Authors:Martin Halla; Lothar Nannen Pages: 611 - 643 Abstract: We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9549-5 Issue No:Vol. 44, No. 3 (2018)

Authors:M. Esmaeilbeigi; O. Chatrabgoun; M. Shafa Pages: 673 - 691 Abstract: In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9555-7 Issue No:Vol. 44, No. 3 (2018)

Authors:Linghua Chen; Espen Robstad Jakobsen; Arvid Naess Pages: 693 - 721 Abstract: We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in application oriented fields. In this paper we provide a rigorous analysis of the method that covers systems of equations with unbounded coefficients. Working in a natural space for densities, L 1, we obtain stability, consistency, and new convergence results for the method, new well-posedness and semigroup generation results for the related Fokker-Planck-Kolmogorov equation, and a new and rigorous connection to the corresponding probability density functions for both the approximate and the exact problems. To prove the results we combine semigroup and PDE arguments in a new way that should be of independent interest. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9558-4 Issue No:Vol. 44, No. 3 (2018)

Authors:Tao Sun; Penghang Yin; Lizhi Cheng; Hao Jiang Pages: 723 - 744 Abstract: In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM often leads to difficult nonconvex subproblems. To address this issue, we propose to convexify the subproblems through a linearization technique as done in the difference of convex functions algorithm (DCA). By assuming the Kurdyka-Łojasiewicz property, we prove that the resulting algorithm sequentially converges to a critical point. It turns out that in the applications of signal and image processing such as compressed sensing and image denoising, the proposed algorithm usually enjoys closed-form solutions of the subproblems and thus can be very efficient. We provide numerical experiments to demonstrate the effectiveness of our algorithm. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9559-3 Issue No:Vol. 44, No. 3 (2018)

Authors:Sølve Eidnes; Brynjulf Owren; Torbjørn Ringholm Pages: 815 - 839 Abstract: A framework for constructing integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The approach can be used with both finite difference and partition of unity methods, thereby including finite element methods. The schemes are then extended to accommodate r-, h- and p-adaptivity. To illustrate the ideas, the method is applied to the Korteweg–de Vries equation and the sine-Gordon equation. Results from numerical experiments are presented. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9562-8 Issue No:Vol. 44, No. 3 (2018)

Authors:Guillermo Navas-Palencia Pages: 841 - 859 Abstract: We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9565-5 Issue No:Vol. 44, No. 3 (2018)

Authors:Hong Lu; Peter W. Bates; Wenping Chen; Mingji Zhang Pages: 861 - 878 Abstract: We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (−△) α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9564-6 Issue No:Vol. 44, No. 3 (2018)

Authors:Fusheng Lv; Wenchang Sun Pages: 879 - 896 Abstract: We study the signal recovery from unordered partial phaseless frame coefficients. To this end, we introduce the concepts of m-erasure (almost) phase retrievable frames. We show that with an m-erasure (almost) phase retrievable frame, it is possible to reconstruct (almost) all n-dimensional real signals up to a sign from their arbitrary N − m unordered phaseless frame coefficients, where N stands for the element number of the frame. We give necessary and sufficient conditions for a frame to be m-erasure (almost) phase retrievable. Moreover, we give an explicit construction of such frames based on prime numbers. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9566-4 Issue No:Vol. 44, No. 3 (2018)

Authors:Qi Hong; Jiming Wu Pages: 897 - 922 Abstract: In this paper, we study a so-called modified Q 1-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q 1-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h 1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H 1 error estimate of the modified Q 1-finite volume element scheme is obtained without the traditional h 1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9567-3 Issue No:Vol. 44, No. 3 (2018)

Authors:Huadong Gao; Weiwei Sun Pages: 923 - 949 Abstract: A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg-Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = c u r l A as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional nonconvex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains. PubDate: 2018-06-01 DOI: 10.1007/s10444-017-9568-2 Issue No:Vol. 44, No. 3 (2018)

Authors:Scott Congreve; Paul Houston; Ilaria Perugia Abstract: In this article, we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h–refinement) and local basis enrichment (p–refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples. PubDate: 2018-07-16 DOI: 10.1007/s10444-018-9621-9

Authors:Maolin Che; Yimin Wei Abstract: Randomized algorithms provide a powerful tool for scientific computing. Compared with standard deterministic algorithms, randomized algorithms are often faster and robust. The main purpose of this paper is to design adaptive randomized algorithms for computing the approximate tensor decompositions. We give an adaptive randomized algorithm for the computation of a low multilinear rank approximation of the tensors with unknown multilinear rank and analyze its probabilistic error bound under certain assumptions. Finally, we design an adaptive randomized algorithm for computing the tensor train approximations of the tensors. Based on the bounds about the singular values of sub-Gaussian matrices with independent columns or independent rows, we analyze these randomized algorithms. We illustrate our adaptive randomized algorithms via several numerical examples. PubDate: 2018-07-05 DOI: 10.1007/s10444-018-9622-8

