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Publisher: Springer-Verlag (Total: 2355 journals)

 Advances in Computational Mathematics   [SJR: 1.255]   [H-I: 44]   [15 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1572-9044 - ISSN (Online) 1019-7168    Published by Springer-Verlag  [2355 journals]
• Theoretical investigations of the new Cokriging method for
variable-fidelity surrogate modeling
• Authors: Anna Bertram; Ralf Zimmermann
Abstract: Abstract Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling: A mandatory requirement for the well-posedness of the Cokriging emulator is the positive definiteness of the associated Cokriging correlation matrix. Spatial correlations are usually modeled by positive definite correlation kernels, which are guaranteed to yield positive definite correlation matrices for mutually distinct sample points. However, in applications, low-fidelity information is often available at high-fidelity sample points and the Cokriging predictor may benefit from the additional information provided by such an inclusive sampling. We investigate the positive definiteness of the Cokriging covariance matrix in both of the aforementioned cases and derive sufficient conditions for the well-posedness of the Cokriging predictor. The approximation quality of the Cokriging predictor is highly dependent on a number of model- and hyper-parameters. These parameters are determined by the method of maximum likelihood estimation. For standard Kriging, closed-form optima of the model parameters along hyper-parameter profile lines are known. Yet, these do not readily transfer to the setting of Cokriging, since additional parameters arise, which exhibit a mutual dependence. In previous work, this obstacle was tackled via a numerical optimization. Here, we derive closed-form optima for all Cokriging model parameters along hyper-parameter profile lines. The findings are illustrated by numerical experiments.
PubDate: 2018-01-11
DOI: 10.1007/s10444-017-9585-1

• Decomposition and reconstruction of multidimensional signals by radial
functions with tension parameters
• Authors: Mira Bozzini; Christophe Rabut; Milvia Rossini
Abstract: Abstract The aim of the paper is to construct a multiresolution analysis of L2(IR d ) based on generalized kernels which are fundamental solutions of differential operators of the form $$\boldsymbol {\prod }_{\ell = 1}^{m}(-{\Delta }+\kappa _{\ell }^{2}\,I)$$ . We study its properties and provide a set of pre-wavelets associated with it, as well as the filters which are indispensable to perform decomposition and reconstruction of a given signal, being very useful in applied problems thanks to the presence of the tension parameters κ ℓ .
PubDate: 2018-01-03
DOI: 10.1007/s10444-017-9571-7

• A fully diagonalized spectral method using generalized Laguerre functions
on the half line
• Authors: Fu-Jun Liu; Zhong-Qing Wang; Hui-Yuan Li
Pages: 1227 - 1259
Abstract: Abstract A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9522-3
Issue No: Vol. 43, No. 6 (2017)

• Solving F ( z + 1) = b F(z) in the complex plane
• Authors: William Paulsen; Samuel Cowgill
Pages: 1261 - 1282
Abstract: Abstract The generalized tetration, defined by the equation F(z+1) = b F(z) in the complex plane with F(0) = 1, is considered for any b > e 1/e . By comparing other solutions to Kneser’s solution, natural conditions are found which force Kneser’s solution to be the unique solution to the equation. This answers a conjecture posed by Trappmann and Kouznetsov. Also, a new iteration method is developed which numerically approximates the function F(z) with an error of less than 10−50 for most bases b, using only 180 nodes, with each iteration gaining one or two places of accuracy. This method can be applied to other problems involving the Abel equation.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9524-1
Issue No: Vol. 43, No. 6 (2017)

• A simple finite element method for the Stokes equations
• Authors: Lin Mu; Xiu Ye
Pages: 1305 - 1324
Abstract: Abstract The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9526-z
Issue No: Vol. 43, No. 6 (2017)

• Computing ultra-precise eigenvalues of the Laplacian within polygons
• Authors: Robert Stephen Jones
Pages: 1325 - 1354
Abstract: Abstract The classic eigenvalue problem of the Laplace operator inside a variety of polygons is numerically solved by using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper. It is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, often to well over a hundred digits, or even thousands of digits. To work well, geometric symmetry must be exploited. The de-symmetrized fundamental domains (usually triangular) considered here have at most one non-analytic vertex. Dirichlet, Neumann, and periodic-type edge conditions are independently imposed on each symmetry-reduced polygon edge. The method of particular solutions is used whereby an eigenfunction is expanded in an N-term Fourier-Bessel series about the non-analytic vertex and made to match at a set of N points on the boundary. Under the right conditions, the so-called point-matching determinant has roots that approximate eigenvalues. A key observation is that by increasing the number of terms in the expansion, the approximate eigenvalue may be made to alternate above and below, while approaching what is presumed to be the exact eigenvalue. This alternation effectively provides a new method to bound eigenvalues, by inspection. Specific examples include Dirichlet and Neumann eigenvalues within polygons with re-entrant angles (L-shape, cut-square, 5-point star) and the regular polygons. Thousand-digit results are reported for the lowest Dirichlet eigenvalues of the L-shape, and regular pentagon and hexagon.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9527-y
Issue No: Vol. 43, No. 6 (2017)

