Authors:Miloud Sadkane; Roger B. Sidje Pages: 609 - 628 Abstract: Abstract Although the principle of alternating maximization is well known in the optimization literature, it has not been used before in the context of calculating the hump of the matrix exponential. We propose a method that applies alternating maximization in this particular context, and we show that it has a number of advantages over traditional Newton-like methods. We establish convergence results that fit this context with mild assumptions than would otherwise be the case in general optimization problems. We conduct numerical tests to complement the theory and they show convergence in just a few iterations. PubDate: 2017-09-01 DOI: 10.1007/s10543-016-0644-7 Issue No:Vol. 57, No. 3 (2017)
Authors:Helle Hallik; Peeter Oja Pages: 629 - 648 Abstract: Abstract The histopolation with quadratic/linear rational splines of class \(C^2\) is studied. Such kind of splines keep the sign of its second derivative on the whole interval and, consequently, the given histogram should be strictly convex or strictly concave. The grid points of the histogram and suitable number of the spline knots between them are supposed to place arbitrarily. The uniqueness of such an histopolant is established. It is shown that the histopolant may not exist but some sufficient conditions for the existence are given. Presented numerical results confirm their adequacy. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0645-1 Issue No:Vol. 57, No. 3 (2017)
Authors:Xiaofei Zhao Pages: 649 - 683 Abstract: Abstract We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0646-0 Issue No:Vol. 57, No. 3 (2017)
Authors:S. Yeganeh; R. Mokhtari; J. S. Hesthaven Pages: 685 - 707 Abstract: Abstract This paper is devoted to determining a space-dependent source term in an inverse problem of the time-fractional diffusion equation. We use a method based on a finite difference scheme in time and a local discontinuous Galerkin method in space and investigate the numerical stability and convergence of the proposed method. Finally, various numerical examples are used illustrate the effectiveness and accuracy of the method. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0648-y Issue No:Vol. 57, No. 3 (2017)
Authors:Xingjun Luo; Chunmei Zeng; Suhang Yang; Rong Zhang Pages: 709 - 730 Abstract: Abstract Multiscale collocation methods are developed for solving ill-posed Fredholm integral equations of the first kind in Banach spaces, if the associated resolvent integral operator fulfils a condition with respect to a interval. We apply a multiscale collocation method with a matrix compression strategy to discretize the integral equation of the second kind obtained by using the Lavrentiev regularization from the original ill-posed integral equation and then use the multilevel augmentation method to solve the resulting discrete equation. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0649-x Issue No:Vol. 57, No. 3 (2017)
Authors:Guang-Da Hu; Taketomo Mitsui Pages: 731 - 752 Abstract: Abstract We are concerned with stability of numerical methods for delay differential systems of neutral type. In particular, delay-dependent stability of numerical methods is investigated. By means of the H-matrix norm, a necessary and sufficient condition for the asymptotic stability of analytic solution of linear neutral differential systems is derived. Then, based on the argument principle, sufficient conditions for delay-dependent stability of Runge–Kutta and linear multi-step methods are presented, respectively. Furthermore, two algorithms are provided for checking delay-dependent stability of analytical and numerical solutions, respectively. Numerical examples are given to illustrate the main results. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0650-4 Issue No:Vol. 57, No. 3 (2017)
Authors:Ludwig Gauckler; Harry Yserentant Pages: 753 - 770 Abstract: Abstract The chemical master equation is a differential equation to model stochastic reaction systems. Its solutions are nonnegative and \(\ell ^1\) -contractive which is inherently related to their interpretation as probability densities. In this note, numerical discretizations of arbitrarily high order are discussed and analyzed that preserve both of these properties simultaneously and without any restriction on the discretization parameters. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0651-3 Issue No:Vol. 57, No. 3 (2017)
Authors:Sebastian Franz Pages: 771 - 786 Abstract: Abstract We present a minor modification of of the stabilisation term in the local projection stabilisation (LPS) method for stabilising singularly perturbed problems on layer-adapted meshes, such that the error can be estimated uniformly and in optimal order in the associated LPS-norm, which is stronger than the standard energy norm. Numerical results confirm the theoretical findings. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0652-2 Issue No:Vol. 57, No. 3 (2017)
Authors:Axel Målqvist; Tony Stillfjord Pages: 787 - 810 Abstract: Abstract We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and first-order in time. The theoretical results are verified by several numerical experiments in two and three dimensions. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0653-1 Issue No:Vol. 57, No. 3 (2017)
Authors:Christian Mehl; Volker Mehrmann; Punit Sharma Pages: 811 - 843 Abstract: Abstract We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0654-0 Issue No:Vol. 57, No. 3 (2017)
