Spectral Condition Number Research Articles

Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals

A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the condition number depends on three quantities, two of which can cause ill-conditioning. The numerical linear algebra literature presents several estimates of various instances of these condition numbers. All the prior values exceed the formula introduced here, sometimes by large factors.

A finite element method for surface PDEs: matrix properties

We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. The main idea of this method is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface, for example, two-phase flow problems. It has been proved that the method has optimal order of convergence both in the H 1 and in the L 2-norm. In this paper, we address linear algebra aspects of this new finite element method. In particular the conditioning of the mass and stiffness matrix is investigated. For the two-dimensional case we present an analysis which proves that the (effective) spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h −3| ln h| and h −2| ln h|, respectively, where h is the mesh size of the outer triangulation.

Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs

This paper presents the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the first kind on open arcs. The spectral condition number of MQMs is only O ( h − 1 ) , where h is the maximal mesh width. The errors of MQMs have multivariate asymptotic expansions, accompanied with O ( h i 3 ) for all mesh widths h i . Hence, once discrete equations with coarse meshes are solved in parallel, the accuracy order of numerical approximations can be greatly improved by splitting extrapolation algorithms (SEAs). Moreover, a posteriori asymptotic error estimates are derived, which can be used to formulate self-adaptive algorithms. Numerical examples are also provided to support our algorithms and analysis. Furthermore, compared with the existing algorithms, such as Galerkin and collocation methods, the accuracy order of the MQMs is higher, and the discrete matrix entries are explicit, to prove that the MQMs in this paper are more promising and beneficial to practical applications.

Implicit standard Jacobi gives high relative accuracy

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by $${O(\epsilon \kappa (X))}$$, where $${\epsilon}$$is the machine precision and $${\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2}$$is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.

A scalable FETI–DP algorithm with non-penetration mortar conditions on contact interface

By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI–DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm.

Locally optimized [formula omitted] preconditioning of Rannacher–Turek FEM systems

In this paper Rannacher–Turek non-conforming rotated bilinear finite elements are applied for the numerical solution of second order elliptic boundary value problems. The preconditioned conjugate gradient method is used for the iterative solution of the arising linear algebraic system A u = f . A locally optimized construction for an M-matrix approximation B of the global stiffness matrix A is the first step of the proposed algorithm. Then, the preconditioner is obtained by modified incomplete Cholesky factorization of the auxiliary M-matrix B. A comparative analysis concerning three different approaches for construction of such matrices B is presented. The related spectral condition number estimates are derived. The most important contributions of the paper is the developed original robust preconditioning scheme for strongly anisotropic problems based on properly skewed meshes. A set of numerical tests is presented to illustrate the theoretical investigations.

A fast Fourier-Galerkin method for solving integral equations of second kind with weakly singular kernels

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a weakly singular kernel under the Fourier basis. This compression leads to a sparse matrix with only ${\mathcal{O}}(n\log n)$ number of nonzero entries, where 2n+1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving second kind integral equations with weakly singular kernels. We prove that the approximate order of the truncated equation remains optimal and that the spectral condition number of the coefficient matrix of the truncated linear system is uniformly bounded. Furthermore, we develop a fast algorithm for solving the corresponding truncated linear system, which preserves the optimal order of the approximate solution with only ${\mathcal{O}}(n\log^{2}n)$ number of multiplications required. Numerical examples complete the paper.

Superrelaxation and the rate of convergence in minimizing quadratic functions subject to bound constraints

The paper resolves the problem concerning the rate of convergence of the working set based MPRGP (modified proportioning with reduced gradient projection) algorithm with a long steplength of the reduced projected gradient step. The main results of this paper are the formula for the R-linear rate of convergence of MPRGP in terms of the spectral condition number of the Hessian matrix and the proof of the finite termination property for the problems whose solution does not satisfy the strict complementarity condition. The bound on the R-linear rate of convergence of the projected gradient is also included. For shorter steplengths these results were proved earlier by Dostal and Schoberl. The efficiency of the longer steplength is illustrated by numerical experiments. The result is an important ingredient in developming scalable algorithms for numerical solution of elliptic variational inequalities and substantiates the choice of parameters that turned out to be effective in numerical experiments.

Positive Definite Hankel Matrices Using Cholesky Factorization

AbstractReal positive definite Hankel matrices arise in many important applications. They have spectral condition numbers which exponentially increase with their orders. We give a structural algorithm for finding positive definite Hankel matrices using the Cholesky factorization, compute it for orders less than or equal to 30, and compare our result with earlier results.

On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces

The performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation (PDE) models. It is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix. In this paper, for typical second-order elliptic problems, some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes. The derivation of such relations for general high-order elements is based on a new trace formula for the element stiffness matrix. It is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces. These results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing.

