Direct Matrix Inversion Method Research Articles

Near trapped wave phenomena in an array of truncated cylinders in a perpendicular arrangement in front of a vertical breakwater

The present paper deals with the investigation of the interaction phenomena of regular waves with arrays of circular cylinders either bottom-mounted or floating in front of a vertical breakwater of infinite and finite length. The acting loads on the array have been modeled using an analytical solution to the diffraction problem. The method of images is applied to define the fluid flow around the bodies of the array in front of a breakwater of infinite length, whereas the finite-length breakwater is represented by an elliptical cylinder with zero semi–minor axis. The multiple scattering approach and the direct matrix inversion method are compared to describe the hydrodynamic interaction phenomena among the members of the array. Focus is placed on the so called “near trapped” mode phenomena, created by an array of finite length, associated to the presence of the breakwater supplemented by representative numerical results concerning the exciting loads on the cylindrical bodies.

Image domain dual material decomposition for dual-energy CT using butterfly network.

Dual-energy CT (DECT) has been increasingly used in imaging applications because of its capability for material differentiation. However, material decomposition suffers from magnified noise from two CT images of independent scans, leading to severe degradation of image quality. Existing algorithms exhibit suboptimal decomposition performance because they fail to fully depict the mapping relationship between DECT images and basis materials under noisy conditions. Convolutional neural network exhibits great promise in the modeling of data coupling and has recently become an important technique in medical imaging application. Inspired by its impressive potential, we developed a new Butterfly network to perform the image domain dual material decomposition. The Butterfly network is derived from the model of image domain DECT decomposition by exploring the geometric relationship between the mapping functions of the data model and network components. The network is designed as the double-entry double-out crossover architecture based on the decomposition formulation. It enters a pair of dual-energy images as inputs and defines the ground true decomposed images as each label. The crossover architecture, which plays an important role in material decomposition, is designed to implement the information exchange between the two material generation pathways in the network. The proposed network is further applied on the digital phantom and clinical data to evaluate its performance. The qualitative and quantitative evaluations of the material decomposition of digital phantoms and clinical data indicate that the proposed network outperforms its counterparts. For the digital phantom, the proposed network reduces the standard deviation (SD) of noise in tissue, bone, and mixture regions by an average of 95.75% and 86.58% compared with the direct matrix inversion and the conventional iterative method, respectively. The line profiles and image biases of the decomposition results of digital phantom indicate that the proposed network provides the decomposition results closest to the ground truth. The proposed network reduces the SD of the noise in decomposed images of clinical head data by over 90% and 80% compared with the direct matrix inversion and conventional iterative method, respectively. As the modulation transfer function decreases to 50%, the proposed network increases the spatial resolution by average factors of 1.34 and 1.17 compared with the direct matrix inversion and conventional iterative methods, respectively. The proposed network is further applied to the clinical abdomen data. Among the three methods, the proposed method received the highest score from six radiologists in the visual inspection of noise suppression in the clinical data. We develop a model-based Butterfly network to perform image domain material decomposition for DECT. The decomposition results of digital phantom validate its capability ofdecomposing two basis materials from DECT images. The proposed approach also leads to higherdecomposition quality in noise suppression on clinical datasets as compared with those using conventional schemes.

A tool for designing digital filters for the Hankel and Fourier transforms in potential, diffusive, and wavefield modeling

The open-source code fdesign makes it possible to design digital linear filters for the Hankel and Fourier transforms used in potential, diffusive, and wavefield modeling. Digital filters can be derived for any electromagnetic (EM) method, such as methods in the diffusive limits (direct current, controlled-source EM [CSEM]) as well as methods using higher frequency content (ground-penetrating radar [GPR], acoustic and elastic wavefields). The direct matrix inversion method is used for the derivation of the filter values, and a brute-force minimization search is carried out over the defined spacing and shifting values of the filter basis. Included or user-provided theoretical transform pairs are used for the inversion. Alternatively, one can provide layered subsurface models that will be computed with a precise quadrature method using the EM modeler empymod to generate numerical transform pairs. The comparison of the presented 201 pt filter with previously presented filters indicates that it performs better for some standard CSEM cases. The derivation of a longer 2001 pt filter for a GPR example with a 250 MHz center frequency proves that the filter method works not only for diffusive EM fields but also for wave phenomena. The presented algorithm provides a tool to create problem specific digital filters. Such purpose-built filters can be made shorter and can speed up consecutive potential, diffusive, and wavefield inversions.

