Showing papers in "Calcolo in 2014"

Shape preservation of scientific data through rational fractal splines

Abstract: Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

Efficient Jarratt-like methods for solving systems of nonlinear equations

Abstract: We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.

Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation

Abstract: We analyze an adaptive boundary element method for Symm's integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data $$g$$ g into the adaptive scheme. We prove quasi-optimal convergence rates for any $$H^{1/2}$$ H 1 / 2 -stable projection used for data approximation.

Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation

Abstract: This paper aims to develop a fully discrete local discontinuous Galerkin finite element method for numerical simulation of the time-fractional telegraph equation, where the fractional derivative is in the sense of Caputo. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. The stability and convergence of this discontinuous approach are discussed and theoretically proven. Finally numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

Finite difference method for a fractional porous medium equation

Abstract: We formulate a numerical method to solve the porous medium type equation with fractional diffusion $$\begin{aligned} \frac{\partial u}{\partial t}+(-\Delta )^{1/2} (u^m)=0. \end{aligned}$$ ? u ? t + ( - Δ ) 1 / 2 ( u m ) = 0 . The problem is posed in $$x\in {\mathbb {R}}^N$$ x ? R N , $$m\ge 1$$ m ? 1 and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli---Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data as well as a section that summarizes and concludes the proposed method.

Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers

Abstract: The $$\varepsilon $$ ? -weighted energy norm is the natural norm for singularly perturbed convection-diffusion problems with exponential layers. But, this norm is too weak to recognise features of characteristic layers. We present an error analysis in a differently weighted energy norm--a balanced norm--that overcomes this drawback.

A new family of Newton-type iterative methods with and without memory for solving nonlinear equations

Abstract: In this paper, we present a new family of two-step iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to Kung and Traubs conjecture Kung and Traub (J Appl Comput Math 21:643---651, 1974). Further accelerations of convergence speed are obtained by varying a free parameter in per full iteration. This self-accelerating parameter is calculated by using information available from the current and previous iteration. The corresponding R-order of convergence is increased form 4 to $$\frac{5+\sqrt{17}}{2}\approx 4.5616, \frac{5+\sqrt{21}}{2}\approx 4.7913$$ and 5. The increase of convergence order is attained without any additional calculations so that the family of the methods with memory possesses a very high computational efficiency. Another advantage of the new methods is that they remove the severe condition $$f^{\prime }(x)$$ in a neighborhood of the required root imposed on Newtons method. Numerical comparisons are made to show the performance of our methods, as shown in the illustration examples.

Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior

Abstract: In this paper, we present a family of optimal, in the sense of Kung---Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methods.

Convergence and comparison theorems for single and double decompositions of rectangular matrices

Abstract: Different convergence and comparison theorems for proper regular splittings and proper weak regular splittings are discussed. The notion of double splitting is also extended to rectangular matrices. Finally, convergence and comparison theorems using this notion are presented.

Interpolation splines minimizing a semi-norm

Abstract: Using S.L. Sobolev's method, we construct the interpolation splines minimizing the semi-norm in $$K_2(P_2)$$ , where $$K_2(P_2)$$ is the space of functions $$\phi $$ such that $$\phi ^{\prime } $$ is absolutely continuous, $$\phi ^{\prime \prime } $$ belongs to $$L_2(0,1)$$ and $$\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $$ . Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the trigonometric functions $$\sin x$$ and $$\cos x$$ . Finally, in a few numerical examples the qualities of the defined splines and $$D^2$$ -splines are compared. Furthermore, the relationship of the defined splines with an optimal quadrature formula is shown.

Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa

Abstract: For a given $$\theta \in (a,b)$$ , we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa $$\theta $$ plus possibly $$a$$ and/or $$b$$ , the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive quadrature formulae are useful in studying problems in one-sided polynomial $$L_1$$ approximation.

Legendre spectral-collocation method for Volterra integral equations with non-vanishing delay

Abstract: The main purpose of this paper is to propose the Legendre spectral-collocation method to solve the Volterra integral equations of the second kind with non-vanishing delay. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay. In each subinterval, where the solution is smooth enough, we can apply Legendre spectral-collocation method to approximate the solution. The provided convergence analysis shows that the numerical errors decay exponentially. Numerical examples are presented to confirm this theoretical predict.

A note on the growth factor in Gaussian elimination for accretive-dissipative matrices

Abstract: This short note proves that if $$A$$ is accretive-dissipative, then the growth factor for such $$A$$ in Gaussian elimination is less than $$4$$ . If $$A$$ is a Higham matrix, i.e., the accretive-dissipative matrix $$A$$ is complex symmetric, then the growth factor is less than $$2\sqrt{2}$$ . The result obtained improves those of George et al. in [Numer. Linear Algebra Appl. 9, 107---114 (2002)] and is one step closer to the final solution of Higham's conjecture.