• New approximations to the principal real-valued branch of the Lambert W
-function
• Authors: Roberto Iacono; John P. Boyd
Pages: 1403 - 1436
Abstract: Abstract The Lambert W-function is the solution to the transcendental equation W(x)e W(x) = x. It has two real branches, one of which, for x ∈ [−1/e, ∞], is usually denoted as the principal branch. On this branch, the function grows from − 1 to infinity, logarithmically at large x. The present work is devoted to the construction of accurate approximations for the principal branch of the W-function. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximum relative error smaller than 5 × 10−3. Starting from it, machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme, here examined for the first time, which is more efficient than standard Newton’s iteration at large x. Analytic bounds for W are also constructed, for x > e, which are much tighter than those currently available. It is noted that the exponential of the upper bounding function yields an upper bound for the prime counting function π(n) that is better than the well-known Chebyshev’s estimates at large n. Finally, the construction of accurate approximations to W based on Chebyshev spectral theory is discussed; the difficulties involved are highlighted, and methods to overcome them are presented.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9530-3
Issue No: Vol. 43, No. 6 (2017)

• An h - p version of the continuous Petrov-Galerkin method for Volterra
delay-integro-differential equations
• Authors: Lina Wang; Lijun Yi
Pages: 1437 - 1467
Abstract: Abstract We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L 2-, H 1- and L ∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9531-2
Issue No: Vol. 43, No. 6 (2017)

• Inverted finite elements for div-curl systems in the whole space
• Authors: Tahar Z. Boulmezaoud; Keltoum Kaliche; Nabil Kerdid
Pages: 1469 - 1489
Abstract: Abstract We use inverted finite element method (IFEM) for computing three-dimensional vector potentials and for solving div-curl systems in the whole space $$\mathbb {R}^{3}$$ . IFEM is substantially different from the existing approaches since it is a non truncature method which preserves the unboundness of the domain. After developping the method, we analyze its convergence in term of weighted norms. We then give some three-dimensional numerical results which demonstrate the efficiency and the accuracy of the method and confirm its convergence.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9532-1
Issue No: Vol. 43, No. 6 (2017)

• A numerical method for solving the time fractional Schrödinger
equation
• Authors: Na Liu; Wei Jiang
Abstract: Abstract In this article, we proposed a new numerical method to obtain the approximation solution for the time-fractional Schrödinger equation based on reproducing kernel theory and collocation method. In order to overcome the weak singularity of typical solutions, we apply the integral operator to both sides of differential equation and yield a integral equation. We divided the solution of this kind equation into two parts: imaginary part and real part, and then derived the approximate solutions of the two parts in the form of series with easily computable terms in the reproducing kernel space. New bases of reproducing kernel spaces are constructed and the existence of approximate solution is proved. Numerical examples are given to show the accuracy and effectiveness of our approach.
PubDate: 2017-12-27
DOI: 10.1007/s10444-017-9579-z

• Local and parallel finite element algorithm based on the partition of
unity method for the incompressible MHD flow
• Authors: Xiaojing Dong; Yinnian He; Hongbo Wei; Yuhong Zhang
Abstract: Abstract Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.
PubDate: 2017-12-27
DOI: 10.1007/s10444-017-9582-4

• Low dissipative entropy stable schemes using third order WENO and TVD
reconstructions
• Authors: Biswarup Biswas; Ritesh Kumar Dubey
Abstract: Abstract A low dissipative framework is given to construct high order entropy stable flux by addition of suitable numerical diffusion operator into entropy conservative flux. The framework is robust in the sense that it allows the use of high order reconstructions which satisfy the sign property only across the discontinuities. The third order weighted essentially non-oscillatory (WENO) interpolations and high order total variation diminishing (TVD) reconstructions are shown to satisfy the sign property across discontinuities. Third order accurate entropy stable schemes are constructed by using third order WENO and high order TVD reconstructions procedures in the diffusion operator. These schemes are efficient and less diffusive since the diffusion is actuated only in the sign stability region of the used reconstruction which includes discontinuities. Numerical results with constructed schemes for various test problems are given which show the third order accuracy and less dissipative nature of the schemes.
PubDate: 2017-12-27
DOI: 10.1007/s10444-017-9576-2

• Adaptive mesh selection asymptotically guarantees a prescribed local error
for systems of initial value problems
• Authors: Bolesław Kacewicz
Abstract: 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: 2017-12-26
DOI: 10.1007/s10444-017-9584-2

