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

Authors:Nils Henrik Risebro; Christoph Schwab; Franziska Weber Abstract: Abstract An error in [4, Theorem 4.1, 4.5, Corollary 4.5] is corrected. There, in the Monte Carlo error bounds for front tracking for scalar conservation laws with random input data, 2-integrability in a Banach space of type 1 was assumed. In providing the corrected convergence rate bounds and error versus work analysis of multilevel Monte Carlo front-tracking methods, we also generalize [4] to q-integrability of the random entropy solution for some \(1<q\le 2\) , allowing possibly infinite variance of the random entropy solutions of the scalar conservation law. PubDate: 2017-08-23 DOI: 10.1007/s10543-017-0670-0

Authors:Iwona Skalna; Milan Hladík Abstract: Abstract We propose a new approach to computing a parametric solution (the so-called p-solution) to parametric interval linear systems. Solving such system is an important part of many scientific and engineering problems involving uncertainties. The parametric solution has many useful properties. It permits to compute an outer solution, an inner estimate of the interval hull solution, and intervals containing the lower and upper bounds of the interval hull solution. It can also be employed for solving various constrained optimisation problems related to the parametric interval linear system. The proposed approach improves both outer and inner bounds for the parametric solution set. In this respect, the new approach is competitive to most of the existing methods for solving parametric interval linear systems. Improved bounds on the parametric solution set guarantees improved bounds for the solutions of related optimisation problems. PubDate: 2017-08-18 DOI: 10.1007/s10543-017-0679-4

Authors:Sotirios E. Notaris Abstract: Abstract It is well known that the Gauss–Kronrod quadrature formula does not always exist with real and distinct nodes and positive weights. In 1996, in an attempt to find an alternative to the Gauss–Kronrod formula for estimating the error of the Gauss quadrature formula, Laurie constructed the anti-Gaussian quadrature formula, which always has real and distinct nodes and positive weights. First, we give a description and prove the most important properties of the anti-Gaussian formula, by applying a different approach than that of Laurie. Then, we consider a measure such that the respective (monic) orthogonal polynomials, above a specific index, satisfy a three-term recurrence relation with constant coefficients. We show that for a measure of this kind the nodes of the anti-Gaussian formula are the zeros of the respective Stieltjes polynomial, while the resulting averaged Gaussian quadrature formula is precisely the corresponding Gauss–Kronrod formula, having elevated degree of exactness. Moreover, we show, by a new method, that a symmetric Gauss–Lobatto quadrature formula is a modified anti-Gaussian formula, and we specialize our results to the measures with constant recurrence coefficients. PubDate: 2017-08-08 DOI: 10.1007/s10543-017-0672-y

Authors:Dario Fasino; Antonio Fazzi Abstract: Abstract The Total Least Squares solution of an overdetermined, approximate linear equation \(Ax \approx b\) minimizes a nonlinear function which characterizes the backward error. We devise a variant of the Gauss–Newton iteration with guaranteed convergence to that solution, under classical well-posedness hypotheses. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix A is modified by a rank-one term. In exact arithmetics, the method is equivalent to an inverse power iteration to compute the smallest singular value of the complete matrix \((A\mid b)\) . Geometric and computational properties of the method are analyzed in detail and illustrated by numerical examples. PubDate: 2017-07-29 DOI: 10.1007/s10543-017-0678-5

Authors:Liviu I. Ignat; Alejandro Pozo Abstract: Abstract In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation. PubDate: 2017-07-26 DOI: 10.1007/s10543-017-0673-x

Authors:Rakesh Kumar Abstract: Abstract In this article, we have proposed a septic B-spline quasi-interpolation (SeBSQI) based numerical scheme for the modified Burgers’ equation. The SeBSQI scheme maintains eighth order accuracy for the smooth solution, but fails to maintain a non-oscillatory profile when the solution has discontinuities or sharp variations. To ensure the non-oscillatory profile of the solution, we have proposed an adaptive SeBSQI (ASeBSQI) scheme for the modified Burgers’ equation. The ASeBSQI scheme maintains higher order accuracy in the smooth regions using SeBSQI approximation and in regions with discontinuities or sharp variations, 5th order weighted essentially non-oscillatory (WENO) reconstruction is used to preserve a non-oscillatory profile. To identify discontinuous or sharp variation regions, a weak local truncation error based smooth indicator is proposed for the modified Burgers’ equation. For the temporal derivative, we have considered the Runge–Kutta method of order four. We have shown numerically that the ASeBSQI scheme preserves the convergence rate of the SeBSQI and it converges to the exact solution with convergence rate eight. We have performed numerical experiments to validate the proposed scheme. The numerical experiments demonstrate an improvement in accuracy and efficiency of the proposed schemes over the WENO5 and septic B-spline collocation schemes. The ASeBSQI scheme is also tested for one-dimensional Euler equations. PubDate: 2017-07-25 DOI: 10.1007/s10543-017-0675-8