Optimal quadrature formulas in the sense of Sard in $$W_2^{(m,m-1)}$$ space

Abstract: This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $$W_2^{(m,m-1)}(0,1)$$ space. Using the Sobolev's method we obtain new optimal quadrature formulas of such type for $$N+1\ge m$$ , where $$N+1$$ is the number of the nodes. Moreover, explicit formulas of the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $$m=1$$ and prove an asymptotic optimality of such a formula in the Sobolev space $$L_2^{(1)}(0,1)$$ . It turns out that the error of the optimal quadrature formula in $$W_2^{(1,0)}(0,1)$$ is less than the error of the optimal quadrature formula given in the $$L_2^{(1)}(0,1)$$ space. The obtained optimal quadrature formula in the $$W_2^{(m,m-1)}(0,1)$$ space is exact for $$\exp (-x)$$ and $$P_{m-2}(x)$$ , where $$P_{m-2}(x)$$ is a polynomial of degree $$m-2$$ . Furthermore, some numerical results, which confirm the obtained theoretical results of this work, are given.

Block SOR methods for the solution of indefinite least squares problems

Abstract: This paper describes a technique for constructing block SOR methods for the solution of the large and sparse indefinite least squares problem which involves minimizing a certain type of indefinite quadratic form. Two block SOR-based algorithms and convergence results are presented. The optimum parameters for the methods are also given. It has been shown both theoretically and numerically that the optimum block SOR methods have a faster convergence than block Jacobi and Gauss---Seidel methods.

Fractional convolution quadrature based on generalized Adams methods

Abstract: In this paper we present a product quadrature rule for Volterra integral equations with weakly singular kernels based on the generalized Adams methods. The formulas represent numerical solvers for fractional differential equations, which inherit the linear stability properties already known for the integer order case. The numerical experiments confirm the valuable properties of this approach.

Sinc approximation of eigenvalues of Sturm---Liouville problems with a Gaussian multiplier

Abstract: Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. The sinc Gaussian sampling technique derived by Qian (Proc Am Math Soc 131:1169---1176, 2002) establishes a sampling technique which converges faster than the classical sampling technique. Schmeisser and Stenger (Sampl Theory Signal Image Process 6:199---221, 2007) studied the associated error analysis. In the present paper we apply a sinc Gaussian technique to compute approximate values of the eigenvalues of Sturm---Liouville problems with eigenvalue parameter in one or two boundary conditions. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.

Superconvergence analysis for parabolic optimal control problems

Abstract: In this paper, we investigate the superconvergence of a variational discretization approximation for parabolic optimal control problems with control constraints. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. The time discretization is based on difference methods. We derive the superconvergence between the numerical solution and elliptic projection for the state and the adjoint state and present a numerical example for illustrating our theoretical results.

Eigenvalue approximation by mixed non-conforming finite element methods

Abstract: In this paper we give a theory for the approximation of eigenvalue problems in mixed form by non-conforming methods. We then apply this theory to analyze the problem of determining the vibrational modes of a linear elastic structure using the classical Hellinger---Reissner mixed formulation. We show that a numerical method based on the lowest-order Arnold---Winther non-conforming space provides a spectrally correct approximation of the eigenvalue/eigenvector pairs. Moreover, the method is proven to converge with optimal order.

Two-Grid Methods for Hermitian positive definite linear systems connected with an order relation

Abstract: Given a multigrid procedure for linear systems with coefficient matrices $$A_n,$$ we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $$B_n$$ : we assume that both $$A_n$$ and $$B_n$$ are Hermitian positive definite with $$A_n\le \vartheta B_n,$$ for some positive $$\vartheta $$ independent of $$n.$$ In this context we prove the Two-Grid Method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured linear systems, in which the Hermitian positive definite coefficient matrix is banded in a multilevel sense. As structured matrices, Toeplitz, circulants, Hartley, sine ( $$\tau $$ class) and cosine algebras are considered. In such a way, several linear systems arising from the approximation of integro---differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.

Some representations for the Drazin inverse of a modified matrix

Abstract: In this paper, we give some results for the Drazin inverse of a modified matrix $$M=A-CD^dB$$ M = A - C D d B with the generalized Schur complement $$Z=D-BA^dC$$ Z = D - B A d C under some conditions. Further, we present some new results for the Drazin inverse of the modified matrix $$M=A-CD^dB$$ M = A - C D d B , when the generalized Schur complement $$Z=0$$ Z = 0 under some conditions. As a result, some conclusions are obtained directly from our results.

Duality for multiobjective variational control problems with (Φρ)-invexity

Abstract: In this paper, Mond-Weir and Wolfe type duals for multiobjective variational control problems are formulated. Several duality theorems are established relating efficient solutions of the primal and dual multiobjective variational control problems under $$(\Phi , \rho )$$ ( ? , ? ) -invexity. The results generalize a number of duality results previously established for multiobjective variational control problems under other generalized convexity assumptions.

