Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

Abstract: A combination of variable-metric second-order update schemes and the DIIS method for both geometry and Hartree-Fock wave function optimization is described. A recursive procedure for updating large Hessians is presented. The performances of geometry optimizations with respect to the choice of the coordinate system (symmetry-adapted, internal, and Cartesian coordinates), the initial nuclear Hessian, and the optimization procedure have been investigated by a series of benchmark molecules. Formulas for the generation of initial nuclear Hessians are given

Abstract: The main purpose of this paper is to give a systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works...

Abstract: In this note a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established. Several such approximations are considered. The main idea of these techniquesis to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.

Abstract: The fast multipole method (FMM) developed by V. Rokhlin (1990) to efficiently solve acoustic scattering problems is modified and adapted to the second-kind-integral-equation formulation of electromagnetic scattering problems in two dimensions. The present implementation treats the exterior Dirichlet problem for two-dimensional, closed, conducting objects of arbitrary geometry. The FMM reduces the operation count for solving the second-kind integral equation from O(n/sup 3/) for Gaussian elimination to O(n/sup 4/3/) per conjugate-gradient iteration, where n is the number of sample points on the boundary of the scatterer. A sample technique for accelerating convergence of the iterative method, termed complexifying k, the wavenumber, is also presented. This has the effect of bounding the condition number of the discrete system; consequently, the operation count of the entire FMM (all iterations) becomes O(n/sup 4/3/). Computational results for moderate values of ka, where a is the characteristic size of the scatterer, are given. >

Abstract: A preconditioned iterative method for indefinite linear systems corresponding to certain saddlepoint problems is suggested. The block structure of the systems is utilized in order to design effective preconditioners, while the governing iterative solver is a standard minimum residual method. The method is applied to systems derived from discretizations of the Stokes problem and mixed formulations of second-order elliptic problems.

Abstract: A fully discrete finite element method for the Cahn-Hilliard equation with a logarithmic free energy based on the backward Euler method is analysed. Existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved. Two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented.

Abstract: A nonlinear projection method is presented to visualize high-dimensional data as a 2D image. The proposed method is based on the topology preserving mapping algorithm of Kohonen. The topology preserving mapping algorithm is used to train a 2D network structure. Then the interpoint distances in the feature space between the units in the network are graphically displayed to show the underlying structure of the data. Furthermore, we present and discuss a new method to quantify how well a topology preserving mapping algorithm maps the high-dimensional input data onto the network structure. This is used to compare our projection method with a well-known method of Sammon (1969). Experiments indicate that the performance of the Kohonen projection method is comparable or better than Sammon's method for the purpose of classifying clustered data. Its time-complexity only depends on the resolution of the output image, and not on the size of the dataset. A disadvantage, however, is the large amount of CPU time required. >

Abstract: We propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation. For this purpose, we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods. The different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image. In the linearized inversion, we use the background velocity model obtained by velocity analysis. We determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model. Because the inverse operator is not exact, and because the source and station distribution is limited, the first iteration of our asymptotic inversion technique is not exact. We improve the images by an iterative procedure. Since the background velocity does not change between iterations. There is no need to retrace rays,...

Abstract: A simple iterative algorithm which can be used to find array weights that produce array patterns with a given look direction and an arbitrary sidelobe specification is presented. The method can be applied to nonuniform array geometries in which the individual elements have arbitrary (and differing) radiation patterns. The method is iterative and uses sequential updating to ensure that peak sidelobe levels in the array meet the specification. Computation of each successive pattern is based on the solution of a linearly constrained least-squares problem. The constraints ensure that the magnitude of the sidelobes at the locations of the previous peaks takes on the prespecified values. Phase values for the sidelobes do not change during this process, and problems associated with choosing a specific phase value are therefore avoided. Experimental evidence suggests that the procedure terminates in remarkably few iterations, even for arrays with significant numbers of elements. >

Abstract: SUMMARY An asymptotic linearized iterative elastic inversion method is proposed to invert 2-D Earth parameters from multicomponent data and is tested numerically. The forward problem is solved by a combination of the Born approximation and ray theoretical methods. We express the perturbed seismogram in terms of perturbations of P- and S-wave impedances and density. The inversion method is based on generalized least squares. We introduce a special form of the ρ2 norm with a weighting function that corrects for geometrical spreading and obliquity of the reflectors. The Hessian for this norm could be estimated in a closed form that is asymptotically valid at high frequencies. We propose a quasi-Newtonian iterative method for the solution of the inverse problem. The first iteration of this inversion method resembles the operator proposed by Beylkin (1985) and Beylkin & Burridge (1990) for the asymptotic inversion of seismic data. Our method is more general than theirs because it can handle arbitrary discrete distributions of sources and receivers. Elastic inversion is generally ill-posed because the problem is overdetermined but undersampled. We study the resolution of the asymptotic inversion method for general sets of sources and receivers. We show that simultaneous inversion for both P- and S-wave impedance is generally ill-conditioned if data for a single scattering mode are available. In particular, it seems that only one parameter can be reliably resolved from marine data. Simultaneous inversion for a finite set of parameters can be resolved only for multicomponent elastic data containing both P-wave and S-wave information. Inversion tests using synthetic data calculated by finite-differences demonstrates that it is possible to invert simultaneously for P and S impedances.

