An algorithm for addressing the real interval eigenvalue problem

In this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices. Such matrices could be used to model real-life problems, where data sets suffer from bounded variations such as uncertainties (e.g. tolerances on parameters, measurement errors), or to study problems for given states.The algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis. The quality of the computed approximation and the running time of the algorithm depend on a given input accuracy. We also present an efficient C++ implementation and illustrate its efficiency on various data sets. In most of the cases we manage to compute efficiently the exact boundary points (limited by floating point representation).     

Non-separable splines and numerical computation of evolution equations by the Galerkin methods

We construct new non-separable splines and we apply the spline sampling approximation to the computation of numerical solutions of evolution equations. The non-separable splines are basis functions which give a fine sampling approximation which enables us to compute numerical solutions by means of the method of lines combined with the Galerkin method. To demonstrate our approach we compute numerical solutions of the Burgers equation and the Kadomtsev–Petviashvili equation.     

The spectral method for high order problems with proper simulations of asymptotic behaviors at infinity

In this paper, we investigate new spectral and multidomain spectral methods for high order problems. We introduce a family of new generalized Laguerre functions, which are mutually orthogonal with the weight function x^α(δ+x)^−γ$x^{\alpha }{(\delta +x)}^{-\gamma }$, δ>0$\delta >0$,α$\alpha$ and γ$\gamma$ being arbitrary real numbers. The corresponding quasi-orthogonal approximation and Laguerre–Gauss–Radau type interpolation are proposed. The spectral and multidomain spectral schemes are provided for several model problems, which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly, but also match the asymptotic behaviors at infinity reasonably. Numerical results demonstrate the efficiency of suggested algorithms, and confirm the analysis well.     

Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem

Numerical approximation of the coupled system of compressible miscible displacement problem in porous media is considered in this paper. A continuous in time discontinuous Galerkin scheme is developed. The symmetric interior penalty discontinuous Galerkin method is used to solve both the flow and transport equations. Upwind technique is used to treat the convection term in the transport equation. The hp-a priori error bounds are derived.     

Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrödinger form

     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, II

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains by using an implicit scheme for the time discretization. After the analysis of stability of the fully discrete scheme, L^∞ in time, L² in space a priori error estimates for the approximation of the displacement, the strain, the pressure and the rotational are derived. Numerical tests are presented which confirm our theoretical results.     

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, I

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains. In this formulation the symmetry of the strain tensor is relaxed by the rotation of the displacement. For the time discretization of this new dual mixed formulation, we use an explicit scheme. After the analysis of stability of the fully discrete scheme, L^∞ in time, L² in space a priori error estimates are derived for the approximation of the displacement, the strain, the pressure and the rotation. Numerical experiments confirm our theoretical predictions.     

An inverse eigenvalue problem and an associated approximation problem for generalized K-centrohermitian matrices

A partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized K-centrohermitian matrices are considered. It is shown under which conditions the inverse eigenproblem has a solution. An expression of its general solution is given. In case a solution of the inverse eigenproblem exists, the optimal approximation problem can be solved. The formula of its unique solution is given.     

A remark on least-squares Galerkin procedures for pseudohyperbolic equations

In this paper, we introduce two split least-squares Galerkin finite element procedures for pseudohyperbolic equations arising in the modelling of nerve conduction process. By selecting the least-squares functional properly, the procedures can be split into two sub-procedures, one of which is for the primitive unknown variable and the other is for the flux. The convergence analysis shows that both the two methods yield the approximate solutions with optimal accuracy in L²(Ω)$L^2\left(\Omega \right)$ norm for u$u$ and _ut$u_t$ and (L²(Ω))²${\left(L^2\left(\Omega \right)\right)}^2$ norm for the flux σ$\sigma$. Moreover, the two methods get approximate solutions with first-order and second-order accuracy in time increment, respectively. A numerical example is given to show the efficiency of the introduced schemes.     

Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

Rate of convergence estimates for the spectral approximation of a generalized eigenvalue problem

Summary. The aim of this work is to derive rate of convergence estimates for the spectral approximation of a mathematical model which describes the vibrations of a solid-fluid type structure. First, we summarize the main theoretical results and the discretization of this variational eigenvalue problem. Then, we state some well known abstract theorems on spectral approximation and apply them to our specific problem, which allow us to obtain the desired spectral convergence. By using classical regularity results, we are able to establish estimates for the rate of convergence of the approximated eigenvalues and for the gap between generalized eigenspaces. Received February 6, 1996 / Revised version received November 28, 1996     

Numerical enclosure for each eigenvalue in generalized eigenvalue problem

An algorithm for enclosing all eigenvalues in generalized eigenvalue problem Ax=λBx is proposed. This algorithm is applicable even if A∈C^n×n is not Hermitian and/or B∈C^n×n is not Hermitian positive definite, and supplies nerror bounds while the algorithm previously developed by the author supplies a single error bound. It is proved that the error bounds obtained by the proposed algorithm are equal or smaller than that by the previous algorithm. Computational cost for the proposed algorithm is similar to that for the previous algorithm. Numerical results show the property of the proposed algorithm.     

A generalized inverse eigenvalue problem in structural dynamic model updating

This paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint. Such a problem arises in structural dynamic model updating. By using the Moore–Penrose generalized inverse and the singular value decomposition (SVD) matrices, the solvability condition and the expression for the solution of the problem are presented. A numerical algorithm for solving the problem is developed.     

Characterization of total ill-posedness in linear semi-infinite optimization

This paper deals with the stability of linear semi-infinite programming (LSIP, for short) problems. We characterize those LSIP problems from which we can obtain, under small perturbations in the data, different types of problems, namely, inconsistent, consistent unsolvable, and solvable problems. The problems of this class are highly unstable and, for this reason, we say that they are totally ill-posed. The characterization that we provide here is of geometrical nature, and it depends exclusively on the original data (i.e., on the coefficients of the nominal LSIP problem). Our results cover the case of linear programming problems, and they are mainly obtained via a new formula for the subdifferential mapping of the support function.     

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.     

Irregular sampling and the inverse spectral problem

We are interested in finding necessary and sufficient conditions for irregular sampling to hold. We shall show that the inverse spectral problem can be used to construct sampling type theorems from the knowledge of the sampling points only. This improves Kramer's theorem as it reveals all possible distributions of the sampling points together with a construction of the sampling functions.     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.     

A rational spectral collocation method for solving a class of parameterized singular perturbation problems

A new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two-point boundary value problems with one boundary layer. By means of the sinh transformation, the original Chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem. The results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation. Numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method.     

Finite element approximation of the volume-matching problem

Summary Optimal orderH ¹ andL error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem. This problem consists of minimising |v| _1, ² overvH ¹() subject to the inequality constraintv0 and a number of linear equality constraints. The presence of the equality constraints leads to Lagrange multipliers, which in turn lead to complications with the standard error analysis for variational inequalities. Finally we consider an algorithm for solving the resulting algebraic problem.Supported by a SERC research studentship