Spectral Condition Number Research Articles

Nodal auxiliary space preconditioners for mixed virtual element methods

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.

Block diagonal Calderón preconditioning for scattering at multi-screens

A preconditioner is proposed for Laplace exterior boundary value problems on multi-screens. To achieve this, the quotient-space boundary element method and operator preconditioning are combined. For a fairly general subclass of multi-screens, it is shown that this approach paves the way for block diagonal Calderón preconditioners which achieve a spectral condition number that grows only logarithmically with decreasing mesh size, just as in the case of simple screens. Since the resulting scheme contains many more degrees of freedom than strictly required, strategies are presented to remove almost all redundancy without significant loss of effectiveness of the preconditioner. The performance of this method is verified by providing representative numerical results. Further numerical experiments suggest that these results can be extended to a much wider class of multi-screens that cover essentially all geometries encountered in practice, leading to a significantly reduced simulation cost.

Two new Hager–Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing

Notwithstanding its efficiency and nice attributes, most research on the Hager–Zhang (HZ) iterative scheme are focused on unconstrained minimization problems. Inspired by this and recent extensions of the one-parameter HZ scheme to system of nonlinear monotone equations, two new HZ-type iterative methods are developed in this paper for solving system of monotone equations with convex constraint. This is achieved by developing two HZ-type search directions with new parameter choices combined with the popular projection method. The first parameter choice is obtained by minimizing the condition number of a modified HZ direction matrix, while the second choice is realized using singular value analysis and minimizing the spectral condition number of a nonsingular HZ search direction matrix. Interesting properties of the schemes include solving non-smooth problems and satisfying the inequality that is vital for global convergence. Using standard assumptions, global convergence of the schemes are proven and numerical experiments with recent methods in the literature, indicate that the methods proposed are promising. The effectiveness of the schemes are further demonstrated by their applications to sparse signal and image reconstruction problems, where they outperform some recent schemes in the literature.

Tikhonov regularization with MTRSVD method for solving large-scale discrete ill-posed problems

Regularization is possibly the most popular method for solving discrete ill-posed problems, whose solution is less sensitive to the error in the observed vector in the right hand than the original solution. This paper presents a new modified truncated randomized singular value decomposition (TR-MTRSVD) method for large Tikhonov regularization in standard form. The proposed TR-MTRSVD algorithm introduces the idea of randomized algorithm into the improved truncated singular value decomposition (MTSVD) method to solve large Tikhonov regularization problems. The approximation matrix Ãℓ produced by randomized SVD is replaced by the closest matrix Ãk̃ in a unitarily invariant matrix norm with the same spectral condition number. The regularization parameters are determined by the discrepancy principle. Numerical examples show the effectiveness and efficiency of the proposed TR-MTRSVD algorithm for large Tikhonov regularization problems.

On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusion Problem

The mixed formulation of the Local Discontinuous Galerkin (LDG) method is presented for a two boundary value problem that involves the Riesz operator with fractional order $$1< \alpha < 2$$ . Well posedness of the stabilized and non stabilized LDG method is proved. Using a penalty term of order $${{\mathcal {O}}}\left( h^{1-\alpha }\right) $$ a sharp error estimate in a mesh dependent energy semi-norm is developed for sufficiently smooth solutions. Error estimates in the $$L^2$$ -norm are obtained for two auxiliary variables which characterize the LDG formulation. Our analysis indicates that the non stabilized version of the method achieves higher order of convergence for all fractional orders. A numerical study suggests a less restrictive, $${{\mathcal {O}}}\left( h^{-\alpha }\right) $$ , spectral condition number of the stiffness matrix by using the proposed penalty term compared to the $${{\mathcal {O}}}\left( h^{-2}\right) $$ growth obtained when the traditional $${{\mathcal {O}}}\left( h^{-1}\right) $$ penalization term is chosen. The sharpness of our error estimates is numerically validated with a series of numerical experiments. The present work is the first attempt to elucidate the main differences between both versions of the method.

Parametric eigenvalue assignment by constant output feedback – a cascaded approach

Abstract In this paper a numerical efficient approach to the problem of eigenvalue assignment by constant output feedback is presented. It improves the well known Kimura’s condition by 2, i. e., it is shown that if m + p ≥ n − 1 m+p\ge n-1 generically a solution to this design problem exists where n , m n,m and p denote the dimensions of the system states, inputs and outputs, respectively. The algorithm is based on a cascaded control scheme with up to three design steps. The first two steps merely require standard methods from linear algebra while the last step only in case of m + p = n − 1 m+p=n-1 demands for the numerical solution of a system of three polynomial equations each of order two. The design procedure explicitly embodies all degrees of freedom beyond eigenvalue assignment. Thus, they can be used to account for other design it is shown goals, e. g., to minimize the spectral condition number of the closed-loop system or a norm of the feedback gain as it is shown by numerical examples from literature.

