Ill-conditioned Equations Research Articles

Designing shielded radio-frequency phased-array coils for magnetic resonance imaging

In this paper, an approach to the design of shielded radio-frequency (RF) phased-array coils for magnetic resonance imaging (MRI) is proposed. The target field method is used to find current densities distributed on primary and shield coils. The stream function technique is used to discretize current densities and to obtain the winding patterns of the coils. The corresponding highly ill-conditioned integral equation is solved by the Tikhonov regularization with a penalty function related to the minimum curvature. To balance the simplicity and smoothness with the homogeneity of the magnetic field of the coil's winding pattern, the selection of a penalty factor is discussed in detail.

Application of Tikhonov regularisation method in bounded chord subjected to enforcement

Inverse problem of identifying initial condition about forced vibration equation of chord is investigated. First, we separate the forward problem into two determinate problems by the superposition principle, and obtain the solution of direct problem by separate variable method and latent function method. Second, discrete it by trapezoidal formula method, and then obtain a severely ill-conditioned linear equation, which is sensitive to the disturbance of the right data. Thirdly, identify the initial condition by Tikhonov regularisation method. The numerical results confirm that this method is feasible and effective.

Solution and simulation algorithm of microseismic events location to three-dimensional model by comprehensive location method based on Matlab

The essential for microseismic monitoring is fast and accurate calculation of seismic wave source location. The precision of most traditional microseismic monitoring processes of mines, using TDOA location method in two-dimensional space to position the microseismic events, as well as the accuracy of positioning microseismic events, may be reduced by the two-dimensional model and simple method, and ill-conditioned equations produced by TDOA location method will increase the positioning error. This article, based on inversion theory, studies the mathematical model of TDOA location method, polarization analysis location method, and comprehensive difference location method of adding angle factor in the traditional TDOA location method. The feasibility of three methods is verified by numerical simulation and analysis of the positioning error of them. The results show that the comprehensive location method of adding angle difference has strong positioning stability and high positioning accuracy, and it may reduce the impact effectively about ill-conditioned equations to positioning results. Comprehensive location method with the data of actual measure may get better positioning results.

A Novel Dynamical Method for Solving Ill-Conditioned Nonlinear Ordinary Differential Equations

A Novel Dynamical Method for Solving Ill-Conditioned Nonlinear Ordinary Differential Equations

A new developed method for the vibration analysis of a beam with variable cross section

In this paper, one method which combined the Improved Fourier series method and Differential Quadrature method (DQM) is firstly introduced to analyze the vibration of a beam with variable cross section. For DQM, the weighting coefficients is traditionally obtained by Lagrange interpolation, which would lead the equation system to be ill-conditioned when points are equidistantly chosen. In this paper, Fourier series with several auxiliary functions is applied to obtain the weighting coefficients which would converge at a fast speed and the more points included, the more accurate would be the results. Application of Fourier series to the DQM would avoid the appearance of the ill-conditioned equation system when the points are chosen equidistantly.

Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects

The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM.

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.

APPLICATION OF THE DISCRETE REGULARIZATION METHOD TO THE INVERSE OF THE CHORD VIBRATION EQUATION

The inverse problem of the initial condition about the boundary value of the chord vibration equation is ill-posed. First, we transform it into a Fredholm integral equation. Second, we discretize it by the trapezoidal formula method, and then obtain a severely ill-conditioned linear equation, which is sensitive to the disturbance of the data. In addition, the tiny error of right data causes the huge concussion of the solution. We cannot obtain good results by the traditional method. In this paper, we solve this problem by the Tikhonov regularization method, and the numerical simulations demonstrate that this method is feasible and effective.

Exact expression for the effective acoustics of patchy-saturated rocks

Seismic effects of a partially gas-saturated subsurface have been known for many years. For example, patches of nonuniform saturation occur at the gas-oil and gas-water contacts in hydrocarbon reservoirs. Open-pore boundary conditions are applied to the quasi-static Biot equations of poroelasticity to derive an exact analytic expression of the effective bulk modulus for partially saturated media with spherical gas patches larger than the typical pore size. The pore fluid and the rock properties can have different values in the central sphere and in the surrounding region. An analytic solution prevents loss of accuracy from ill-conditioned equations as encountered in the numerical solution for certain input. For a sandstone saturated with gas and water, we found that the P-wave velocity and attenuation in conventional models differ as much as 15% from the exact solution at seismic frequencies. This makes the use of present exact theory necessary to describe patchy saturation, although (more realistic) complex patch shapes and distributions were not considered. We found that, despite earlier corrections, the White conventional model does not yield the correct low-frequency asymptote for the attenuation.

