Authors:Pedro G. Massey; Noelia B. Rios; Demetrio Stojanoff Pages: 51 - 86 Abstract: Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\) . We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g i ∥2 = a i for \(i\in \mathbb {I}_{k}\) , and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\ g_{i}-\tilde {g}_{i}\ : \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\) . In this context we show that local minimizers on the set of completions of a convex potential P φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x 2 then P φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD. PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9535-y Issue No:Vol. 44, No. 1 (2018)

Authors:Chang-Yeol Jung; Thien Binh Nguyen Pages: 147 - 174 Abstract: The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-ðœƒ6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-ðœƒ6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-ðœƒ6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem. PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9538-8 Issue No:Vol. 44, No. 1 (2018)

Authors:Liqun Wang; Songming Hou; Liwei Shi Pages: 175 - 193 Abstract: Elliptic interface problems with multi-domains have wide applications in engineering and science. However, it is challenging for most existing methods to solve three-dimensional elliptic interface problems with multi-domains due to local geometric complexity, especially for problems with matrix coefficient and sharp-edged interface. There are some recent work in two dimensions for multi-domains and in three dimensions for two domains. However, the extension to three dimensional multi-domain elliptic interface problems is non-trivial. In this paper, we present an efficient non-traditional finite element method with non-body-fitting grids for three-dimensional elliptic interface problems with multi-domains. Numerical experiments show that this method achieves close to second order accurate in the L ∞ norm for piecewise smooth solutions. PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9539-7 Issue No:Vol. 44, No. 1 (2018)

Authors:Tatyana Sorokina Pages: 227 - 244 Abstract: Bernstein-Bézier techniques for analyzing polynomial spline fields in n variables and their divergence are developed. Dimension and a minimal determining set for continuous piecewise divergence-free spline fields on the Alfeld split of a simplex in ℝ n are obtained using the new techniques, as well as the dimension formula for continuous piecewise divergence-free splines on the Alfeld refinement of an arbitrary simplicial partition in ℝ n . PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9541-0 Issue No:Vol. 44, No. 1 (2018)

Authors:José L. López Pages: 277 - 294 Abstract: We consider the Bessel functions J ν (z) and Y ν (z) for R ν > −1/2 and R z ≥ 0. We derive a convergent expansion of J ν (z) in terms of the derivatives of \((\sin z)/z\) , and a convergent expansion of Y ν (z) in terms of derivatives of \((1-\cos z)/z\) , derivatives of (1 − e −z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments. PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9543-y Issue No:Vol. 44, No. 1 (2018)

Authors:Pedro R. S. Antunes Pages: 351 - 365 Abstract: The method of fundamental solutions (MFS) is a meshless method for solving boundary value problems with some partial differential equations. It allows to obtain highly accurate approximations for the solutions assuming that they are smooth enough, even with small matrices. As a counterpart, the (dense) matrices involved are often ill-conditioned which is related to the well known uncertainty principle stating that it is impossible to have high accuracy and good conditioning at the same time. In this work, we propose a technique to reduce the ill conditioning in the MFS, assuming that the source points are placed on a circumference of radius R. The idea is to apply a suitable change of basis that provides new basis functions that span the same space as the MFS’s, but are much better conditioned. In the particular case of circular domains, the algorithm allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of points sources and R. PubDate: 2018-02-01 DOI: 10.1007/s10444-017-9548-6 Issue No:Vol. 44, No. 1 (2018)

Authors:Zoran Tomljanović; Christopher Beattie; Serkan Gugercin Abstract: We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the \(\mathcal {H}_{2}\) system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the \(\mathcal {H}_{2}\) system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments. PubDate: 2018-04-11 DOI: 10.1007/s10444-018-9605-9

Authors:Erchuan Zhang; Lyle Noakes Abstract: Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C ∞ curve x : [a, b] → G/H, let \(\tilde {x}:[a,b]\rightarrow G\) be the horizontal lifting of x with \(\tilde {x}(a)=e\) , where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction \(V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)\) of \(\dot {\tilde x}(t)\) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector \(\dot {x}(t)\) from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S3. PubDate: 2018-04-02 DOI: 10.1007/s10444-018-9601-0

Authors:Xin Liu; Zhangxin Chen Abstract: In this paper a unified nonconforming virtual element scheme for the Navier-Stokes equations with different dimensions and different polynomial degrees is described. Its key feature is the treatment of general elements including non-convex and degenerate elements. According to the properties of an enhanced nonconforming virtual element space, the stability of this scheme is proved based on the choice of a proper velocity and pressure pair. Furthermore, we establish optimal error estimates in the discrete energy norm for velocity and the L2 norm for both velocity and pressure. Finally, we test some numerical examples to validate the theoretical results. PubDate: 2018-03-28 DOI: 10.1007/s10444-018-9602-z

