Optimal Condition Number Research Articles

Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

Hardy’s multiquadric and its related interpolators have been found to be highly efficient for interpolating continuous, multivariate functions, as well as for the solution of partial differential equations. Particularly, the interpolation error can be dramatically reduced by varying the shape parameter to make the interpolator optimally flat. This improvement of accuracy is accomplished without reducing the fill distance of collocation points, that is, without the increase of computational cost. There exist a number of mathematical theories investigating the multiquadric family of radial basis functions. These theories are often not fully tested due to the computation difficulty associated with the ill-conditioning of the interpolation matrix. This paper overcomes this difficulty by utilizing arbitrary precision arithmetic in the computation. The issues investigated include conditional positive definiteness, error estimate, optimal shape parameter, traditional and effective condition numbers, round-off error, derivatives of interpolator, and the edge effect of interpolation.

Integrated Design for Configuration/Vibration Control of Hexapod Platform

The Hexapod configuration and vibration control integrated design is treated as an optimal problem of choosing minimum control energy index among a family of Hexapod with optimal Jacobian condition number. Based on the theory of optimal Jacobian condition number, a programming problem is given, which takes control parameter of nonlinear feedback adaptive disturbance canceller(ADC) method, radius of Hexapod base plate and half separation angle between adjacent actuator attachment points on the up plate as design variables, control convergence condition, workspace, and space between attachment points as constrained conditions, and weighted sum of six actuators’ average energy consumption and its variance component as objective function. As the optimal model and its sensitivities contain implicit functions of design variables, the primal problem is approached by 2-rank elliptical response surface method with D-optimal test design. The original programming problem is settled by solving sequence of approximate problems. The example result gives out rules and key points of controller and configuration design of Hexapod, and demonstrates validity and feasibility of the proposed method. Compared with original developed Hexapod, the optimized Hexapod has the minimum condition number, and the advantages in control energy consumption and distribution.

Generalized nonsmooth invexity over cones in vector optimization

In this paper K-nonsmooth quasi-invex and (strictly or strongly) K-nonsmooth pseudo-invex functions are defined. By utilizing these new concepts, the Fritz–John type and Kuhn–Tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein Clarke’s generalized gradient is used. Further a Mond Weir type dual is associated and weak and strong duality results are obtained.

저속 유동 계산의 수렴성에 미치는 특성 조건수의 영향 II : 나비어스톡스 방정식

예조건화 나비어스톡스 방정식의 수렴 특성에 미치는 특성 조건수의 영향을 조사하였다. Choi와 Merkle 예조건화를 적용한 경우와 온도 예조건화를 적용한 경우의 수렴 특성을 분석하였다. 공간차분을 위해 예조건화 Roe의 FDS 기법을 적용하였고, 시간적분을 위해 LU-SGS 기법을 적용하였다. 나비어스톡스 방정식의 수렴 특성은 특성 조건수에 크게 영향을 받으며, 최적의 특성 조건수가 존재하는 것을 보였다. 그리고 점성 유동의 최적 특성 조건수는 비점성 유동에 비해 큰 것으로 나타났다. The effects of characteristic condition number on the convergence of preconditioned Navier-Stokes equations were investigated. The two-dimensional preconditioned Navier-Stokes adopting Choi and Merkle's preconditioning and the temperature preconditioning are considered. Preconditioned Roe's FDS scheme was adopted for spatial discretization and preconditioned LU-SGS scheme was used for time integration. It is shown that the convergence characteristics of the Navier-Stokes equations are strongly affected by the characteristic condition number. Also it is shown that the optimal characteristic condition numbers for viscous flows are larger than that in inviscid flows.