• Nonlinear approximation with nonstationary Gabor frames
• Authors: Emil Solsbæk Ottosen; Morten Nielsen
Abstract: Abstract We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions generalizes to the nonstationary case, providing perfect reconstruction and an FFT based implementation for compactly supported window functions sampled at a certain density. It is also known that for some signal classes, NSGFs with flexible time resolution tend to provide sparser expansions than can be obtained with classical Gabor frames. In this article we show, for the continuous case, that sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an associated decomposition space. In this way we characterize signals with sparse expansions relative to NSGFs with flexible time resolution. Based on this characterization we prove an upper bound on the approximation error occurring when thresholding the coefficients of the corresponding frame expansions. We complement the theoretical results with numerical experiments, estimating the rate of approximation obtained from thresholding the coefficients of both stationary and nonstationary Gabor expansions.
PubDate: 2017-12-26
DOI: 10.1007/s10444-017-9577-1

• A boundary integral equation method for mode elimination and vibration
confinement in thin plates with clamped points
• Authors: Alan E. Lindsay; Bryan Quaife; Laura Wendelberger
Abstract: Abstract We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green’s functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.
PubDate: 2017-12-26
DOI: 10.1007/s10444-017-9580-6

• Atomistic-continuum multiscale modelling of magnetisation dynamics at
non-zero temperature
• Authors: Doghonay Arjmand; Mikhail Poluektov; Gunilla Kreiss
Abstract: Abstract In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.
PubDate: 2017-12-23
DOI: 10.1007/s10444-017-9575-3

• A note on “Reguralizers for structured sparsity”
• Authors: F. Lara
Abstract: Abstract In this note, the notion of admissible sets contained in the strictly positive orthant introduced in Micchelli et al. (Adv. Comp. Math. 38(3), 455–489 2013) is analyzed. This notion was used to generalize theoretical results and optimization methods for structured sparsity. Unfortunately, we will prove that there is no generalization using admissible sets.
PubDate: 2017-12-19
DOI: 10.1007/s10444-017-9583-3

• Fast elliptic curve point multiplication based on binary and binary
• Authors: Denis Khleborodov
Abstract: Abstract This article presents two methods for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. Two new effective algorithms have been presented: one of them is based on a binary Non-Adjacent Form of scalar representation and another one on a binary of scalar representation method. All algorithms were developed based on simple and composite operations with point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. Theorems concerning their computational complexity are formulated and proved for these new algorithms. In the end of this article comparative analysis of both new algorithms among themselves and previously known algorithms are represented.
PubDate: 2017-12-16
DOI: 10.1007/s10444-017-9581-5

• Cardinal interpolation with general multiquadrics: convergence rates
• Authors: Keaton Hamm; Jeff Ledford
Abstract: Abstract This article pertains to interpolation of Sobolev functions at shrinking lattices $$h\mathbb {Z}^{d}$$ from L p shift-invariant spaces associated with cardinal functions related to general multiquadrics, ϕ α, c (x) := ( x 2 + c 2) α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, L p error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h k ) for functions with derivatives of order up to k in L p , $$1<p<\infty$$ ). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.
PubDate: 2017-12-09
DOI: 10.1007/s10444-017-9578-0

• A partially isochronous splitting algorithm for three-block separable
convex minimization problems
• Authors: Hongjin He; Liusheng Hou; Hong-Kun Xu
Abstract: Abstract During the last decade, the state-of-the-art alternating direction method of multipliers (ADMM) has successfully been used to solve many two-block separable convex minimization problems arising from several applied areas such as signal/image processing and statistical and machine learning. It however remains an interesting problem of how to implement ADMM to three-block separable convex minimization problems as required by the situation where many objective functions in the above-mentioned areas are actually more conveniently decomposed to the sum of three convex functions, due also to the observation that the straightforward extension of ADMM from the two-block case to the three-block case is apparently not convergent. In this paper, we shall introduce a new algorithm that is called a partially isochronous splitting algorithm (PISA) in order to implement ADMM for the three-block separable model. The main idea of our algorithm is to incorporate only one proximal term into the last subproblem of the extended ADMM so that the resulting algorithm maximally inherits the promising properties of ADMM. A remarkable superiority over the extended ADMM is that we can simultaneously solve two of the subproblems, thereby taking advantages of the separable structure and parallel architectures. Theoretically, we will establish the global convergence of our algorithm under standard conditions, and also the O(1/t) rate of convergence in both ergodic and nonergodic senses, where t is the iteration counter. The computational competitiveness of our algorithm is shown by numerical experiments on an application to the well-tested robust principal component analysis model.
PubDate: 2017-12-01
DOI: 10.1007/s10444-017-9574-4

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