Authors:Bolesław Kacewicz Pages: 1325 - 1344 Abstract: We study potential advantages of adaptive mesh point selection for the solution of systems of initial value problems. For an optimal order discretization method, we propose an algorithm for successive selection of the mesh points, which only requires evaluations of the right-hand side function. The selection (asymptotically) guarantees that the maximum local error of the method does not exceed a prescribed level. The usage of the algorithm is not restricted to the chosen method; it can also be applied with any method from a general class. We provide a rigorous analysis of the cost of the proposed algorithm. It is shown that the cost is almost minimal, up to absolute constants, among all mesh selection algorithms. For illustration, we specify the advantage of the adaptive mesh over the uniform one. Efficiency of the adaptive algorithm results from automatic adjustment of the successive mesh points to the local behavior of the solution. Some numerical results illustrating theoretical findings are reported. PubDate: 2018-10-01 DOI: 10.1007/s10444-017-9584-2 Issue No:Vol. 44, No. 5 (2018)

Authors:Harald Garcke; Michael Hinze; Christian Kahle; Kei Fong Lam Pages: 1345 - 1383 Abstract: We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization. PubDate: 2018-10-01 DOI: 10.1007/s10444-018-9586-8 Issue No:Vol. 44, No. 5 (2018)

Authors:Michael O’Neil Pages: 1385 - 1409 Abstract: The Laplace-Beltrami problem ΔΓψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderón projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles. PubDate: 2018-10-01 DOI: 10.1007/s10444-018-9587-7 Issue No:Vol. 44, No. 5 (2018)

Authors:Ana Alonso Rodríguez; Enrico Bertolazzi; Riccardo Ghiloni; Ruben Specogna Pages: 1411 - 1440 Abstract: We present an efficient algorithm for the construction of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of \(\overline {\Omega }\) the idea is to find, first g different 1-boundaries of \(\overline {\Omega }\) with supports contained in ∂Ω whose homology classes in \(\mathbb {R}^{3} \setminus {\Omega }\) form a basis of \(H_{1}(\mathbb {R}^{3} \setminus {\Omega };\mathbb {Z})\) , and then to construct a set of 2-chains in \(\overline {\Omega }\) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in \(\overline {\Omega }\) modulo ∂Ω form a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) . We devise a simple procedure for the construction of the required set of 1-boundaries of \(\overline {\Omega }\) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms. PubDate: 2018-10-01 DOI: 10.1007/s10444-018-9588-6 Issue No:Vol. 44, No. 5 (2018)

Authors:T. M. Dunster; A. Gil; J. Segura Pages: 1441 - 1474 Abstract: Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials \(L_{n}^{(\alpha )}(x)\) , as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of \(L_{n}^{(\alpha )}(x)\) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided. PubDate: 2018-10-01 DOI: 10.1007/s10444-018-9589-5 Issue No:Vol. 44, No. 5 (2018)

Authors:R. Hiptmair; L. Scarabosio; C. Schillings; Ch. Schwab Pages: 1475 - 1518 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: 2018-10-01 DOI: 10.1007/s10444-018-9594-8 Issue No:Vol. 44, No. 5 (2018)

Authors:Ole Christensen; Augustus J. E. M. Janssen; Hong Oh Kim; Rae Young Kim Pages: 1519 - 1535 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: 2018-10-01 DOI: 10.1007/s10444-018-9595-7 Issue No:Vol. 44, No. 5 (2018)

Authors:Nisha Sharma; Amiya K. Pani; Kapil K. Sharma Pages: 1537 - 1571 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: 2018-10-01 DOI: 10.1007/s10444-018-9596-6 Issue No:Vol. 44, No. 5 (2018)

Authors:Yuezheng Gong; Jia Zhao; Qi Wang Pages: 1573 - 1600 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: 2018-10-01 DOI: 10.1007/s10444-018-9597-5 Issue No:Vol. 44, No. 5 (2018)

Authors:Urs Vögeli; Khadijeh Nedaiasl; Stefan A. Sauter Pages: 1601 - 1626 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: 2018-10-01 DOI: 10.1007/s10444-018-9598-4 Issue No:Vol. 44, No. 5 (2018)

Authors:Rida T. Farouki; Hwan Pyo Moon Pages: 1627 - 1650 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: 2018-10-01 DOI: 10.1007/s10444-018-9599-3 Issue No:Vol. 44, No. 5 (2018)

Authors:Huamin Li; Yuval Kluger; Mark Tygert Pages: 1651 - 1672 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: 2018-10-01 DOI: 10.1007/s10444-018-9600-1 Issue No:Vol. 44, No. 5 (2018)

Authors:Erchuan Zhang; Lyle Noakes Pages: 1673 - 1686 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: 2018-10-01 DOI: 10.1007/s10444-018-9601-0 Issue No:Vol. 44, No. 5 (2018)