Fitting discrete aspherical surface sag data using orthonormal polynomials.

Characterizing real-life optical surfaces usually involves finding the best-fit of an appropriate surface model to a set of discrete measurement data. This process can be greatly simplified by choosing orthonormal polynomials for the surface description. In case of rotationally symmetric aspherical surfaces, new sets of orthogonal polynomials were introduced by Forbes to replace the numerical unstable standard description. From these, for the application of surface retrieval using experimental ray tracing, the sag orthogonal Q(con)-polynomials are of particular interest. However, these are by definition orthogonal over continuous data and may not be orthogonal for discrete data. In this case, the simplified solution is not valid. Hence, a Gram-Schmidt orthonormalization of these polynomials over the discrete data set is proposed to solve this problem. The resulting difference will be presented by a performance analysis and comparison to the direct matrix inversion method.

Random Response Analysis of Curvilinearly Stiffened Plates Using Element-Free Galerkin Method

An Element Free Galerkin (EFG) formulation is presented for studying the random response of curvilinearly-stiffened plates. The structure is subjected to stationary random stochastic loading. The random loads are assumed as stationary in time but can be nonhomogeneous in space. The spectral density of the nodal force vector is formulated using the displacement shape functions. The direct complex matrix inversion method and modal superposition method are used to obtain the random response of structure. The spectral density of displacement for un-stiffened plate under white noise and plate with straight stiffeners subjected to jet noise are compared with those available in the literature. The power spectral density of displacement of curvilinearly-stiffened plate under white noise is also verified using the commercial finite element software ANSYS. Excellent agreement is seen in all cases. The effect of stiffener stiffness and curvature and in-plane load on spectral density of deflection of curvilinearly-stiffened plate is also investigated.

SU-GG-T-428: Photon Beam Spectrum Characterization Using Scatter Radiation Analysis

Purpose: A method is described to characterize a linac bremsstrahlung spectrum. The method employs a combination of monoenergetic scatter simulations, polyenergetic scatter measurements, and matrix solutions. Method and Materials:Monte Carlo simulations were used to predict the ratio of primary to scatteredphotons for narrow mono‐energetic photon beams at 9 different locations, with 10 degree increments and 15 cm from a scattering material. Measurements were performed in the same geometry using the photon beams produced by linear accelerators. A linear matrix system, A×F=T, was developed to describe the scattering interactions and their relationship to the primary spectrum. Matrix “A” is the monoenergetic scatter kernel determined from the Monte Carlo simulations, “F” is the incident photon spectrum, and “T” represents the scatter distribution characterized by empirical measurement. Direct matrix inversion methods produce results that are not physically consistent due to errors inherent in the system. Tikhonov regularization methods were applied to address the effects of these errors and solve the system for physically consistent bremsstrahlung spectra. The technique was applied to two photon energy spectra ‐a helical tomotherapy accelerator operating at therapeutic energy and at imaging energy. Results: The resulting spectra were qualitatively and quantitatively similar to other published data and consistent with expected behavior such as maximum photon energy. Conclusion: The results of this research provide a method to empirically characterize the primary radiation energy spectrum produced by a linear accelerator.Key words: photon energy, spectrum, scatteranalysis, regularization, Tikhonov

Simulating deformation of objects with multi‐materials using boundary element method

AbstractIn this paper, an algorithm for simulating the elastostatic deformation of multiple objects with different material properties using boundary element method is introduced. By tessellating the surface of a geometric model into elements and classifying all the element nodes into different groups with different attributes, and partitioning the stiffness matrix into several sub‐matrices according to these attributes, a compact expression for the unknown variables is obtained. Comparing with the direct matrix inversion method, the dimension of the system matrix in the expression has been effectively reduced. Besides, this expression shows that the deformation of a multi‐component object can be simulated in a way similar to that of a single‐component object. The capacitance method is adopted for evaluating the deformation of the object. Experimental results illustrate that the proposed method is practical and efficient. Copyright © 2007 John Wiley & Sons, Ltd.