AN IMPROVED LOCAL BOUNDARY INTEGRAL EQUATION METHOD FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS

Combining the local boundary integral equation with the improved moving least-squares (IMLS) approximation, an improved local boundary integral equation (ILBIE) method for two-dimensional potential problems is presented in this paper. In the IMLS approximation, the weighted orthogonal functions are used as basis functions. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation and does not lead to an ill-conditioned equations system. The corresponding formulae of the ILBIE method are obtained. Comparing with the conventional local boundary integral equation (LBIE) method, the ILBIE method is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be implemented directly and easily as in the finite element method. The ILBIE method has greater computational efficiency and precision. Some numerical examples to demonstrate the efficiency of the method are presented in this paper.

An Improved Super-Resolution Source Reconstruction Method

The source reconstruction method (SRM) is a technique developed for antenna diagnostics and for carrying out near-field (NF) to far-field (FF) transformation. The SRM is based on the application of the electromagnetic equivalence principle, in which one establishes an equivalent current distribution that radiates the same fields as the actual currents induced in the antenna under test (AUT). The knowledge of the equivalent currents allows the determination of the antenna radiating elements, as well as the prediction of the AUT-radiated fields outside the equivalent currents domain. The unique feature of the novel methodology presented in this paper is that it can resolve equivalent currents that are smaller than half a wavelength in size, thus providing super-resolution. Furthermore, the measurement field samples can be taken at field spacings greater than half a wavelength, thus going beyond the classical sampling criteria. These two distinctive features are possible due to the choice of a model-based parameter estimation methodology where the unknowns are approximated by a continuous basis and, secondly, through the use of the analytic Green's function. The latter condition also guarantees the invertibility of the electric field operator and provides a stable solution for the currents even when evanescent waves are present in the measurements. In addition, the use of the singular value decomposition in the solution of the matrix equations provides the user with a quantitative tool to assess the quality and the quantity of the measured data. Alternatively, the use of the iterative conjugate gradient (CG) method in solving the ill-conditioned matrix equations can also be implemented. Two examples of an antenna diagnostics method are presented to illustrate the applicability and accuracy of the proposed methodology.

NOVEL METHODS FOR SOLVING SEVERELY ILL-POSED LINEAR EQUATIONS SYSTEM

We treat an ill-posed system of linear equations by trans forming it into a linear system of stiff ordinary differential equations (SODEs), adding a differential term on the left-hand side. In order to overcome the difficulty of numerical instability when integrating the SODEs, Liu [20] has combined nonstandard finite difference method and group-preserving scheme, namely the nonstandard group-preserving scheme (NGPS), to obtain an unconditional stable numerical method for SODEs This paper applies the NGPS to the SODEs resulting from the ill-posed linear equations, and proves that the new algorithms are unconditional stable. To strengthen accuracy, an L-curve is used to select a suitable regularization parameter. Moreover, we also combine the NGPS with a newly developed fictitious time integration method (FTIM) from Liu and Atluri [29] to solve the ill-posed linear equations. Several numerical examples are examined and compared with exact solutions, revealing that the new algorithms have better computational efficiency and accuracy even for the highly ill-conditioned linear equations with a large disturbance on the given data.

Factorization-based sparse solvers and preconditioners

Efficient solution of large-scale, ill-conditioned and highly-indefinite algebraic equations often relies on high quality preconditioners together with iterative solvers. Because of their robustness, the factorization-based algorithms could play a significant role when they are combined with iterative methods, particularly in the development of scalable solvers. We present our recent work in using the direct solver SuperLU code base to develop a new supernode-based ILU preconditioner and a domain-decomposition hybrid solver. Our ILU preconditioner is a modification of the classic ILUTP approach, incorporating a number of techniques to improve robustness and performance, which include new dropping strategies that accommodate the use of supernodal structure in the factored matrix. Our hybrid solver is based on the Schur complement method. We use parallel graph partitioning to obtain hierarchical interface/domain decomposition, and multiple parallel direct solvers to solve the subdomain problems simultaneously, and parallel preconditioned iterative solvers to solve the interface problem. We will demonstrate the effectiveness of our new techniques by applying them to two SciDAC applications, modeling next-generation particle accelerators and fusion devices.