Abstract: Methods are discussed for the solution of sparse linear equations $Ky = z$, where K is symmetric and indefinite. Since exact solutions are not always required, direct and iterative methods are both...

Abstract: A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point...

Abstract: This work studies the smoothness of the solutions of dilation equations, which are encountered in the multiresolution analysis and iterative interpolation processes. Sharp limit of the Sobolev exponent of the solution is given as a function of the spectral radius of an associated finite-dimensional positive operator. In addition, tools are given to get good explicit upper and lower bounds for the exponent.

Abstract: The theoretical foundations and the numerical performance of an advanced nonlinear circuit simulator based on the piecewise harmonic-balance (HB) technique are discussed. The exact computation of the Jacobian matrix for Newton-iteration based HB simulation and the related conversion-matrix technique for fast mixer analysis are formulated in a general form. Convergence problems at high drive levels are solved by a parametric formulation of the device models coupled with an advanced norm-reducing iteration. A physics-based approximation is shown to allow the HB equations to be effectively decoupled in many practical cases, bringing large-sized jobs, such as pulsed-RF analysis, within the reach of ordinary workstations. The exact Jacobian is used in conjunction with an exact formula for the gradient of the objective function, to implement an efficient broadband nonlinear circuit optimization capability. Examples are presented. >

Abstract: The authors report novel methods for determining switching angles for selective-harmonics-eliminated pulse-width modulation (SHE PWM) inverters. Such switching angles are defined by a set of nonlinear equations, and to solve these equations a predicting algorithm is used to calculate initial values which are first-order approximations of the exact solutions. With these predicted initial values, the Newton algorithm can be used to find the solutions within usually only one or two iterations. The authors also suggest another approach for solving the SHE PWM problem: an ordinary differential equations approach. The advantages of this approach are discussed, and its applications are demonstrated by some examples. >

Abstract: An iterative discrete optimization algorithm that works with Monte Carlo estimation of the objective function is developed Two algorithms, the simulated annealing algorithm and the stochastic ruler algorithm, are considered The authors examine some of the problems of their use and combine the advantages of both algorithms to form an iterative random search algorithm called the stochastic comparison (SC) algorithm The SC algorithm actually solves an alternative optimization problem, and it is shown under symmetry assumption that the alternative problem is equivalent to the original one The convergence of the SC algorithm is proved based on time-inhomogeneous Markov chain theory Results of numerical experiments on a testbed problem with randomly generated objective function are presented >

Abstract: Three conjugate gradient accelerated row projection (RP) methods for nonsymmetric linear systems are presented and their properties described. One method is based on Kaczmarz’s method and has an iteration matrix that is the product of orthogonal projectors; another is based on Cimmino’s method and has an iteration matrix that is the sum of orthogonal projectors. A new RP method, which requires fewer matrix-vector operations, explicitly reduces the problem size, is error reducing in the two-norm, and consistently produces better solutions than other RP algorithms, is also introduced. Using comparisons with the method of conjugate gradient applied to the normal equations, the properties of RP methods are explained.A row partitioning approach is described that yields parallel implementations suitable for a wide range of computer architectures, requires only a few vectors of extra storage, and allows computing the necessary projections with small errors. Numerical testing verifies the robustness of this appro...

Abstract: An iterative constrained inversion technique is used to find the control inputs to the plant. That is, rather than training a controller network and placing this network directly in the feedback or feedforward paths, the forward model of the plant is learned, and iterative inversion is performed on line to generate control commands. The control approach allows the controllers to respond online to changes in the plant dynamics. This approach also attempts to avoid the difficulty of analysis introduced by most current neural network controllers, which place the highly nonlinear neural network directly in the feedback path. A neural network-based model reference adaptive controller is also proposed for systems having significant dynamics between the control inputs and the observed (or desired) outputs and is demonstrated on a simple linear control system. These results are interpreted in terms of the need for a dither signal for on-line identification of dynamic systems. >

Abstract: Sequential imaging cameras are designed to record objects in motion. When the speed of the objects exceeds the temporal resolution of the shutter, the image is blurred. Because objects in a scene are often moving in different directions at different speeds, the degradation of a recorded image may be characterized by a space-variant point spread function (PSF). The sequential nature of such images can be used to determine the relative motion of various parts of the image. This information can be used to estimate the space-variant PSF. A modification of the Landweber iteration is developed to utilize the space-variant PSF to produce an estimate of the original image. >