Authors:Giampaolo Mele; Elias Jarlebring Abstract: Abstract An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated. PubDate: 2017-07-13 DOI: 10.1007/s10543-017-0671-z

Authors:Patrick Zulian; Teseo Schneider; Kai Hormann; Rolf Krause Abstract: Abstract The discretization of the computational domain plays a central role in the finite element method. In the standard discretization the domain is triangulated with a mesh and its boundary is approximated by a polygon. The boundary approximation induces a geometry-related error which influences the accuracy of the solution. To control this geometry-related error, iso-parametric finite elements and iso-geometric analysis allow for high order approximation of smooth boundary features. We present an alternative approach which combines parametric finite elements with smooth bijective mappings leaving the choice of approximation spaces free. Our approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parameterization domain to the complex computational domain, thus creating a curved mesh of the latter. Numerical examples provide evidence that our method has lower approximation error for domains with complex shapes than the standard finite element method, because we are able to solve the problem directly on the exact domain without having to approximate it. PubDate: 2017-07-04 DOI: 10.1007/s10543-017-0669-6

Authors:Yuji Nakatsukasa Abstract: Abstract The standard approach to computing an approximate SVD of a large-scale matrix is to project it onto lower-dimensional trial subspaces from both sides, compute the SVD of the small projected matrix, and project it back to the original space. This results in a low-rank approximate SVD to the original matrix, and we can then obtain approximate left and right singular subspaces by extracting subsets from the approximate SVD. In this work we assess the quality of the extraction process in terms of the accuracy of the approximate singular subspaces, measured by the angle between the exact and extracted subspaces (relative to the angle between the exact and trial subspaces). The main message is that the extracted approximate subspaces are optimal usually to within a modest constant. PubDate: 2017-06-19 DOI: 10.1007/s10543-017-0665-x

Authors:M. Irene Falcão; Fernando Miranda; Ricardo Severino; M. Joana Soares Abstract: Abstract In this paper we focus on computational aspects associated with polynomial problems in the ring of one-sided quaternionic polynomials. The complexity and error bounds of quaternion arithmetic are considered and several evaluation schemes are analyzed from their complexity point of view. The numerical stability of generalized Horner’s and Goertzel’s algorithms to evaluate polynomials with quaternion floating-point coefficients is addressed. Numerical tests illustrate the behavior of the algorithms from the point of view of performance and accuracy. PubDate: 2017-06-19 DOI: 10.1007/s10543-017-0667-8

Authors:Mohammad Asadzadeh; Christoffer Standar Abstract: Abstract This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov–Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge. PubDate: 2017-06-16 DOI: 10.1007/s10543-017-0666-9

Authors:Xinyan Niu; Jianbo Cui; Jialin Hong; Zhihui Liu Abstract: Abstract We construct stochastic pseudo-symplectic methods and analyze their pseudo-symplectic orders for stochastic Hamiltonian systems with additive noises in this paper. All of these methods are explicit so that the numerical implementations become much easier than implicit methods. Through the numerical experiments, we find that these methods have desired properties in accuracy and stability as well as the preservation of the symplectic structure of the systems. PubDate: 2017-06-15 DOI: 10.1007/s10543-017-0668-7

Authors:Tomoaki Okayama Abstract: Abstract A Sinc-collocation method was proposed by Stenger, who also gave a theoretical analysis of the method in the case of a “scalar” equation. This paper extends the theoretical results to the case of a “system” of equations. Furthermore, this paper proposes a more efficient method by replacing the variable transformation employed in Stenger’s method. The efficiency was confirmed by both a theoretical analysis and some numerical experiments. In addition to the existing and newly proposed Sinc-collocation methods, this paper also gives similar theoretical results for the Sinc-Nyström methods proposed by Nurmuhammad et al. In terms of computational cost, the newly proposed Sinc-collocation method is the most efficient among these methods. PubDate: 2017-06-08 DOI: 10.1007/s10543-017-0663-z