Authors:Carmen Gräßle; Michael Hinze Abstract: The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing proper orthogonal decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembly of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced-order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility of our approach, we present a test case of the Cahn–Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids. PubDate: 2018-06-27 DOI: 10.1007/s10444-018-9620-x

Authors:Radu Ioan Boţ; Ernö Robert Csetnek Abstract: We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper, convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on suitable choice of dynamical step sizes. The numerical performances, which can be obtained for different dynamical step size strategies, are compared in the context of solving an image denoising problem. PubDate: 2018-06-25 DOI: 10.1007/s10444-018-9619-3

Authors:Xudong Yao Abstract: In Yao (J. Sci. Comput. 66, 19–40 2016), two Ljusternik-Schnirelman minimax algorithms for capturing multiple free saddle points are developed from well-known Ljusternik-Schnirelman critical point theory, numerical experiment is carried out and global convergence is established. In this paper, a Ljusternik-Schnirelman minimax algorithm for calculating multiple equality constrained saddle points is presented. The algorithm is applied to numerically solve eigen problems. Finally, global convergence for the algorithm is verified. PubDate: 2018-06-22 DOI: 10.1007/s10444-018-9616-6

Authors:Xiaodong Cheng; Jacquelien M. A. Scherpen Abstract: This paper considers the network structure preserving model reduction of power networks with distributed controllers. The studied system and controller are modeled as second-order and first-order ordinary differential equations, which are coupled to a closed-loop model for analyzing the dissimilarities of the power units. By transfer functions, we characterize the behavior of each node (generator or load) in the power network and define a novel notion of dissimilarity between two nodes by the \(\mathcal {H}_{2}\) -norm of the transfer function deviation. Then, the reduction methodology is developed based on separately clustering the generators and loads according to their behavior dissimilarities. The characteristic matrix of the resulting clustering is adopted for the Galerkin projection to derive explicit reduced-order power models and controllers. Finally, we illustrate the proposed method by the IEEE 30-bus system example. PubDate: 2018-06-16 DOI: 10.1007/s10444-018-9617-5

Authors:Hakop Hakopian; Vahagn Vardanyan Abstract: An n-poised node set \(\mathcal {X}\) in the plane is called GC n set if the (bivariate) fundamental polynomial of each node is a product of n linear factors. A line is called k-node line if it passes through exactly k-nodes of \(\mathcal {X}\) . An (n + 1)-node line is called a maximal line. In 1982, Gasca and Maeztu conjectured that every GC n set has a maximal line. Until now the conjecture has been proved only for n ≤ 5. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. It is a simple fact that any maximal line λ is used by all \(\binom {n + 1}{2}\) nodes in \(\mathcal {X} \setminus \lambda \) . We consider the main result of the paper—Bayramyan and Hakopian (Adv. Comput. Math. 43, 607–626, 2017) stating that any n-node line of GC n set is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes, provided that the Gasca-Maeztu conjecture is true. Here, we show that this result is not correct in the case n = 3. Namely, we bring an example of a GC3 set and a 3-node line there which is not used at all. Fortunately, then we were able to establish that this is the only possible counterexample, i.e., the abovementioned result is true for all n ≥ 4. We also characterize the exclusive case n = 3 and present some new results on the maximal lines and the usage of n-node lines in GC n sets. PubDate: 2018-06-12 DOI: 10.1007/s10444-018-9618-4

Authors:Patrick Kürschner Abstract: In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region. PubDate: 2018-06-05 DOI: 10.1007/s10444-018-9608-6

Authors:Cédric Gérot Abstract: When a subdivision scheme is factorised into lifting steps, it admits an in–place and invertible implementation, and it can be the predictor of many multiresolution biorthogonal wavelet transforms. In the regular setting where the underlying lattice hierarchy is defined by ℤ s and a dilation matrix M, such a factorisation should deal with every vertex of each subset in ℤ s /Mℤ s in the same way. We define a subdivision scheme which admits such a factorisation as being uniformly elementary factorable. We prove a necessary and sufficient condition on the directions of the Box spline and the arity of the subdivision for the scheme to admit such a factorisation, and recall some known keys to construct it in practice. PubDate: 2018-06-05 DOI: 10.1007/s10444-018-9612-x

Authors:William Paulsen Abstract: In this paper we will consider the tetration, defined by the equation F(z + 1) = bF(z) in the complex plane with F(0) = 1, for the case where b is complex. A previous paper determined conditions for a unique solution the case where b is real and b > e1/e. In this paper we extend these results to find conditions which determine a unique solution for complex bases. We also develop iteration methods for numerically approximating the function F(z), both for bases inside and outside the Shell-Thron region. PubDate: 2018-06-02 DOI: 10.1007/s10444-018-9615-7