Authors:Xianyi Zeng Abstract: We propose a general hybrid-variable (HV) framework to solve linear advection equations by utilizing both cell-average approximations and nodal approximations. The construction is carried out for 1D problems, where the spatial discretization for cell averages is obtained from the integral form of the governing equation whereas that for nodal values is constructed using hybrid-variable discrete differential operators (HV-DDO); explicit Runge-Kutta methods are employed for marching the solutions in time. We demonstrate the connection between the HV-DDO and Hermite interpolation polynomials, and show that it can be constructed to arbitrary order of accuracy. In particular, we derive explicit formula for the coefficients to achieve the optimal order of accuracy given any compact stencil of the HV-DDO. The superconvergence of the proposed HV methods is then proved: these methods have one-order higher spatial accuracy than the designed order of the HV-DDO; in contrast, for conventional methods that only utilize one type of variables, the two orders are the same. Hence, the proposed method can potentially achieve higher-order accuracy given the same computational cost, comparing to existing finite difference methods. We then prove the linear stability of sample HV methods with up to fifth-order accuracy in the case of Cauchy problems. Next, we demonstrate how the HV methods can be extended to 2D problems as well as nonlinear conservation laws with smooth solutions. The performance of the sample HV methods are assessed by extensive 1D and 2D benchmark tests of linear advection equations, the nonlinear Euler equations, and the nonlinear Buckely-Leverett equation. PubDate: 2018-11-10 DOI: 10.1007/s10444-018-9647-z

Authors:Hui Liang; Martin Stynes Abstract: General Riemann-Liouville linear two-point boundary value problems of order αp, where n − 1 < αp < n for some positive integer n, are investigated on the interval [0,b]. It is shown first that the natural degree of regularity to impose on the solution y of the problem is \(y\in C^{n-2}[0,b]\) and \(D^{\alpha _{p}-1}y\in C[0,b]\) , with further restrictions on the behavior of the derivatives of y(n− 2) (these regularity conditions differ significantly from the natural regularity conditions in the corresponding Caputo problem). From this regularity, it is deduced that the most general choice of boundary conditions possible is \(y(0) = y^{\prime }(0) = {\dots } = y^{(n-2)}(0) = 0\) and \({\sum }_{j = 0}^{n_{1}}\beta _{j}y^{(j)}(b_{1}) =\gamma \) for some constants βj and γ, with b1 ∈ (0,b] and \(n_{1}\in \{0, 1, \dots , n-1\}\) . A wide class of transformations of the problem into weakly singular Volterra integral equations (VIEs) is then investigated; the aim is to choose the transformation that will yield the most accurate results when the VIE is solved using a collocation method with piecewise polynomials. Error estimates are derived for this method and for its iterated variant. Numerical results are given to support the theoretical conclusions. PubDate: 2018-11-08 DOI: 10.1007/s10444-018-9645-1

Authors:Parag Bobade; Suprotim Majumdar; Savio Pereira; Andrew J. Kurdila; John B. Ferris Abstract: This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown or uncertain nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. Under the assumption that full state measurements are available, this paper (1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, (2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and (3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis. PubDate: 2018-10-30 DOI: 10.1007/s10444-018-9639-z

Authors:Jun Wang; Leslie Greengard Abstract: We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes. PubDate: 2018-10-24 DOI: 10.1007/s10444-018-9641-5

Authors:Xiao Wang; Wenpeng Shang; Xiaofan Li; Jinqiao Duan; Yanghong Huang Abstract: Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology, and more. In this work, we consider the Fokker-Planck equation (FPE) due to one-dimensional asymmetric Lévy motion, which is a non-local partial differential equation. We present an accurate numerical quadrature for the singular integrals in the non-local FPE and develop a fast summation method to reduce the order of the complexity from O(J2) to \(O(J\log J)\) in one time step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises, and the domain size. PubDate: 2018-10-24 DOI: 10.1007/s10444-018-9642-4

Authors:Xiao Tang; Aiguo Xiao Abstract: This study concerns the approximation of some stochastic integrals used in the strong second-order methods for several classes of stochastic differential equations. An explicit construction of the asymptotically optimal approximation (in the mean-square sense) to these stochastic integrals is proposed based on a Karhunen-Loève expansion of a Wiener process. This asymptotically optimal approximation is more efficient by comparison with the Fourier series approximation introduced by Kloeden and Platen (1992) and the Taylor approximation introduced by Milstein and Tretyakov (2004). In the numerical test part, we replace the stochastic integrals appearing in the strong second-order methods with our corresponding approximations. The numerical results show that those strong second-order methods can perform very well by using our approximation method. PubDate: 2018-10-24 DOI: 10.1007/s10444-018-9638-0