Hankel transform filters for dipole antenna radiation in a conductive medium

ABSTRACTWe discuss the Hankel transforms related to a particular application, i.e. the dipole antenna radiation in conductive media, such as the antenna radiation in sea‐bed electromagnetic applications. In this application, the electromagnetic wavefields decay very rapidly with distance. A good filter means that it can be used to evaluate weak fields.Exponential sampling transforms a Hankel transform into a convolution equation, which must be solved to obtain the filter coefficients. Here, we use a direct matrix inversion method to solve the convolution equation in the sample domain, instead of the Fourier transform method and the Wiener–Hopf method, previously used to solve the convolution equation. This direct method is conceptually simple and is suitable for our optimization process: by using the Sommerfeld identity, we search for the optimum sampling interval, which corresponds to the minimum wavefield, evaluated for a given length filter.The performances of the new filters obtained are compared with some well‐known filters. We find that our filters perform better for our application; that is, for the same length filters, our filters are able to calculate weaker fields.For users working in similar applications, three sets of filters with lengths 61, 121, 241 are available from the author.

Decomposed direct matrix inversion for fast non‐cartesian SENSE reconstructions

A new k-space direct matrix inversion (DMI) method is proposed here to accelerate non-Cartesian SENSE reconstructions. In this method a global k-space matrix equation is established on basic MRI principles, and the inverse of the global encoding matrix is found from a set of local matrix equations by taking advantage of the small extension of k-space coil maps. The DMI algorithm's efficiency is achieved by reloading the precalculated global inverse when the coil maps and trajectories remain unchanged, such as in dynamic studies. Phantom and human subject experiments were performed on a 1.5T scanner with a standard four-channel phased-array cardiac coil. Interleaved spiral trajectories were used to collect fully sampled and undersampled 3D raw data. The equivalence of the global k-space matrix equation to its image-space version, was verified via conjugate gradient (CG) iterative algorithms on a 2x undersampled phantom and numerical-model data sets. When applied to the 2x undersampled phantom and human-subject raw data, the decomposed DMI method produced images with small errors (< or = 3.9%) relative to the reference images obtained from the fully-sampled data, at a rate of 2 s per slice (excluding 4 min for precalculating the global inverse at an image size of 256 x 256). The DMI method may be useful for noise evaluations in parallel coil designs, dynamic MRI, and 3D sodium MRI with fixed coils and trajectories.

Solution to the Bethe-Faddeev equation within the continuous version of the hole-line expansion

The contribution of the three-body correlations to the binding energy of symmetric nuclear matter is calculated, within the continuous version of the hole-line expansion, by means of the direct matrix inversion method. For the nucleon-nucleon interactions considered in this paper, we find that at the three-hole-line level, the saturation curve is fairly close to the saturation curve obtained within the standard version. The role of the various contributions to the binding energy is discussed in detail.

On the inversion of geodetic integrals defined over the sphere using 1-D FFT

An iterative method is presented which performs inversion of integrals defined over the sphere. The method is based on one-dimensional fast Fourier transform (1-D FFT) inversion and is implemented with the projected Landweber technique, which is used to solve constrained least-squares problems reducing the associated 1-D cyclic-convolution error. The results obtained are as precise as the direct matrix inversion approach, but with better computational efficiency. A case study uses the inversion of Hotine’s integral to obtain gravity disturbances from geoid undulations. Numerical convergence is also analyzed and comparisons with respect to the direct matrix inversion method using conjugate gradient (CG) iteration are presented. Like the CG method, the number of iterations needed to get the optimum (i.e., small) error decreases as the measurement noise increases. Nevertheless, for discrete data given over a whole parallel band, the method can be applied directly without implementing the projected Landweber method, since no cyclic convolution error exists.

The ground and excitation levels of the trimer

The ground and excited levels of the trimer (4He)3 are obtained theoretically by solving the 2-D Faddeev integral equation in the momentum space. In solving the equation, the direct matrix inversion method (I. M.) is used instead of the iteration and Pade approximate method. The new HFD-B Aziz potential ν is used in the calculation.

