Ill-conditioned System Of Linear Equations Research Articles

Multi-scale method for ill-conditioned linear systems

A multi-scale method is presented for solving ill-conditioned linear systems of equations. Numerical examples are provided to show the effectiveness and efficiency of the proposed method compared with traditional iterative methods.

Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions

Due to the usually complicated and anisotropic nature of the fast-ion distribution function, diagnostic velocity-space weight functions, which indicate the sensitivity of a diagnostic to different fast-ion velocities, are used to facilitate the analysis of experimental data. Additionally, when velocity-space weight functions are discretized, a linear equation relating the fast-ion density and the expected diagnostic signal is formed. In a technique known as velocity-space tomography, many measurements can be combined to create an ill-conditioned system of linear equations that can be solved using various computational methods. However, when velocity-space weight functions (which by definition ignore spatial dependencies) are used, velocity-space tomography is restricted, both by the accuracy of its forward model and also by the availability of spatially overlapping diagnostic measurements. In this work, we extend velocity-space weight functions to a full 6D generalized coordinate system and then show how to reduce them to a 3D orbit-space without loss of generality using an action-angle formulation. Furthermore, we show how diagnostic orbit-weight functions can be used to infer the full fast-ion distribution function, i.e., orbit tomography. In depth derivations of orbit weight functions for the neutron, neutral particle analyzer, and fast-ion D-α diagnostics are also shown.

On the iterative refinement of the solution of ill-conditioned linear system of equations

ABSTRACTRecently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations.

On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation

In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom.

The block Lanczos algorithm for linear ill-posed problems

In the present paper, we propose a new method to inexpensively determine a suitable value of the regularization parameter and an associated approximate solution, when solving ill-conditioned linear system of equations with multiple right-hand sides contaminated by errors. The proposed method is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules to inexpensively approximate matrix-valued function of the form \(W^Tf(A)W\), where \(W\in {\mathbb {R}}^{n\times k}\), \(k\ll n\), and \(A\in {\mathbb {R}}^{n\times n}\) is a symmetric matrix.

A frequency-independent and parallel algorithm for computing the zeros of strictly proper rational transfer functions

We develop an algorithm for computation of the zeros of a strictly proper rational transfer function in partial fraction form, by transforming the problem of finding the roots of the determinant of a frequency-dependent matrix into one of finding the eigenvalues of a companion matrix comprised of the determinants of a binomial-based set of frequency-independent matrices. The proposed algorithm offers a fundamentally new approach that avoids solving severely ill-conditioned system of linear equations, where condition numbers increase rapidly with frequency. The developed algorithm is straightforward, and enables parallel computation of the characteristic polynomial coefficients an′ that comprise the companion matrix to the characteristic polynomial ∑nansn of the frequency-dependent matrix. Additionally, the algorithm allows for relatively inexpensive computation of asymptotically accurate approximants of the transfer function, such that an′ need be computed only for selected powers of s=jω, where the number of required determinant operations are shown to be relatively small. Additionally, limitations of the developed algorithm are highlighted, where the computational cost is shown to be on the order O(2Np) determinant operations on matrices of dimensions (Np+1)×(Np+1), and Np is the number of poles. Illustrative numerical examples are selected and discussed to provide further insight about pros/cons of the proposed method, and to identify potential areas for further research.

A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation

Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into account the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant (multiquadric trigonometric B-spline quasi-interpolant) to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation (requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain). Moreover, the scheme also overcomes the difficulty of the meshless collocation methods (i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points). The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.

Numerical solution of three-dimensional Laplacian problems using the multiple scale Trefftz method

This paper proposes the numerical solution of three-dimensional Laplacian problems based on the multiple scale Trefftz method with the incorporation of the dynamical Jacobian-inverse free method. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 18 independent functions satisfying the governing equation in the cylindrical coordinate system. To mitigate a severely ill-conditioned system of linear equations, this study adopted the newly developed multiple scale Trefftz method and the dynamical Jacobian-inverse free method. Numerical solutions were conducted for problems involving three-dimensional groundwater flow problems enclosed by a cuboid-type domain, a peanut-type domain, a sphere domain, and a cylindrical domain. The results revealed that the proposed method can obtain accurate numerical solutions for three-dimensional Laplacian problems, yielding a superior convergence in numerical stability to that of the conventional Trefftz method.

A new operational matrix method based on the Bernstein polynomials for solving the backward inverse heat conduction problems

Purpose – The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP). Design/methodology/approach – By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system. Findings – The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique. Originality/value – A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.