Non-linear dynamic behaviour of compound planetary gear trains: Model formulation and semi-analytical solution

A discrete, non-linear, time-varying, torsional dynamic model of a multi-stage planetary train that is formed by any number of simple planetary stages is proposed in this study. Each planetary stage has a distinct fundamental mesh frequency and any number of planets spaced in any angular positions. The model allows the analysis of the gear train in all possible power flow configurations suitable for various gear drive ratios. It includes periodic variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities that allow tooth separations. Equations of motion for the general case are formulated and solved semi-analytically using a hybrid harmonic balance method (HBM) in conjugate with inverse Fourier transform. Relative mesh displacements along lines of action of individual gear pairs were used as the continuation parameters to pass singular points and ill-conditioned equations in their proximity. At the end, a case study of a two-stage planetary train is used to demonstrate the effectiveness of the model and solution methods. The HBM solutions are compared to those obtained by a direct numerical integration method to assess their accuracy.

Damage detection using a new regularization method with variable parameter

In this research, a sensitivity approach to finite element model updating is used to determine stiffness reduction factors from measured structural response. The used method causes a set of nonlinear ill-conditioned equations that need to be linearized and regularized in order to find the solution. A new approach to solve the problem is presented using variable regularization parameter. Utilization of variable regularization parameter eliminates dependency on the number of iterations and prevents the loss of regularization effect due to iterations. A new stopping criteria is used which is based on the difference between mean and variance of last iterations. Furthermore the results show that using wavelet transform to update the model yields better results than modal parameters. Expedient performance of the proposed method is shown through a numerical simulation.

Improved robust and accurate camera calibration method used for machine vision application

Camera calibration is an important step for vision-based measurement applications. A well-known flexible camera calibration method is analyzed that uses the checkerboard pattern plane and in which the camera can be moved freely. When using a perspective projection camera model, characteristics of both the objective plane and the image plane are utilized and accurate results can be obtained. However, the method's results may fail when the rotation angles of the planar pattern are small, and the distortion coefficients obtained under the perspective projection model can not be used for a real-time vision application. We solve the ill-conditioned equations using the genetic algorithm, and the correct camera parameters are always obtained. We compute the distortion coefficients of the inverse projection model, which can be used for general vision applications. The influence of the corner detection precision is taken into consideration. Simulation shows that the best results may be obtained when the planar pattern is placed in a close range and its rotation angle is small. Simulations and real-world experiments illustrate that the improved calibration algorithm can always obtain robust and accurate results.

Inverse design involving combined radiative and turbulent convective heat transfer

This work considers an inverse boundary design problem which involves radiation and convection heat transfer. The objective is finding the heat flux distribution required on heaters located on the top and side walls of a two-dimensional enclosure that satisfies both the temperature and heat flux distributions prescribed on the design surface of the enclosure. A turbulent air flow is generated by a fan located inside the chamber. The problem is described by a system of non-linear, ill-conditioned equations, which is solved by an iterative procedure. The solution are obtained by regularizing the system of equations by means of the TSVD method.

Improved Meshless Local Boundary Integral Equation Method

Combining the local boundary integral equation with the improved moving least-square method,an improved meshless local boundary integral equation method is presented.In the improved moving least-square method,the weighted orthogonal functions are used as basis ones so that the matrix inverse at each quadrature point is avoided and the algebra equations system is not ill-conditioned.In addition,the improved meshless local boundary integral equation method is applied to linear elasticity problems, the corresponding discrete equations are derived.Some numerical results to demons trate the efficiency of the method are presented. Compared with the conventional local boundary integral equation method,the present method has higher computational efficiency and precision,and will not form ill-conditioned or singular equations.

An inverse method for radio frequency phased array coil design in magnetic resonance imaging equipment