저속 유동 계산의 수렴성에 미치는 특성 조건수의 영향 I : 오일러 방정식

예조건화 오일러 방정식의 수렴 특성에 미치는 특성 조건수의 영향을 조사하였다. 그리고 Choi와 Merkle 예조건화를 적용한 경우와 온도 예조건화를 적용한 경우의 수렴 특성을 분석하였다. 공간차분을 위해 예조건화 Roe의 FDS 기법을 적용하였고, 시간적분을 위해 예조건화 LU-SGS 기법을 적용하였다. 지배방정식의 수렴 특성은 특성 조건수에 크게 영향을 받는 것으로 나타났으며, 최적의 특성 조건수가 존재하는 것으로 나타났다. 그리고 최적의 특성 조건수는 Choi 와 Merkle 예조건화를 적용한 경우와 온도 예조건화를 적용한 경우가 서로 다른 것으로 나타났다. The effects of characteristic condition number on the convergence of preconditioned Euler equations were investigated. The two-dimensional preconditioned Euler equations adopting Choi and Merkle's preconditioning and the temperature preconditioning are considered. Preconditioned Roe's FDS scheme was adopted for spatial discretization and preconditioned LU-SGS scheme was used for time integration. It is shown that the convergence characteristics of the Euler equations are strongly affected by the characteristic condition number, and there is an optimal characteristic condition number for a problem. The optimal characteristic condition numbers for the Choi and Merkle's preconditioning and temperature preconditioning are different.

Optimality conditions for multiple objective fractional subset programming with ( , ρ,σ,θ )-V-type-I and related non-convex functions

In this paper, we introduce a new class of generalized convex n -set functions, called ( ▪, ρ,σ,θ)-V-Type-I and related non-convex functions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of [G.J. Zalmai, Efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ( ▪, α , ρ , θ )-V-convex functions, Comput. Math. Appl. 43 (2002) 1489–1520] to a wider class of functions.

Singular controls in a given stepwise control problem

The Goursat-Darboux stepwise problem of systems control, i.e., the multi-stage problem of control of systems with distributed parameters, is considered. Second-order formulas for the increment in the quality functional are constructed. In investigating these formulas on needle variations in the control, a number of necessary Pontryagin maximum principle-type optimality conditions is obtained and a singular case is studied.

Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements

This paper analyses two-level Schwarz methods for matrices arising from the p-version finite-element method on triangular and tetrahedral meshes. The coarse level consists of the lowest-order finite-element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a simple model problem and an elasticity problem on a complex geometry.

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

In this work the (Extrapolated) Alternating Direction Implicit ((E)ADI) Preconditioners for the Conjugate Gradient (CG) Method are applied for the solution of the second order elliptic partial differential equation (PDE) in a rectangle under Dirichlet boundary conditions. It is shown that the spectral condition number for the discretized model problem, when the ADI-CG Method is used, is an order of magnitude smaller than that of the CG Method. The PDE is approximated by uniform 5- and 9-point stencils and it is shown that the determination of the optimal condition number is equivalent to that of the determination of the optimal EADI scheme. Analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases of the 5- and 9-point schemes; those for the 5-point complete known ones while those for the 9-point are novel. Finally, numerical examples show that the EADI preconditioners proposed make the CG methods compete well against most of the fastest well-known methods.

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ,d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.

Solution of the Problem of Optimal Diagonal Scaling for Quasireal Hermitian Positive-Definite 3×3 Matrices

The paper completely solves the problem of optimal diagonal scaling for quasireal Hermitian positive-definite matrices of order 3. In particular, in the most interesting irreducible case, it is demonstrated that for any matrix C from the class considered there is a uniquely determined optimally scaled matrix D0*CD0 of one of the four canonical types. Formulas for the entries of the diagonal matrix D0 are presented, as well as formulas for the eigenvalues and eigenvectors of D0*CD0 and for the optimal condition number of C, which is equal to k(D0*CD0). The optimality of the Jacobi scaling is analyzed. Bibliography: 10 titles.

Generalized (𝔉, b, ϕ, ρ, θ)‐univex n‐set functions and global parametric sufficient optimality conditions in minmax fractional subset programming

We first define a new class of generalized convex n‐set functions, called (𝔉, b, ϕ, ρ, θ)‐univex functions, and then establish a fairly large number of global parametric sufficient optimality conditions under a variety of generalized (𝔉, b, ϕ, ρ, θ)‐univexity assumptions for a discrete minmax fractional subset programming problem.