Authors:Philip Greengard; Vladimir Rokhlin Abstract: We introduce an algorithm for the evaluation of the Incomplete Gamma Function, P(m, x), for all m, x > 0. For small m, a classical recursive scheme is used to evaluate P(m, x), whereas for large m a newly derived asymptotic expansion is used. The number of operations required for evaluation is O(1) for all x and m. Nearly full double and extended precision accuracies are achieved in their respective environments. The performance of the scheme is illustrated via several numerical examples. PubDate: 2018-03-23 DOI: 10.1007/s10444-018-9604-x

Authors:Odysseas Kosmas; Sigrid Leyendecker Abstract: In this work, we present a new derivation of higher order variational integration methods that exploit the phase lag properties for numerical integrations of systems with oscillatory solutions. More specifically, for the derivation of these integrators, the action integral along any curve segment is defined using a discrete Lagrangian that depends on the endpoints of the segment and on a number of intermediate points of interpolation. High order integrators are then obtained by writing down the discrete Lagrangian at any time interval as a weighted sum of the Lagrangians corresponding to a set of the chosen intermediate points. The respective positions and velocities are interpolated using trigonometric functions. The methods derived this way depend on a frequency, which in general needs to be accurately estimated. The new methods, which improve the phase lag characteristics by re-estimating the frequency at every time step, are presented and tested on the general N-body problem as numerical examples. PubDate: 2018-03-23 DOI: 10.1007/s10444-018-9603-y

Authors:Huamin Li; Yuval Kluger; Mark Tygert Abstract: Randomized algorithms provide solutions to two ubiquitous problems: (1) the distributed calculation of a principal component analysis or singular value decomposition of a highly rectangular matrix, and (2) the distributed calculation of a low-rank approximation (in the form of a singular value decomposition) to an arbitrary matrix. Carefully honed algorithms yield results that are uniformly superior to those of the stock, deterministic implementations in Spark (the popular platform for distributed computation); in particular, whereas the stock software will without warning return left singular vectors that are far from numerically orthonormal, a significantly burnished randomized implementation generates left singular vectors that are numerically orthonormal to nearly the machine precision. PubDate: 2018-03-19 DOI: 10.1007/s10444-018-9600-1

Authors:R. Hiptmair; L. Scarabosio; C. Schillings; Ch. Schwab Abstract: We address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain. PubDate: 2018-03-16 DOI: 10.1007/s10444-018-9594-8

Authors:Ole Christensen; Augustus J. E. M. Janssen; Hong Oh Kim; Rae Young Kim Abstract: It is a well-known problem in Gabor analysis how to construct explicitly given dual frames associated with a given frame. In this paper we will consider a class of window functions for which approximately dual windows can be calculated explicitly. The method makes it possible to get arbitrarily close to perfect reconstruction by allowing the modulation parameter to vary. Explicit estimates for the deviation from perfect reconstruction are provided for some of the standard functions in Gabor analysis, e.g., the Gaussian and the two-sided exponential function. PubDate: 2018-03-13 DOI: 10.1007/s10444-018-9595-7

Authors:Urs Vögeli; Khadijeh Nedaiasl; Stefan A. Sauter Abstract: In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation. PubDate: 2018-03-12 DOI: 10.1007/s10444-018-9598-4

Authors:Nisha Sharma; Amiya K. Pani; Kapil K. Sharma Abstract: In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings. PubDate: 2018-03-10 DOI: 10.1007/s10444-018-9596-6

Authors:Rida T. Farouki; Hwan Pyo Moon Abstract: An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent \(\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/ \mathbf {r}^{\prime }(\xi ) \) and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples. PubDate: 2018-03-07 DOI: 10.1007/s10444-018-9599-3

Authors:Peter Benner; Christian Himpe; Tim Mitchell Abstract: The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems. PubDate: 2018-02-27 DOI: 10.1007/s10444-018-9592-x

Authors:Hongjin He; Liusheng Hou; Hong-Kun Xu Abstract: The original publication of this article has an error. PubDate: 2018-02-20 DOI: 10.1007/s10444-018-9591-y

Authors:Bülent Karasözen; Murat Uzunca Abstract: An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD. PubDate: 2018-02-12 DOI: 10.1007/s10444-018-9593-9