Fundamental solution method for reconstructing past climate change from borehole temperature gradients

Deep borehole temperature profiles have successfully been used to reconstruct past ground surface temperature history and the results are dependent on the inversion methods. These methods are tedious and sometimes unstable in iterative computation. In this paper, we propose a new fundamental solution method to reconstruct the past ground surface temperature variation, which depends on the assumption that ground temperature field in a homogeneous region is governed by a one-dimensional heat conductive equation. To regularize the resultant ill-conditioned linear system of equations, we apply successfully both the Tikhonov regularization technique and the generalized cross validation parameter choice rule to obtain a stable approximation solution of the ill-posed inverse problem. Our new method is stable and meshless, and it does not require iteration. We conducted idealized simulations with good results. We also used in-situ borehole data of RU-Yakutia329 from Yakutia, Siberia and CN-XZ-naqu903 from Naqu, Qinghai–Xizang (Tibetan) Plateau to validate our new approach. Results from these borehole studies show a warming of 0.1 and 2.3°C, respectively, in the past 450years. When comparing to the results from previous studies, the RU-Yakutia329 study has the same magnitude of warming, while the magnitude of warming at Naqu is slightly smaller.

Reconstruction of an additive space- and time-dependent heat source

In this paper, we consider the inverse problem of simultaneous determination of an additive space- and time-dependent heat source together with the temperature in the heat equation, with Dirichlet boundary conditions and two over-determination conditions. These latter ones consist of a specified temperature measurement at an internal point and a time-average temperature condition. The mathematical problem is linear but ill-posed since the continuous dependence on the input data is violated. In discretised form, the problem reduces to solving an ill-conditioned system of linear equations. We investigate the performances of several regularisation methods and examine their stability with respect to noise in the input data. The boundary element method combined with either the truncated singular value decomposition, or the Tikhonov regularisation, using various methods for choosing regularisation parameters, e.g. the L-curve method, the generalised cross-validation criterion, the discrepancy principle and the ...

A Novel Method for Solving Ill-conditioned Systems ofLinear Equations with Extreme Physical PropertyContrasts

A Novel Method for Solving Ill-conditioned Systems ofLinear Equations with Extreme Physical PropertyContrasts

Computational strategies for surface fitting using thin plate spline finite element methods

Thin plate spline finite element methods are used to fit a surface to an irregularly scattered dataset [S. Roberts, M. Hegland, and I. Altas. Approximation of a Thin Plate Spline Smoother using Continuous Piecewise Polynomial Functions. SIAM, 1:208--234, 2003]. The computational bottleneck for this algorithm is the solution of large, ill-conditioned systems of linear equations at each step of a generalised cross validation algorithm. Preconditioning techniques are investigated to accelerate the convergence of the solution of these systems using Krylov subspace methods. The preconditioners under consideration are block diagonal, block triangular and constraint preconditioners [M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1--137, 2005]. The effectiveness of each of these preconditioners is examined on a sample dataset taken from a known surface. From our numerical investigation, constraint preconditioners appear to provide improved convergence for this surface fitting problem compared to block preconditioners.

Clock Math — a System for Solving SLEs Exactly

In this paper, we present a GPU-accelerated hybrid system that solves ill-conditioned systems of linear equations exactly. Exactly means without rounding errors due to using integer arithmetics. First, we scale floating-point numbers up to integers, then we solve dozens of SLEs within different modular arithmetics and then we assemble sub-solutions back using the Chinese remainder theorem. This approach effectively bypasses current CPU floating-point limitations. The system is capable of solving Hilbert’s matrix without losing a single bit of precision, and with a significant speedup compared to existing CPU solvers.

On the Use of Rigid Body Modes in the Deflated Preconditioned Conjugate Gradient Method

Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper considers the deflated preconditioned conjugate gradient method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.

A numerical solution of a two-dimensional IHCP

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e., the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method.

On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs

AbstractCavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.

IMPROVING THE SOLVABILITY OF ILL-CONDITIONED SYSTEMS OF LINEAR EQUATIONS BY REDUCING THE CONDITION NUMBER OF THEIR MATRICES

This paper is concerned with the solution of ill-conditioned Systems of Linear Equations (SLE's) via the solution of equivalent SLE's which are well-conditioned. A matrix is rst constructed from that of the given ill-conditioned system. Then, an adequate right-hand side is computed to make up the instance of an equivalent system. Formulae and algorithms for computing an instance of this equivalent SLE and solving it will be given and illustrated. Ill-conditioned systems of linear equations (SLE's) are notoriously difficult to solve to any useful accuracy, (5). Their matrices are characterized by large condition numbers. The difficulty may be negotiated by solving different but equivalent systems which are well-conditioned well-conditioned SLE's have matrices with small condition numbers. The approach we put forward here constructs a new matrix and a new right-hand side that constitute an instance of an equivalent SLE to the one given which is ill-conditioned. Moreover, this new matrix has a small condition number compared to that of the matrix of the initial SLE. This means that solving this equivalent system must be better than solving the original one by virtue of the difference in the magnitude of the condition numbers of their matrices. Before embarking in the derivation of the necessary formulae and algorithms to construct such equivalent SLE's, let usrst motivate this approach. Consider the following system of linear equations (SLE) in two variables.

On the Nyström discretization of integral equations on planar curves with corners

The Nyström method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplaceʼs equation. We explain the origin of this instability and show that a straightforward modification to the Nyström scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly-accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.

A new iterative refinement of the solution of ill-conditioned linear system of equations

In this paper, a new iterative refinement of the solution of an ill-conditioned linear system of equations are given. The convergence properties of the method are studied. Some numerical experiments of the method are given and compared with that of two of the available methods.