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Structured matrices are encountered in various guises in many mathematical models which describe problems of the real world. In fact, they are the algebraic translation of the specific properties at the basis of the physical problem. The analysis and the exploitation of matrix structures is the first fundamental step in the design of highly efficient algorithms for the solution of complex computational problems. [ABSTRACT FROM AUTHOR]

In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T≈ O(1) and Reynolds numbers ν −1≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ ν −1≤200. [ABSTRACT FROM AUTHOR]

For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10 [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [ Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case. [ABSTRACT FROM AUTHOR]

In this paper we solve a conjecture on the zeros of R-Bonacci polynomials in the case when r=3 (Tribonacci polynomials) and determine the zero attractor of the Tribonacci polynomials. Universality of the zero attractor is also established. [ABSTRACT FROM AUTHOR]

In this paper, the variations of both subunitary polar factors and Hermitian positive semidefinite polar factors in the polar decomposition are studied. New perturbation bounds of both polar factors are given without the restriction that A and its perturbed matrix à have the same rank. These improve recent results. [ABSTRACT FROM AUTHOR]

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples. [ABSTRACT FROM AUTHOR]

In order to determine the extinction probability of a Markovian binary tree, it is necessary to find the minimal nonnegative solution of a quadratic vector equation. We apply the modified Newton-Shamanskii method for solving the equation. We show that the sequence of vectors generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

A variable coefficient fourth-order partial differential equation in one space variable that arises in the study of transverse vibrations of a uniform flexible beam is examined. The numerical solution is obtained by using the sinc-Galerkin method. The method is then tested on three problems which have appeared in physical applications. The numerical results demonstrate the reliability and efficiency of the algorithm developed. [ABSTRACT FROM AUTHOR]

We provide explicit expressions for both mixed and componentwise structured condition numbers for several classes of structured rectangular matrices: upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. Mathematics Subject Classification (2000): 15A12, 65F35 [ABSTRACT FROM AUTHOR]

Discusses the growth factor in Gaussian elimination for matrices with sharp angular field of values. Stability of Gaussian elimination for complex matrices A+B+iC where both B and C are Hermitian definite matrices; Value of growth factor.

Variable-step (VS) $$3$$ -stage Hermite-Birkhoff (HB) methods HB $$(p=k+1)$$ of order $$p=8$$ and 9 are constructed as a combination of linear $$k$$ -step methods of order $$(p-2)$$ and a diagonally implicit one-step $$3$$ -stage Runge-Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A( $$\alpha $$ )-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O $$(p^2)$$ operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB( $$p$$ ) of order $$p=8$$ and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval. [ABSTRACT FROM AUTHOR]

We propose and study an iterative substructuring method for an $$h$$ - $$p$$ Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103-134, () for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory. [ABSTRACT FROM AUTHOR]

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1− ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied. [ABSTRACT FROM AUTHOR]

In this note, we present an explicit formula for perturbations of an outer inverse under certain conditions, which extends previous results. [ABSTRACT FROM AUTHOR]

Presents a mixed convergent formulation for the three-dimensional Stokes equations. Focus on the Stokes problem in a bounded domain of R3 with a polyhedral boundary; Involvement of vorticity and vector potential; Basing of discrete problem on stabilized formulation depending on a nonnegative parameter.

Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter ε or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < ε ≪ 1. In this work we develop a rigorous ε-uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit ε → 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation ε-uniform convergence of order one half is proven whereas in the Lagrangian framework ε-uniform convergence of optimal order is established. The estimates are based on ε-uniform a priori estimates for the solution of the continuous problems which are derived first. [ABSTRACT FROM AUTHOR]

A virtual element method is introduced for the mixed approximation of a simple model problem for the Laplace operator on a polyhedron. The method is fully analysed when the meshes are made up of triangular right prisms, pyramids and tetrahedra. The local discrete spaces coincide with the lowest order Raviart–Thomas spaces on tetrahedral and triangular right prismatic elements, and extend them to pyramidal elements. The discrete scheme is well posed and optimal interpolation error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the discrete element space are optimal and anisotropic on anisotropic right prisms. Furthermore, a discretization of the model problem in the presence of edge and vertex singularities is analysed for the proposed method on a family of suitably designed graded meshes, and optimal estimates for the approximation error are obtained, extending in this way the results of Farhloul et al. (ESAIM Math Model Numer Anal 35:907–920, 2001) where cylindrical domains with edge singularities were considered. [ABSTRACT FROM AUTHOR]