A note on the eigenvalues of $$g$$g-circulants (and of $$g$$g-Toeplitz, $$g$$g-Hankel matrices)

Abstract: A matrix $$A$$ A of size $$n$$ n is called $$g$$ g -circulant if $$A=[a_{(r-g s)\text { mod } n}]_{r,s=0}^{n-1}$$ A = [ a ( r - g s ) mod n ] r , s = 0 n - 1 , while a matrix $$A$$ A is called $$g$$ g -Toeplitz if its entries obey the rule $$A=[a_{r-g s}]_{r,s=0}^{n-1}$$ A = [ a r - g s ] r , s = 0 n - 1 . In this note we study the eigenvalues of $$g$$ g -circulants and we provide a preliminary asymptotic analysis of the eigenvalue distribution of $$g$$ g -Toeplitz sequences, in the case where the numbers $$\{a_k\}$$ { a k } are the Fourier coefficients of an integrable function $$f$$ f over the domain $$(-\pi ,\pi )$$ ( - ? , ? ) : while the singular value distribution of $$g$$ g -Toeplitz sequences is nontrivial for $$g>1$$ g > 1 , as proved recently, the eigenvalue distribution seems to be clustered at zero and this completely different behaviour is explained by the high nonnormal character of $$g$$ g -Toeplitz sequences when the size is large, $$g>1$$ g > 1 , and $$f$$ f is not identically zero. On the other hand, for negative $$g$$ g the clustering at zero is proven for essentially bounded $$f$$ f . Some numerical evidences are given and critically discussed, in connection with a conjecture concerning the zero eigenvalue distribution of $$g$$ g -Toeplitz sequences with $$g>1$$ g > 1 and Wiener symbol.

Uniform $$l^{1}$$ behavior in the Crank---Nicolson methods for a linear Volterra equation with convex kernel

Abstract: We study the time discretization of the Cauchy problem $$\begin{aligned} u_{t}+\int _{0}^{t}\,\beta (t-s)\,L\,u\,(s)\;ds = 0,\quad t>0, \quad u(0)=u_{0}, \end{aligned}$$ where $$L$$ is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigen system $$\{\lambda _{m},\; \varphi _{m}\}_{m=1}^{\infty }$$ , and the subscript denotes differentiation with respect to $$t$$ . The real valued kernel $$\beta \in \,C(0,\,\infty )\bigcap \,L^{1}(0,\,1)$$ is assumed to be nonnegative, nonincreasing and convex, and $$-\beta ^{\prime }$$ is convex. The equation is discretized in time by Crank---Nicolson method based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature rule, constructed again from the trapezoidal rule, is used to approximate the integral term. The results and methods extend and simulate numerically those introduced by Carr and Hannsgen (SIAM J Math Anal 10:961---984, 1979) and (SIAM J Math Anal 13:459---483, 1982) for integrability with respect to continuous solutions. The uniform error estimates of the discretization in time are derived in the $$ l^{\infty }_{t}(0,\,\infty ;H) $$ norm. Some simple numerical examples illustrate our theoretical error bounds.

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

Abstract: We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax---Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $$\ell ^2$$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax---Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel's conjectures in Turkel (16(2):109---129, 1977).

Multiplicative perturbation bounds for weighted polar decomposition

Abstract: In this paper, we obtain the multiplicative perturbation bounds for generalized nonnegative polar factor of weighted polar decomposition under the weighted unitarily invariant norm, weighed Frobenius norm and weighted spectral norm, respectively. More sharper bounds than the known ones are also obtained under certain condition. Moreover, new multiplicative perturbation bounds for weighted unitary polar factor are also given under the weighted unitarily invariant norm, which improve the existing multiplicative perturbation bounds.

Generating functions for B-Splines with knots in geometric or affine progression

Abstract: We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio $$q$$ q , we construct a PDE for these generating functions in which classical derivatives are replaced by $$q$$ q -derivatives. We then solve this PDE for the generating functions using $$q$$ q -exponential functions. We apply our generating functions to derive some known and some novel identities for B-splines with knots in geometric or affine progression, including a generalization of the Schoenberg identity, formulas for sums and alternating sums, and an explicit expression for the moments of these B-splines. Special cases include both the uniform B-splines with knots at the integers and the nonuniform B-splines with knots at the $$q$$ q -integers.

Asymptotic solvers for second-order differential equation systems with multiple frequencies

Abstract: In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very effective method of dicretizing the differential equation system in question. Numerical experiments illustrate the effectiveness of the asymptotic method in contrast to the standard Runge---Kutta method.

Accurate similarity transformation derived from the discrete Lotka---Volterra system for bidiagonal singular values

Abstract: In this paper, we derive a new similarity transformation from the discrete Lotka---Volterra system by taking an infinite limit of the discretization parameter. The proposed similarity transformation is shown to be applicable for computing singular values of a bidiagonal matrix. The forward stability of it is also verified in floating point arithmetic.

On a posteriori error estimates for the linear triangular finite element

Abstract: Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.