Authors:Haitao Leng; Yanping Chen Pages: 367 - 394 Abstract: In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9546-8 Issue No:Vol. 44, No. 2 (2018)

Authors:Alberto Gil C. P. Ramos Pages: 395 - 421 Abstract: The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm–Liouville problems via Fer streamers, put forth in Ramos and Iserles (Numer. Math. 131(3), 541—565 2015). In particular, the discretization schemes are shown to enjoy large step sizes uniform over the entire eigenvalue range and tight error estimates uniform for every eigenvalue. They are made explicit for global orders 4,7,10. In addition, the present paper provides total error estimates that quantify the interplay between the truncation and the discretization in the approach by Fer streamers. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9547-7 Issue No:Vol. 44, No. 2 (2018)

Authors:Roman Chapko; Drossos Gintides; Leonidas Mindrinos Pages: 453 - 476 Abstract: In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9550-z Issue No:Vol. 44, No. 2 (2018)

Authors:Guo-Dong Zhang; Jinjin Yang; Chunjia Bi Pages: 505 - 540 Abstract: In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L 2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9552-x Issue No:Vol. 44, No. 2 (2018)

Authors:Sudeep Kundu; Amiya Kumar Pani Pages: 541 - 570 Abstract: In this article, we discuss global stabilization results for the Burgers’ equation using nonlinear Neumann boundary feedback control law. As a result of the nonlinear feedback control, a typical nonlinear problem is derived. Then, based on C 0-conforming finite element method, global stabilization results for the semidiscrete solution are analyzed. Further, introducing an auxiliary projection, optimal error estimates in \(L^{\infty }(L^{2})\) , \(L^{\infty }(H^{1})\) and \(L^{\infty }(L^{\infty })\) -norms for the state variable are obtained. Moreover, superconvergence results are established for the first time for the feedback control laws, which preserve exponential stabilization property. Finally, some numerical experiments are conducted to confirm our theoretical findings. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9553-9 Issue No:Vol. 44, No. 2 (2018)

Authors:Stéphane Clain; Raphaël Loubère; Gaspar J. Machado Pages: 571 - 607 Abstract: We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9556-6 Issue No:Vol. 44, No. 2 (2018)

Authors:Stéphane Clain; Raphaël Loubère; Gaspar J. Machado Pages: 609 - 609 Abstract: During typesetting, Figs. 8 and 21 got corrupted and the images shown in the online published version are not correct. The original publication was updated. PubDate: 2018-04-01 DOI: 10.1007/s10444-017-9563-7 Issue No:Vol. 44, No. 2 (2018)

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

Authors:Jichun Li; Meng Chen; Min Chen Abstract: In this paper, we develop both a fourth order explicit scheme and a compact implicit scheme for solving the metamaterial Maxwell’s equations. A systematic technique is introduced to prove stability and error estimate for both schemes. Numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel, and provide the first error estimate for the fourth order finite difference methods for Maxwell’s equations. PubDate: 2018-05-28 DOI: 10.1007/s10444-018-9614-8

Authors:Ion Victor Gosea; Mihaly Petreczky; Athanasios C. Antoulas; Christophe Fiter Abstract: We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians. PubDate: 2018-05-21 DOI: 10.1007/s10444-018-9610-z

Authors:James Bremer Abstract: We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel’s equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between 0 and 1,000,000,000 at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has at least three advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel’s differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions is publicly available. PubDate: 2018-05-07 DOI: 10.1007/s10444-018-9613-9

Authors:Yuezheng Gong; Jia Zhao; Qi Wang Abstract: We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions. PubDate: 2018-05-05 DOI: 10.1007/s10444-018-9597-5

Authors:Andrea Carracedo Rodriguez; Serkan Gugercin; Jeff Borggaard Abstract: Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian. PubDate: 2018-05-01 DOI: 10.1007/s10444-018-9611-y

Authors:Shinya Miyajima Abstract: Two numerical algorithms for computing interval matrices containing the matrix exponential are proposed. The first algorithm is based on a numerical spectral decomposition and requires only cubic complexity under some assumptions. The second algorithm is based on a numerical Jordan decomposition and applicable even for defective matrices. Numerical results show the effectiveness and robustness of the algorithms. PubDate: 2018-04-26 DOI: 10.1007/s10444-018-9609-5

Authors:Mariano Franco-de-Leon; John Lowengrub Abstract: In this paper, we implement interface tracking methods for the evolution of 2-D curves that follow Airy flow, a curvature-dependent dispersive geometric evolution law. The curvature of the curve satisfies the modified Korteweg de Vries equation, a dispersive non-linear soliton equation. We present a fully discrete space-time analysis of the equations (proof of convergence) and numerical evidence that confirms the accuracy, convergence, efficiency, and stability of the methods. PubDate: 2018-04-26 DOI: 10.1007/s10444-018-9607-7

Authors:Lucia Romani; Francesca Montagner Abstract: We introduce a new class of Pythagorean-Hodograph (PH) space curves - called Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) space curves - that are defined over a six-dimensional space mixing algebraic and trigonometric polynomials. After providing a general definition for this new class of curves, their quaternion representation is introduced and the fundamental properties are discussed. Then, as previously done with their quintic polynomial counterpart, a constructive approach to solve the first-order Hermite interpolation problem in ℝ3 is provided. Comparisons with the polynomial case are illustrated to point out the greater flexibility of ATPH curves with respect to polynomial PH curves. PubDate: 2018-04-23 DOI: 10.1007/s10444-018-9606-8

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