A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions

Continuously differentiable radial basis functions (C∞-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C∞-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations

A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.

Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics

In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often ill-conditioned.When the conjugate gradient method is used,their ill-conditioning effects to the solutions must be overcome,which is studied in this paper.Through the regularization ideas,the conjugate gradient method is improved,and the regularization iterative solution based on controlling condition number is put forward.Firstly by constructing the interference source vector,a new equation is derived with ill-condition diminished greatly,which has the same solution to the original normal equation.Then the new equation is solved by conjugate gradient method.Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface.In the numerical experiments the new method is compared with LS,CG and Tikhonov methods,and its accuracy is the highest.

Modeling dam deformation using independent component regression method

In the application of regression analysis method to model dam deformation, the ill-condition problem occurred in coefficient matrix always prevents an accurate modeling mainly due to the multicollinearity of the variables. Independent component regression （ICR） was proposed to model the dam deformation and identify the physical origins of the deformation. Simulation experiment shows that ICR can successfully resolve the problem of ill-condition and produce a reliable deformation model. After that, the method is applied to model the deformation of the Wuqiangxi Dam in Hunan province, China. The result shows that ICR can not only accurately model the deformation of the dam, but also help to identify the physical factors that affect the deformation through the extracted independent components.

Application of the Monte Carlo Method in Simulative Generation Test Problems Simulative Generation Test Problems

An approach for generating test problems by a computer using the Monte Carlo method based upon user-given characterizations is described.A single point X~* is prespocified by the user to be a solution of the test problems.The approach is flex- ible enough to specify function values,gradients,Hesse,degeneracy degree and ill- conditioned degree at the point X~*.Other numerical features such as indefiniteness, convexity are also under user's control.

Solving Load Flow Problems of Power System by Explicit Pseudo-Transient Continuation (E-ψtc) Method

This paper is a further study of two papers [1] and [2], which were related to Ill-Conditioned Load Flow Problems and were published by IEEE Trans. PAS. The authors of this paper have some different opinions, for example, the 11-bus system is not an ill-conditioned system. In addition, a new approach to solve Load Flow Problems, E-ψtc, is introduced. It is an explicit method;solving linear equations is not needed. It can handle very tough and very large systems. The advantage of this method has been fully proved by two examples. The authors give this new method a detailed description of how to use it to solve Load Flow Problems and successfully apply it to the 43-bus and the 11-bus systems. The authors also propose a strategy to test the reliability, and by solving gradient equations, this new method can answer if the solution exists or not.

An Improved Velocity Volume Processing Method

Velocity volume processing （VVP） retrieval of single Doppler radar is an effective method which can be used to obtain many wind parameters. However, due to the problem of an ill-conditioned matrix arising from the coefficients of equations not being easily resolved, the VVP method has not been applied adequately and effectively in operation. In this paper, an improved scheme, SVVP （step velocity volume processing）, based on the original method, is proposed. The improved algorithm retrieves each group of components of the wind field through a stepwise procedure, which overcomes the problem of an ill-conditioned matrix, which currently limits the application of the VVP method. Variables in a six-parameter model can be retrieved even if the analysis volume is very small. In addition, the source and order of errors which exist in the traditional method are analyzed. The improved method is applied to real cases, which show that it is robust and has the capability to obtain the wind field structure of the local convective system. It is very helpful for studying severe storms.

Robust Sensor Bias Estimation for Ill-Conditioned Scenarios

Sensor bias estimation is an inherent problem in multi-sensor data fusion systems. Classical methods such as the Generalized Least Squares （GLS） method can have numerical problems with ill-conditioned sets which are common in practical applications. This paper describes an azimuth-GLS method that provides a solution to the ill-conditioning problem while maintaining reasonable accuracy com- pared with the classical GLS method. The mean square error is given for both methods as a criterion to de- termine when to use this azimuth-GLS method. Furthermore, the separation boundary between the azi- muth-GLS favorable region and that of the GLS method is explicitly plotted. Extensive simulations show that the azimuth-GLS approach is preferable in most scenarios.

Solving single-frequency phase ambiguity using parameter weights fitting and constrained equation ambiguity resolution methods

Based on the structural characteristics of the double-differenced normal equation, a new method was proposed to resolve the ambiguity float solution through a selection of parameter weights to construct an appropriate regularized matrix, and a singular decomposition method was used to generate regularization parameters. Numerical test results suggest that the regularized ambiguity float solution is more stable and reliable than the least-squares float solution. The mean square error matrix of the new method possesses a lower correlation than the variancecovariance matrix of the least-squares estimation. The size of the ambiguity search space is reduced and the search efficiency is improved. The success rate of the integer ambiguity searching process is improved significantly when the ambiguity resolution by using constraint equation method is used to determine the correct ambiguity integervector. The ambiguity resolution by using constraint equation method requires an initial input of the ambiguity float solution candidates which are obtained from the LAMBDA method in the new method. In addition, the observation time required to fix reliable integer ambiguities can he significantly reduced.

Some Remarks on the Method of Fundamental Solutions for Two- Dimensional Elasticity

In this paper,some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made.First,the effects of the distance between pseudo and main boundaries on the solution are investigated and by a numerical study a lower bound for the distance of each source point to the main boundary is suggested.In some cases,the resulting system of equations becomes ill-conditioned for which,the truncated singular value decomposition with a criterion based on the accuracy of the imposition of boundary conditions is used.Moreover,a procedure for normalizing the shear modulus is presented that significantly reduces the condition number of the system of equations.By solving two example problems with stress concentration,the effectiveness of the proposed methods is demonstrated.