On the condition number of integral equations in linear elasticity using the modified Green's function

AbstractIn this work the modified Green's function technique for an exterior Dirichlet and Neumann problem in linear elasticity is investigated. We introduce a modification of the fundamental solution in order to remove the lack of uniqueness for the solution of the boundary integral equations describing the problems, and to simultaneously minimise their condition number. In view of this procedure the cases of the sphere and perturbations of the sphere are examined. Numerical results that demonstrate the effect of increasing the number of coefficients in the modification on the optimal condition number are also presented.

Optimal scaling for p-norms and componentwise distance to singularity

We give lower and upper bounds for the optimal p‐norm condition number achievable by two‐sided diagonal scalings. There are no assumptions on the irreducibility of certain matrices. The bounds are shown to be optimal for the 2‐norm. For the 1‐norm and inf‐norm the (known) exact value of the optimal condition number is confirmed. We give means how to calculate the minimizing diagonal matrices. Furthermore, a class of new lower bounds for the componentwise distance to the nearest singular matrix is given. They are shown to be superior to existing ones.

Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs

In this paper, consider biharmonic equations and 3D blending surfaces, and choose the bi-cubic Hermite elements to seek their approximate solutions. We pursue not only the global superconvergence originated in Lin and his colleagues (Lin, 1994; Lin and Luo, 1995; Lin and Yan, 1996), but also better numerical stability. The boundary penalty plus hybrid integrals are employed to satisfy the normal derivative boundary conditions. Compared with the penalty finite methods (BP-FEM) of bi-cubic Hermite elements in (Li, 1998, 1999; Li and Chang, 1999), the merit of the new methods in this paper is reduction of σ's values thus to improve numerical stability. Suppose that the solution domain Ω can be split into small rectangles □ ij . The global superconvergence O( h 2.5) and O( h 3.5) in H 2 norms are achieved for quasiuniform and uniform □ ij , respectively. Both cases yield the optimal condition number O( h −4), compared with O( h −6) and O( h −8) in (Li, 1999; Li and Chang, 1999). This is an important improvement of stability for biharmonic solutions. However, for 3D blending surfaces, only the global superconvergence O( h 3) is achieved for uniform □ ij . Morever, numerical experiments are provided for biharmonic equations to support the high superconvergence O( h 3.5) involving the natural boundary condition for uniform □ ij with parameter μ∈[0,1). This paper manifests a great flexibility of global superconvegence in applications.

Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Structured perturbations and symmetric matrices

For a given n × n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal Bauer-Skeel condition number cannot be larger than (3 + 2√2) n. In this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation, for relative perturbations of the individual matrix components. It is shown that a symmetric linear system can be arbitrarily ill-conditioned, while any symmetric and entrywise relative perturbation of the matrix of less than 100% does not produce a singular matrix. That means that the inverse of the condition number and the distance to the nearest ill-posed problem can be arbitrarily far apart. Finally we prove that restricting structured perturbations to symmetric (entrywise) perturbations cannot change the condition number by more than a factor (3 + 2√2) n.

On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods

A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the two-level methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomially-based inner iteration method. The latter gives rise to hybrid-type multilevel cycles. This is the so-called (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the so-called wavelet multilevel decomposition based on $L^2$-projections, which in practice must be approximated. Both approaches---the AMLI one and the one based on approximate wavelet decompositions---lead to optimal relative condition numbers of the multilevel preconditioners.

On the Condition Behaviour in the Jacobi Method

The aim of this note is to show that the matrix $S(n,\alpha)=(1-\alpha) I + \alpha ee^{\tau}$, $e=(1,\ldots,1)^{\tau}$, $\alpha\in(0,1)$ is not a counterexample for the accuracy properties of the Jacobi method for computing the singular and eigenvalue decomposition, as might be understood from a recent article of Mascarenhas in this journal. In fact, the Jacobi process on $S(n,\alpha)$ is an example of the perfect behaviour of the algorithm. It is shown that Jacobi rotations preserve the optimal (with respect to diagonal scalings) spectral condition number of $S(n,\alpha)$.