Deprojection of Planetary Nebula Images

We present first results of the deprojection of some optical and radio images of planetary nebulae using an iterative technique based on the Lucy method, as discussed in Leahy (1991). This approach allows a greater range of possible geometries to be investigated and is less sensitive to the effects of noise compared with the direct matrix-inversion method which can be used for the spherical case. We assume that the emissivity function can be represented as

Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

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 iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

Performance of DMI and eigenspace-based beamformers

The performance of the direct matrix inversion (DMI) method for antenna arrays of arbitrary geometry is analyzed by asymptotic statistical techniques. The effects of eigenspace disturbance caused by finite samples on the output interference and noise powers are examined under the unit gain constraint in the direction of the desired signal. The results show that the performance of the DMI method is degraded mostly by the disturbed noise subspace. That suggests the use of an eigenspace-based beamformer in which the weight vector is computed by using the signal-plus-interference subspace component of the sample correlation matrix. Convergence properties of the eigenspace-based beamformer are evaluated for the cases in which the source number is known and in which it is overestimated. Theoretical analyses validated by computer simulations indicate that the eigenspace-based beamformer has faster convergence rate than the DMI method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

New algorithm for solving block matrix equations with applications in 2-D AR spectral estimation

The author presents an iterative algorithm to solve Toeplitz and non-Toeplitz block matrix equations. The development is based upon some well-known matrix iterative techniques. The algorithm is developed for the ideal case where the individual blocks in the autocorrelation matrix are Toeplitz, and it is then extended to a more general least squares data formulation case. It is shown that the algorithm requires fewer computations than the direct matrix inversion methods and is very simple to implement. The algorithm is applied to compute the spectral estimates of 2-D data of very small size based on the least squares data formulation with a quarter plane support.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the analysis of frequency-selective surfaces using subdomain basis functions

The problem of scattering from frequency-selective surfaces (FSSs) has been investigated by expanding the unknown current distribution with three different sets of basis functions, namely the roof top, surface patch, and triangular patch. The boundary condition on the total electric field on the FSS due to this current distribution is tested either by a line integral or by the Galerkin procedure. This results in an operator equation that can be solved either by a direct matrix inversion method or by an iterative procedure, namely the conjugate gradient method (CGM). The performance of each of these basis and testing functions is evaluated. It is found that the roof-top and the surface-patch basis functions in conjunction with the Galerkin testing are superior in computational efficiency to other combinations of basis and testing functions that have been studied. Comparison of the CPU times on a Cray X-MP/48 supercomputer in solving the operator equation by the direct matrix inversion method and the CGM is provided. Frequency responses of free-standing, periodic arrays of conducting and resistive plates are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comparison of FFT, direct inversion, and conjugate gradient methods for use in pseudo-spectral methods

Direct matrix inversion methods and the conjugate gradient method are compared with the FFT for speed when used in pseudo-spectral calculations. It is demonstrated that in some cases either of these methods is competitive with an FFT. This is especially true when using direct methods in two dimensions.

An Efficient Fully Implicit Simulator

Abstract This paper describes Program for Oil Reservoir Simulation (pores), an efficient general-purpose black-oil simulator now in production use for modeling North Sea fields. The fully implicit finite-difference equations are solved for each timestep with a Newton-Raphson procedure. The resulting large sets of linear equations are usually solved simultaneously by a new and powerful iterative method that uses a preconditioned conjugate gradient algorithm with an enforced column-sum condition to accelerate convergence. A sequential solution option is available, and direct matrix inversion methods also are provided. Gas condensate problems are handled by a variable switching technique. Four examples are presented to illustrate the power and efficiency of the program.

Incomplete Block Designs for Three or Four Replicates of Many Treatments

Some practical situations are described where many treatments have to be compared in trials with only a small amount of replication. Appropriate designs are given for three or four replicates with most of the attention being given to designs having from 25 to 100 treatments, some of which may be duplicated. Some of the designs are partially balanced incomplete block designs with two associate classes (PBIBD(2)), most of which are already known, while others are duals of PBIBD (2) or balanced incomplete block designs. WVhen a trial has been conducted, not all treatments may be worth analyzing, and the results are best analyzed by direct matrix inversion methods.