Ill-conditioned problems of dam safety monitoring models and their processing methods

The focus of this paper is the ill-conditioned problems in the dam safety monitoring model. The reasons to give rise to the ill-conditioned problems in statistical models,deterministic models and hybrid models are analyzed in detail,and the criterions for ill-conditioned models are investigated. It is shown that safety monitoring models are not easy to be ill-conditioned if the number of influence factors is less than seven. Moreover,the models have a high accuracy and can meet the engineering requirements. Another frequently encountered problem in establishing a safety monitoring model is the existence of inflection points,which are often present in the mathematical model for the hydraulic components in deterministic models and hybrid models. The conditions for inflection points are studied and their treatments are suggested. Numerical example indicates that the treatments proposed in this paper are effective in removing the ill-conditioned problems.

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned.Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic.Surprisingly,we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices.We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations.Our approach combines message passing and shared memory,achieving near-perfect scalability and high tolerance for network latency.We are thus able to find solutions for much larger matrices than previously possible,with the potential for extending this work to systems with greater levels of parallelism.The contributions of this work are in three areas:determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations;the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster,and the numerical results themselves.

A ROBUST INTERIOR POINT METHOD FOR COMPUTING THE ANALYTIC CENTER OF AN ILL-CONDITIONED POLYTOPE WITH ERRORS

In this paper we propose an efficient and robust method for computing the analytic center of the polyhedral set P={x€R^n|Ax=b,x>0},where the matrix A€ Rm×n is ill-conditioned,and there are errors in A and b.Besides overcoming the difficulties caused by ill-cond计ioning of the matrix A and errors in A and b,our method can also detect the infeasibility and the unboundedness of the polyhedral set P automatically during the compu tation.Det ailed mat hematical analyses for our method are presen ted and the worst case complexity of the algorithm is also given.Finally some numerical results are presented to show the robustness and effectiveness of the new method.

Double-Epoch Algorithm in Rapid Positioning Using Single Frequency GPS Receivers

A new algorithm, called as Double-Epoch Algorithm CDEA) is proposed in GPSrapid positioning using two epoch single frequency phase data in this paper. Firstly, the structurecharacteristic of the normal matrix in GPS rapid positioning is analyzed. Then, in the light of thecharacteristic, based on TIK-HONOV regularization theorem, a new regularizer is designed to mitigatethe ill-condition of the normal matrix. The accurate float ambiguity solutions and their MSEM (MeanSquared Error Matrix) are obtained, u-sing two epoch single frequency phase data. Combined withLAMBDA method, DEA can fix the integer ambiguities correctly and quickly using MSEM instead of thecovariance matrix of the ambiguities. Compared with the traditional methods, DEA can improve theefficiency obviously in rapid positioning. So, the new algorithm has an extensive applicationoutlook in deformation monitoring, pseudokinematic relative positioning and attitude determination,etc.

SPLITTING MODULUS FINITE ELEMENT METHOD FOR ORTHOGONAL ANISOTROPIC PLATE BENGING

Splitting modulus variational principle in linear theory of solid mechanics was introduced, the principle for thin plate was derived, and splitting modulus finite element method of thin plate was established too. The distinctive feature of the splitting model is that its functional contains one or more arbitrary additional parameters, called splitting factors, so stiffness of the model can be adjusted by properly selecting the splitting factors. Examples show that splitting modulus method has high precision and the ability to conquer some ill-conditioned problems in usual finite elements. The cause why the new method could transform the ill-conditioned problems into well-conditioned problem, is analyzed finally.

IMPROVED PRECONDITIONED CONJUGATE GRADIENT METHOD AND ITS APPLICATION IN F.E.A.FOR ENGINEERING

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.

Optimal Static State Estimation Using hybrid Particle Swarm-Differential Evolution Based Optimization

In this paper, swarm optimization hybridized with differential evolution (PSO-DE) technique is proposed to solve static state estimation (SE) problem as a minimization problem. The proposed hybrid method is tested on IEEE 5-bus, 14-bus, 30-bus, 57-bus and 118-bus standard test systems along with 11-bus and 13-bus ill-conditioned test systems under different simulated conditions and the results are compared with the same, obtained using standard weighted least square state estimation (WLS-SE) technique and general particle swarm optimization (GPSO) based technique. The performance of the proposed optimization technique for SE, in terms of minimum value of the objective function and standard deviations of minimum values obtained in 100 runs, is found better as compared to the GPSO based technique. The statistical error analysis also shows the superiority of the proposed PSO-DE based technique over the other two techniques.

A Modified Augemented Lagrangian Method for a Class of Nonlinear Ill-Posed Problems

A class of nonlinear problems with real parameters is defined. Generally, in this class of problems, when the parametric values are very large, the problems become ill-posed and numerical difficulties are encountered when trying to solve these problems. In this paper, the nonlinear problems are reformulated to overcome the numerical difficulties associated with large parametric values. A novel iterative algorithm, which is suitable for large scale problems and can be easily parallelized, is proposed to solve the reformulated problems. Numerical tests indicate that the proposed algorithm gives stable solutions. Convergence properties of the proposed algorithm are investigated. In the limiting case, when the corresponding constraint is exactly satisfied, the proposed method is equivalent to the standard augmented Lagrangian method.