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  • Artikel  (257)
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  • Artikel  (257)
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  • 1
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    On the dense unbounded divergence of interpolatory product integration on Jacobi nodes (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: This paper deals with interpolatory product integration rules based on Jacobi nodes, associated with the Banach space of all s -times continuously differentiable functions, and with a Banach space of absolutely integrable functions, on the interval \([-1,1]\) of the real axis. In order to highlight the topological structure of the set of unbounded divergence for the corresponding product quadrature formulas, a family of continuous linear operators associated with these product integration procedures is pointed out, and the unboundedness of the set of their norms is established, by means of some properties involving the theory of Jacobi polynomials. The main result of the paper is based on some principles of Functional Analysis, and emphasizes the phenomenon of double condensation of singularities with respect to the considered interpolatory product quadrature formulas, by pointing out large subsets (in topological meaning) of the considered Banach spaces, on which the quadrature procedures are unboundedly divergent.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 2
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    Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time \(H^1\) stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity , thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 3
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    Image reconstruction from scattered Radon data by weighted positive definite kernel functions (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 4
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    A real structure-preserving method for the quaternion LU decomposition, revisited (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: In a paper published in 2013, Wang and Ma proposed a structure-preserving algorithm for computing the quaternion LU decomposition. They claimed that it was faster than the LU decomposition implemented in the quaternion Toolbox for Matlab (QTFM). But in 2015, Sangwine, one of the authors of QTFM, pointed out that the tests carried out by him did not support Wang and Ma’s claim. We studied the structure-preserving algorithm of Wang and Ma, and found that the computations were based on element to element operations. In this paper, we re-propose real structure-preserving methods for the quaternion LU decomposition and partial pivoting quaternion LU decomposition, which make full use of high-level operations, and relation of operations between quaternion matrices and their real representation matrices. These algorithms are more efficient than that in QTFM using quaternion arithmetics. Numerical experiments are provided to demonstrate the efficiency of the real structure-preserving method.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 5
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    An inexact Newton method for solving complementarity problems in hydrodynamic lubrication (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: We present an iterative procedure based on a damped inexact Newton iteration for solving linear complementarity problems. We introduce the method in the framework of a popular problem arising in mechanical engineering: the analysis of cavitation in lubricated contacts. In this context, we show how the perturbation and the damping parameter are chosen in our method and we prove the global convergence of the entire procedure. A Fortran implementation of the method is finally analyzed. First, we validate the procedure and analyze all its components, performing also a comparison with a recently proposed technique based on the Fischer–Burmeister–Newton iteration. Then, we solve a 2D problem and provide some insights on an efficient implementation of the method exploiting routines of the Lapack and of the PETSc packages for the solution of inner linear systems.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 6
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    A numerical method for solving shortest path problems (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: Chebyshev pseudo-spectral method is one of the most efficient methods for solving continuous-time optimization problems. In this paper, we utilize this method to solve the general form of shortest path problem. Here, the main problem is converted into a nonlinear programming problem and by solving of which, we obtain an approximate shortest path. The feasibility of the nonlinear programming problem and the convergence of the method are given. Finally, some numerical examples are considered to show the efficiency of the presented method over the other methods.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 7
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    An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: In this paper, we propose an extended block Krylov process to construct two biorthogonal bases for the extended Krylov subspaces \(\mathbb {K}_{m}^e(A,V)\) and \(\mathbb {K}_{m}^e(A^{T},W)\) , where \(A \in \mathbb {R}^{n \times n}\) and \(V,~W \in \mathbb {R}^{n \times p}\) . After deriving some new theoretical results and algebraic properties, we apply the proposed algorithm with moment matching techniques for model reduction in large scale dynamical systems. Numerical experiments for large and sparse problems are given to show the efficiency of the proposed method.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 8
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    A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \) . A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left| \hbox {Im }\omega \right| 〈C\) . This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \) . Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 9
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    A locking-free stabilized mixed finite element method for linear elasticity: the high order case (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k -order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\) -order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k -order convergent for the stress in \(H(\mathrm {div})\) -norm and for the displacement in \(L^2\) -norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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  • 10
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    Nonlinear Galerkin methods for a system of PDEs with Turing instabilities (2018)
    Springer
    In: CALCOLO
    Details
    Publikationsdatum: 2018-03-06
    Beschreibung: We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.
    Print ISSN: 0008-0624
    Digitale ISSN: 1126-5434
    Thema: Mathematik
    Publiziert von Springer
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