A novel modified TRSVD method for large-scale linear discrete ill-posed problems

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix Ak of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD method, by replacing A by a matrix that is closer to A than Ak in a unitarily invariant matrix norm and has the same spectral condition number as Ak. Approximations of the SVD of a large matrix A can be computed quite efficiently by using a randomized SVD (RSVD) method. This paper presents a novel modified truncated randomized singular value decomposition (MTRSVD) method for computing approximate solutions to large-scale linear discrete ill-posed problems. The rank, k, is determined with the aid of the discrepancy principle, but other techniques for selecting a suitable rank also can be used. Numerical examples illustrate the effectiveness of the proposed method and compare it to the truncated RSVD method.

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

On a relation of discontinuous Petrov–Galerkin and least-squares finite element methods

The discontinuous Petrov–Galerkin (DPG) method minimizes a residual in a non-standard norm. This paper shows that the minimization of this residual is equivalent to the minimization of a residual in a L2 norm. Since such residuals are well known from least squares finite element methods, this novel interpretation allows to extend results for least squares methods to the DPG method and vice versa. This paper exemplifies the benefits of this possibility by the verification of an asymptotic exactness result for a DPG method for the Helmholtz equation, the design of a locking-free DPG method for linear elasticity, and an investigation of the spectral condition number.

The spectral condition number plot for regularization parameter evaluation

Many modern statistical applications ask for the estimation of a covariance (or precision) matrix in settings where the number of variables is larger than the number of observations. There exists a broad class of ridge-type estimators that employs regularization to cope with the subsequent singularity of the sample covariance matrix. These estimators depend on a penalty parameter and choosing its value can be hard, in terms of being computationally unfeasible or tenable only for a restricted set of ridge-type estimators. Here we introduce a simple graphical tool, the spectral condition number plot, for informed heuristic penalty parameter assessment. The proposed tool is computationally friendly and can be employed for the full class of ridge-type covariance (precision) estimators.

Matrix analyses on the Dai–Liao conjugate gradient method

Some optimal choices for a parameter of the Dai–Liao conjugate gradient method are proposed by conducting matrix analyses of the method. More precisely, first the \(\ell_1\) and \(\ell_\infty\) norm condition numbers of the search direction matrix are minimized, yielding two adaptive choices for the Dai–Liao parameter. Then we show that a recent formula for computing this parameter which guarantees the descent property can be considered as a minimizer of the spectral condition number as well as the well-known measure function for a symmetrized version of the search direction matrix. Brief convergence analyses are also carried out. Finally, some numerical experiments on a set of test problems related to constrained and unconstrained testing environment, are conducted using a well-known performance profile. doi:10.1017/S1446181119000063

MATRIX ANALYSES ON THE DAI–LIAO CONJUGATE GRADIENT METHOD

Some optimal choices for a parameter of the Dai–Liao conjugate gradient method are proposed by conducting matrix analyses of the method. More precisely, first the $\ell _{1}$ and $\ell _{\infty }$ norm condition numbers of the search direction matrix are minimized, yielding two adaptive choices for the Dai–Liao parameter. Then we show that a recent formula for computing this parameter which guarantees the descent property can be considered as a minimizer of the spectral condition number as well as the well-known measure function for a symmetrized version of the search direction matrix. Brief convergence analyses are also carried out. Finally, some numerical experiments on a set of test problems related to constrained and unconstrained testing environment, are conducted using a well-known performance profile.

Spectral condition‐number estimation of large sparse matrices

SummaryWe describe a randomized Krylov‐subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value of A. Our method estimates this value by solving a consistent linear least squares problem with a known solution using a specific Krylov‐subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to . Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running dense singular value decomposition would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR), and it works equally well on square and rectangular matrices.

A Class of Efficient Locally Constructed Preconditioners Based on Coarse Spaces

In this paper we present a class of robust and fully algebraic two-level preconditioners for symmetric positive definite (SPD) matrices. We introduce the notion of algebraic local symmetric positive semidefinite splitting of an SPD matrix and we give a characterization of this splitting. This splitting leads to construct algebraically and locally a class of efficient coarse spaces which bound the spectral condition number of the preconditioned system by a number defined a priori. We also introduce the $\tau$-filtering subspace. This concept helps compare the dimension minimality of coarse spaces. Some PDEs-dependant preconditioners correspond to a special case. The examples of the algebraic coarse spaces in this paper are not practical due to expensive construction. We propose a heuristic approximation that is not costly. Numerical experiments illustrate the efficiency of the proposed method.

A New Dai-Liao Conjugate Gradient Method with Optimal Parameter Choice

A new nonlinear conjugate gradient method is proposed to solve large-scale unconstrained optimization problems. The direction is given by a search direction matrix, which contains a positive parameter. The value of the parameter is calculated by minimizing the upper bound of spectral condition number of the matrix defining it in order to cluster all the singular values. The new search direction satisfies the sufficient descent condition. Under some mild assumptions, the global convergence of the proposed method is proved for uniformly convex functions and the general functions. Numerical experiments show that, for the CUTEr library and the test problem collection given by Andrei, the proposed method is superior to M1 proposed by Babaie-Kafaki and Ghanbari (Eur. J. Oper. Res. 234(3), 625–630, 2014), CG_DESCENT(5.3), and CGOPT.

Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

BackgroundThe single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) method, such as single-step genomic BLUP (ssGBLUP), simultaneously analyses phenotypic, pedigree, and genomic information of genotyped and non-genotyped animals. In contrast to ssGBLUP, SNP effects are fitted explicitly as random effects in the ssSNPBLUP model. Similarly, principal components associated with the genomic information can be fitted explicitly as random effects in a single-step principal component BLUP (ssPCBLUP) model to remove noise in genomic information. Single-step genomic BLUP is solved efficiently by using the preconditioned conjugate gradient (PCG) method. Unfortunately, convergence issues have been reported when solving ssSNPBLUP by using PCG. Poor convergence may be linked with poor spectral condition numbers of the preconditioned coefficient matrices of ssSNPBLUP. These condition numbers, and thus convergence, could be improved through the deflated PCG (DPCG) method, which is a two-level PCG method for ill-conditioned linear systems. Therefore, the first aim of this study was to compare the properties of the preconditioned coefficient matrices of ssGBLUP and ssSNPBLUP, and to document convergence patterns that are obtained with the PCG method. The second aim was to implement and test the efficiency of a DPCG method for solving ssSNPBLUP and ssPCBLUP.ResultsFor two dairy cattle datasets, the smallest eigenvalues obtained for ssSNPBLUP (ssPCBLUP) and ssGBLUP, both solved with the PCG method, were similar. However, the largest eigenvalues obtained for ssSNPBLUP and ssPCBLUP were larger than those for ssGBLUP, which resulted in larger condition numbers and in slow convergence for both systems solved by the PCG method. Different implementations of the DPCG method led to smaller condition numbers, and faster convergence for ssSNPBLUP and for ssPCBLUP, by deflating the largest unfavourable eigenvalues.ConclusionsPoor convergence of ssSNPBLUP and ssPCBLUP when solved by the PCG method are related to larger eigenvalues and larger condition numbers in comparison to ssGBLUP. These convergence issues were solved with a DPCG method that annihilates the effect of the largest unfavourable eigenvalues of the preconditioned coefficient matrix of ssSNPBLUP and of ssPCBLUP on the convergence of the PCG method. It resulted in a convergence pattern, at least, similar to that of ssGBLUP.

On the conditioning of two-equation road traffic models

This article deals with two-equation macroscopic traffic models, which in addition to the standard continuity equation (the LWR model), contain an additional differential equation expressing the traffic dynamics, specifically the process of speed adaptation to road traffic conditions. There are many two-equation models with various forms of the traffic dynamics equation with different possibilities to achieve an accurate, stable and effective solution. The most popular models have been collected here and presented in a unified matrix form. The quality of particular two-equation traffic models, and thus the quality of the results obtained, depends on the quality of the model itself, and on the discretization and approximation of the problem. In this article, the evaluation of particular models is considered only on a continuous level, i.e. at the level of theoretical description of the models. Standard conditioning measures such as the spectral radius and the spectral condition number were used for this purpose. The results of the estimations have been used to compare particular models as well as to characterize two basic classes of models, i.e. isotropic (symmetrical) and anisotropic (asymmetric) models, and to evaluate the stability and potential computational efficiency of solutions of these two-equation models.

A Legendre–Petrov–Galerkin method for solving Volterra integro-differential equations with proportional delays

ABSTRACTIn this paper, a fully discrete Legendre–Petrov–Galerkin method is presented for solving functional Volterra integro-differential equations with vanishing delays. This method produces a fully discrete linear system. We prove that this system has a unique solution for sufficiently large n, where n+1 denotes the order of the system. Moreover, we prove that the approximate solution and its corresponding derivative function arrive at an optimal convergence order and in norm, where m+1 is the regularity of the original solution. In addition, we prove that for sufficiently large n, the spectral condition number of the corresponding linear system is uniformly bounded. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Convergence Analysis of Acceleration Schemes: Spectral Radius versus Condition Number

The theory of matrix splitting plays an important role in convergence analysis for the acceleration schemes. Spectral radius allows us to make a complete description of eigenvalues of a matrix and is independent of any particular matrix norm. The aim of this research article is to find the relationship between spectral radius and condition number in terms of stability of algorithms using various classes of linear systems and determine whether the condition number can be used to predict the convergence of acceleration schemes as well as iterative methods in the solutions of systems of linear equation. From the results obtained using matlab programming language, there is strong relationship between spectral radius and condition number for dense symmetric positive definite (SPD) and tridiagonal system. The relationship is shown as k(A)=||A|| ||A-1||=ρ(A)(A-1) and the value of condition number helps to predict the rate of convergence and iterations for a given linear system.

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where $$n+1$$ denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order $$\mathcal{O}(n^{-m}\log n)$$ in the infinite norm and $$\mathcal{O}(n^{-m})$$ in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is $$\mathcal{O}(\log ^2 n)$$ and its spectral condition number is $$\mathcal{O}(1)$$ . Numerical examples are presented to demonstrate the effectiveness of the proposed method.