Authors:Guangning Tan; Nedialko S. Nedialkov; John D. Pryce Pages: 845 - 865 Abstract: Abstract Structural analysis (SA) of a system of differential-algebraic equations (DAEs) is used to determine its index and which equations to be differentiated and how many times. Both Pantelides’s algorithm and Pryce’s \(\varSigma \) -method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates \(\varSigma \) -method’s failures and presents two conversion methods for fixing them. Under certain conditions, both methods reformulate a DAE system on which the \(\varSigma \) -method fails into a locally equivalent problem on which SA is more likely to succeed. Aiming at achieving global equivalence between the original DAE system and the converted one, we provide a rationale for choosing a conversion from the applicable ones. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0655-z Issue No:Vol. 57, No. 3 (2017)
Authors:Georg Muntingh Pages: 867 - 900 Abstract: Abstract Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc–Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric m-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact Hölder regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0656-y Issue No:Vol. 57, No. 3 (2017)
Authors:Tiago Montanher; Ferenc Domes; Hermann Schichl; Arnold Neumaier Pages: 901 - 926 Abstract: Abstract An interval union is a finite set of closed and disjoint intervals. In this paper we introduce the interval union Gauss–Seidel procedure to rigorously enclose the solution set of linear systems with uncertainties given by intervals or interval unions. We also present the interval union midpoint and Gauss–Jordan preconditioners. The Gauss–Jordan preconditioner is used in a mixed strategy to improve the quality and efficiency of the algorithm. Numerical experiments on interval linear systems generated at random show the capabilities of our approach. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0657-x Issue No:Vol. 57, No. 3 (2017)
Authors:Marko Lange; Siegfried M. Rump Pages: 927 - 941 Abstract: Abstract Standard Wilkinson-type error estimates of floating-point algorithms involve a factor \(\gamma _k:=k\mathbf {u}/(1-k\mathbf {u})\) for \(\mathbf {u}\) denoting the relative rounding error unit of a floating-point number system. Recently, it was shown that, for many standard algorithms such as matrix multiplication, LU- or Cholesky decomposition, \(\gamma _k\) can be replaced by \(k\mathbf {u}\) , and the restriction on k can be removed. However, the arguments make heavy use of specific properties of both the underlying set of floating-point numbers and the corresponding arithmetic. In this paper, we derive error estimates for the summation of real numbers where each sum is afflicted with some perturbation. Recent results on floating-point summation follow as a corollary, in particular error estimates for rounding to nearest and for directed rounding. Our new estimates are sharp and unveil the necessary properties of floating-point schemes to allow for a priori estimates of summation with a factor omitting higher order terms. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0658-9 Issue No:Vol. 57, No. 3 (2017)
Authors:Annika Lang; Andreas Petersson; Andreas Thalhammer Abstract: Abstract The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler–Maruyama, Milstein, Crank–Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory. PubDate: 2017-09-14 DOI: 10.1007/s10543-017-0684-7
Authors:Tomasz Hrycak; Sebastian Schmutzhard Abstract: Abstract This paper studies an approximation to the Chebyshev polynomial \(T_n\) computed via a three-term recurrence in floating-point arithmetic. It is shown that close to either endpoint of the interval \([-1, 1]\) , the numerical approximation coincides with the line tangent to \(T_n\) at that endpoint. From this representation new upper and lower error bounds are derived. PubDate: 2017-09-13 DOI: 10.1007/s10543-017-0683-8
Authors:Jing Gao; Arieh Iserles Abstract: Abstract The Filon–Clenshaw–Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency \(\omega \) it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC \(+\) , which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC \(+\) shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages. PubDate: 2017-09-12 DOI: 10.1007/s10543-017-0682-9
Authors:Huiqing Xie Abstract: Abstract A new method is proposed to compute the eigenvector derivative of a quadratic eigenvalue problem (QEP) analytically dependent on a parameter. It avoids the linearization of the QEP. The proposed method can be seen as an improved incomplete modal method. Only a few eigenvectors of the QEP are required. The contributions of other eigenvectors to the desired eigenvector derivative are obtained by an iterative scheme. From this point of view, our method also can be seen as an iterative method. The convergence properties of the proposed method are analyzed. The techniques to accelerate the proposed method are provided. A strategy is developed for simultaneously computing several eigenvector derivatives by the proposed method. Finally some numerical examples are given to demonstrate the efficiency of our method. PubDate: 2017-08-31 DOI: 10.1007/s10543-017-0680-y
Authors:Markus Wahlsten; Jan Nordström Abstract: Abstract The two dimensional advection–diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry. PubDate: 2017-08-28 DOI: 10.1007/s10543-017-0676-7
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