An Efficient Algorithm for Solving the Generalized Airfoil Equation

In this paper, we develop an efficient Petrov-Galerkin method for the generalized airfoil equation. In general, the Petrov-Galerkin method for this equation leads to a linear system with a dense coefficient matrix. When the order of the coefficient matrix is large, the complexity for solving the corresponding linear system is huge. For this purpose, we propose a matrix truncation strategy to compress the dense coefficient matrix into a sparse matrix. Subsequently, we use a numerical integration method to generate the fully discrete truncated linear system. At last we solve the corresponding linear system by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating the fully discrete truncated linear system and solving it is estimated to be linear up to a logarithm. The spectral condition number of the truncated matrix is proved to be bounded. Numerical examples complete the paper.

Analysis of an extended pressure finite element space for two-phase incompressible flows

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In [6] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.

Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

We study the eigenvalues of the operator generated by using the inverse of the Laplacian as a preconditioner for self-adjoint second-order elliptic partial differential equations with smooth coefficients. It is well-known that the spectral condition number of the preconditioned operator can be bounded by ⁠, where k is the uniformly positive coefficient of the second-order elliptic equation. The purpose of this paper is to study the spectrum of the preconditioned operator. We will show that there is a strong relation between the spectrum of this operator and the range of the coefficient function. In the continuous case, we prove, both for mappings defined on Sobolev spaces and in terms of generalized functions, that the spectrum of the preconditioned operator contains the range of the coefficient function k. In the discrete case, we indicate by numerical examples that the entire discrete spectrum is approximately given by values of k.

Convergence Rate of an Optimization Algorithm for Minimizing Quadratic Functions with Separable Convex Constraints

A new active set algorithm for minimizing quadratic functions with separable convex constraints is proposed by combining the conjugate gradient method with the projected gradient. It generalizes recently developed algorithms of quadratic programming constrained by simple bounds. A linear convergence rate in terms of the Hessian spectral condition number is proven. Numerical experiments, including the frictional three-dimensional (3D) contact problems of linear elasticity, illustrate the computational performance.

Robust multiple‐fault detection and isolation: A gradient flow approach

AbstractThis paper presents a novel solution to the fault detection and isolation observer design problem for linear time‐invariant systems. A gradient flow approach is proposed for synthesizing a residual generator under optimal eigenstructure assignment. This is achieved by minimizing the spectral condition number of the observer eigenvector matrix. The properties of convergence of the gradient flow solution are proved and its efficiency demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

AbstractFinite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd.

A numerical study of a semi‐algebraic multilevel preconditioner for the local discontinuous Galerkin method

AbstractIn this work, we consider the local discontinuous Galerkin (LDG) method applied to second‐order elliptic problems arising in the modeling of single‐phase flows in porous media. It has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of 𝒪(h−2) on structured and unstructured meshes, where h is the mesh size. Thus, efficient preconditioners are mandatory. We present a semi‐algebraic multilevel preconditioner for the LDG method using local Lagrange‐type interpolatory basis functions. We show, numerically, that its performance does not degrade, or at least the number of iterations increases very slowly, as the number of unknowns augments. The preconditioner is tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media. Copyright © 2007 John Wiley & Sons, Ltd.

On Meinardus' examples for the conjugate gradient method

. The conjugate gradient (CG) method is widely used to solve a positive definite linear system Ax = b of order N. It is well known that the relative residual of the kth approximate solution by CG (with the initial approximation x 0 = 0) is bounded above by 2[∇ k k +∇ -k k ] -1 with ∇k=√k+1 √k-1 where K = K ( A) = ∥A∥ 2 ∥A -1 ∥ 2 is A's spectral condition number. In 1963, Meinardus (Numer. Math., 5 (1963), pp. 14-23) gave an example to achieve this bound for k = N - 1 but without saying anything about all other 1 < k < N - 1. This very example can be used to show that the bound is sharp for any given k by constructing examples to attain the bound, but such examples depend on k and for them the (k + 1)th residual is exactly zero. Therefore it would be interesting to know if there is any example on which the CG relative residuals are comparable to the bound for all 1 ≤ k < N - 1. There are two contributions in this paper: (1) A closed formula for the CG residuals for all 1 ≤ k < N- 1 on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of -√2 of the actual residuals; (2) A complete characterization of extreme positive linear systems for which the kth CG residual achieves the bound is also presented.

Almost optimal order approximate inverse based preconditioners for 3-d convection dominated problems on tensor-grids

For a one-dimensional diffusion problem on an refined computational grid we present preconditioners based on the standard approximate inverse technique. Next, we determine its spectral condition number κ2 and perform numerical calculations which corroborate the theoretical results. Then we perform numerical calculations which show that the standard approximate inverse preconditioners and our modified versions behave in a similar manner. To finish with we show that a combination of the standard approximate inverse with an additional incomplete factorisation leads to an almost optimal order preconditioner in 1 3 dimensions on refined grids, with and without dominant convection.

A scalable FETI-DP algorithm for a semi-coercive variational inequality

We develop an optimal algorithm for the numerical solution of semi-coercive variational inequalities by combining dual-primal FETI algorithms with recent results for bound and equality constrained quadratic programming problems. The discretized version of the model problem, obtained by using the FETI-DP methodology, is reduced by the duality theory of convex optimization to a quadratic programming problem with bound and equality constraints, which is solved by a new algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. We present convergence bounds that guarantee the scalability of the algorithm. These results are confirmed by numerical experiments.