abstract: We study a class of circuit-switched wavelength-routing networks with fixed or alternate routing and with random wavelength allocation. We present an iterative path decomposition algorithm to evaluate accurately and efficiently the blocking performance of such networks with and without wavelength converters. Our iterative algorithm analyzes the original network by decomposing it into single-path subsystems. These subsystems are analyzed in isolation, and the individual results are appropriately combined to obtain a solution for the overall network. To analyze individual subsystems, we first construct an exact Markov process that captures the behavior of a path in terms of wavelength use. We also obtain an approximate Markov process which has a closed-form solution that can be computed efficiently for short paths. We then develop an iterative algorithm to analyze approximately arbitrarily long paths. The path decomposition approach naturally captures the correlation of both link loads and link blocking events. Our algorithm represents a simple and computationally efficient solution to the difficult problem of computing call-blocking probabilities in wavelength-routing networks. We also demonstrate how our analytical techniques can be applied to gain insight into the problem of converter placement in wavelength-routing networks.