Magnetic resonance imaging is a popular non-invasive technique for investigating soft tissue structures within the human body. Many design methods now exist for their principle hardware components, such as the radio frequency (RF) coils. A popular RF coil type is the RF phased array, comprised of many closely spaced coils covering a large volume. A time harmonic inverse method is presented for the theoretical design of RF phased arrays. The method allows any array size to be considered where the focus is on optimal coil geometry and position of individual coils. An ill-conditioned integral equation is solved using a regularisation strategy in which the error between induced and target magnetic fields is minimised along with an additional constraint related to the curvature of the coil windings. The method is demonstrated for a number of design considerations and includes the ability to focus the RF field to arbitrary locations within the coil volume. The effect of the choice of magnetic field polarisation direction is also investigated using the model. References Jin J. 1999. Electromagnetic Analysis and Design in Magnetic Resonance Imaging, Florida: CRC Press. Lawrence B. G., Crozier S., Yau D. D. and Doddrell D. M. 2002. A time-harmonic inverse methodology for the design of rf coils in mri, IEEE Transactions on Biomedical Engineering 49(1), 64--71. doi:10.1109/10.972841 Li B. K., Liu F. and Crozier S. 2005. Focused, eight-element transceive phased array coil for parallel magnetic resonance imaging of the chest --- theoretical considerations, Magnetic Resonance in Medicine 53, 1251--1257. doi:10.1002/mrm.20505 Ramo S., Whinnery J. R. and van Duzer T. 1967. Fields and waves in communication electronics, New York: John Wiley and Sons, Inc. Roemer P. B., Edelstein W. A., Hayes C. E., Souza S. P. and Mueller O. M. 1990. The nmr phased array, Magnetic Resonance in Medicine 16(2), 192--225. doi:10.1002/mrm.1910160203 Sodickson D. K., McKenzie C. A., Ohliger M. A., Yeh E. N. and Price M. D. 2002. Recent advances in image reconstruction, coil sensitivity calibration, and coil array design for SMASH and generalized parallel mri, Magnetic Resonance Materials in Physics, Biology and Medicine 13, 158--163. Turner R. 1986. A target field approach to optimal coil design, Journal of Physics D: Applied Physics 19, L147--L151. doi:10.1088/0022-3727/19/8/001 While P. T., Forbes L. K. and Crozier S. 2005a. A time-harmonic target-field method for designing unshielded rf coils in mri, Measurement Science and Technology 16, 997--1006. doi:10.1088/0957-0233/16/4/012 While P. T., Forbes L. K. and Crozier S. 2005b. A time-harmonic target-field method for designing shielded rf coils in mri, Measurement Science and Technology 16, 1381--1393. doi:10.1088/0957-0233/16/6/021 While P. T., Forbes L. K. and Crozier S. 2006. An inverse method for designing loaded rf coils in mri, Measurement Science and Technology 17, 2506--2518. doi:10.1088/0957-0233/17/9/019 Brideson M. A., Forbes L. K. and Crozier S. 2002. Determining complicated winding patterns for shim coils using streamfunctions and the target-field method, Concepts in Magnetic Resonance 14(1), 9--18. doi:10.1002/cmr.10000 While P. T., Forbes L. K. and Crozier S. 2007. An inverse method for designing rf phased array coils in mri --- theoretical considerations, Measurement Science and Technology 18, 245--259. doi:10.1088/0957-0233/18/1/031 Delves L. M. and Mohamed J. L. 1985. Computational methods for integral equations, Cambridge: Cambridge University Press. Forbes L. K. and Crozier S. 2002. A novel target-field method for finite-length magnetic resonance shim coils: II. tesseral shims, Journal of Physics D: Applied Physics 35, 839--849. doi:10.1088/0022-3727/35/9/303 Hardy C. J., Cline H. E., Giaquinto R. O., Niendorf T., Grant A. K. and Sodickson D. K. 2006. 32-element receiver-coil array for cardiac imaging, Magnetic Resonance in Medicine 55, 1142--1149. doi:10.1002/mrm.20870

Experimental and Numerical Investigation of Structural Damage Detection Using Changes in Natural Frequencies

A robust iterative algorithm is used to identify the locations and extent of damage in beams using only the changes in their first several natural frequencies. The algorithm, which combines a first-order, multiple-parameter perturbation method and the generalized inverse method, is tested extensively through experimental and numerical means on cantilever beams with different damage scenarios. If the damage is located at a position within 0–35% or 50–95% of the length of the beam from the cantilevered end, while the resulting system equations are severely underdetermined, the minimum norm solution from the generalized inverse method can lead to a solution that closely represents the desired solution at the end of iterations when the stiffness parameters of the undamaged structure are used as the initial stiffness parameters. If the damage is located at a position within 35–50% of the length of the beam from the cantilevered end, the resulting solution by using the stiffness parameters of the undamaged structure as the initial stiffness parameters deviates significantly from the desired solution. In this case, a new method is developed to enrich the measurement information by modifying the structure in a controlled manner and using the first several measured natural frequencies of the modified structure. A new method using singular value decomposition is also developed to handle the ill-conditioned system equations that occur in the experimental investigation by using the measured natural frequencies of the modified structure.