Abstract: An earlier review paper of analysis methods for numerical weather prediction showed how many methods could be derived from the same basic analysis equation, expressing the Bayesian probability that any state is the true state. Prior probabilities depend on the error covariance of the background prior estimate, and these covariances figure in most solutions. This paper extends the analysis of iterative solution methods (one of which is known as the ‘Successive Correction’ method). In a simple example the methods are compared with a direct solution: the ‘Optimal Interpolation’ method. It is shown that the iterative methods can be equally ‘optimal’. If the number of iterations is limited, or the background weighting is omitted, then the methods normalizing the increments in observation space are shown to be more reliable than those normalizing in grid-point space. the traditional successive-correction method has a grid-point space normalization. The covariances in these methods perform a filtering function. It is possible to replace them by a filter acting on the analysis increments. Iterative methods using such a filter are derived, and are shown to correspond exactly to the iterative methods using covariance functions. A simple and efficient recursive filter is described and applied to the same example. the analysis using a two-pass filter is almost identical to that using a ‘second-order auto-regressive’ covariance function. A filter with many passes corresponds to a Gaussian-shaped covariance function. Approximate filters can be devised to model the effect of observational-error correlations, with an accuracy adequate in view of the lack of knowledge of the real correlations. With this filter, the iterative methods can be extended to deal effectively with data from remote-sensing instruments. Published iterative methods (many originally empirically derived) are reviewed, and fitted into the optimal theory. Practical applications are discussed.

Abstract: In this paper we propose a new iterative method for solving a class of linear complementarity problems:u ≥ 0,Mu + q ≥ 0, uT(Mu + q)=0, where M is a givenl ×l positive semidefinite matrix (not necessarily symmetric) andq is a givenl-vector. The method makes two matrix-vector multiplications and a trivial projection onto the nonnegative orthant at each iteration, and the Euclidean distance of the iterates to the solution set monotonously converges to zero. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. It is pointed out that the method may be used to solve general convex quadratic programming problems. Preliminary numerical experiments indicate that this method may be very efficient for large sparse problems.

Abstract: The aim of medical imaging is to provide in an non - invasive way morphological information about a human patient. The information is obtained by performing an ” experiment ” where the interaction of a source of radiation anf the tissue under consideration is measured. From the measured data the desired information has to be computed, hence we face an inverse problem. It is always ill - posed in the sense that small errors in the data can be amplified to large changes in the reconstruction. For developing efficient and stable software we have to study the mathematical model; i. e., the description of the experiment based on physical and engeneering knowledge. In optimal situations it is possible to derive” inversion formulas” which relate in a constructive way the data to the searched - for information. Reconstruction algorithms can be found by discretisizing these formulas. But of course we have to perform a stability analysis in order to design the software such that the influence of the data noise is reduced as much as possible. If such inversion formulas are unknown or cannot be discretisized in an accurate way direct discretization and iterative methods are used for the computation.

Abstract: An improved dynamic condensation approach is presented for accurate calculation of structural eigenproperties. The approach is iterative. The kept and reduced degrees of freedom in the approach are related through a condensation matrix that is used to form a condensed eigenvalue problem. An initial condensation matrix can be defined in terms of the system submatrices. The eigenproperties calculated in an iterative step are utilized to update the condensation matrix. Matrix inversions or Gaussian eliminations are avoided in the updating process by employing the orthogonality of the eigenvectors. The process of updating the condensation matrix and the eigenvalue problem is repeated until a desired convergence in the eigenvalues is achieved; usually a few iterations are quite adequate. Iterations for both the lower as well as the higher modes can be performed as the condensation matrices for the two sets of modes are very simply related. Numerical examples are presented to show the applicability of the proposed approach. Several methods that have been used in practice to select the kept degrees of freedom for condensing are also evaluated numerically with respect to their effectiveness in providing accurate estimates of the eigenproperties with a minimum number of iterations.

Abstract: Inverse analysis utilizing the conjugate gradient method is used to estimate the timewise varying strength of a line heat source placed at a specified location in a rectangular region with insulated boundaries. No prior information was used on the functional form of the source strength with time. Transient temperature recordings taken at the boundaries of the region served as the simulated experimental data needed as input for the inverse analysis. In order to test the accuracy of the inverse analysis under most strict conditions, timewise variation of the source strength is chosen in the form of rectangular, triangular, and sinusoidal functions. Both the regular iterated and modified conjugate gradient method are used for solving the inverse problem. The modified conjugate gradient method often provided the answer with smaller number of iterations when the initial condition for the unknown function was available.

Abstract: The present paper is concerned with the application of iterative techniques to solve boundary element method (BEM) systems of equations. Initially, a brief description of some algorithms which have been employed for symmetric positive-definite matrices is given (finite element method matrices, for instance). Subsequently, non-symmetric versions of these algorithms are described and their performance is checked for BEM matrices, through various numerical experiments. Preconditioned algorithms were found to work quite well.