Numerical Analysis
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New submissions for Thu, 23 Nov 17
 [1] arXiv:1711.08035 [pdf, ps, other]

Title: C2 continuous time dependent feedrate scheduling with configurable kinematic constraintsComments: 26 pages, 12 figuresSubjects: Numerical Analysis (math.NA)
We present a configurable trajectory planning strategy on planar paths for offline definition of timedependent C2 piecewise quintic feedrates. The more conservative formulation ensures chord tolerance, as well as prescribed bounds on velocity, acceleration and jerk Cartesian components. Since the less restrictive formulations of our strategy can usually still ensure all the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is desired but also when full kinematic control has to be guaranteed. Our approach can be applied to any planar path with a piecewise sufficiently smooth parametric representation. When Pythagoreanhodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited.
 [2] arXiv:1711.08074 [pdf, other]

Title: Mathematical Analysis of the 1D Model and Reconstruction Schemes for Magnetic Particle ImagingAuthors: Wolfgang Erb, Andreas Weinmann, Mandy Ahlborg, Christina Brandt, Gael Bringout, Thorsten M. Buzug, Jürgen Frikel, Christian Kaethner, Tobias Knopp, Thomas März, Martin Möddel, Martin Storath, Alexander WeberComments: This is joint work of the members of the scientific network MathMPI (DFG project ER777/11)Subjects: Numerical Analysis (math.NA)
Magnetic particle imaging (MPI) is a promising new invivo medical imaging modality in which distributions of superparamagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its illposedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.
 [3] arXiv:1711.08187 [pdf, ps, other]

Title: Adomian decomposition method for solving derivativedependent doubly singular boundary value problemsAuthors: Randhir SinghSubjects: Numerical Analysis (math.NA)
In this work, we apply Adomian decomposition method for solving nonlinear derivativedependent doubly singular boundary value problems: $(py')'= qf(x,y,y')$. This method is based on the modification of ADM and new twofold integral operator. The approximate solution is obtained in the form of series with easily determinable components. The effectiveness of the proposed approach is examined by considering three examples and numerical results are compared with known results.
 [4] arXiv:1711.08209 [pdf, ps, other]

Title: A fast multigrid finite element method for the timedependent tempered fractional problemComments: 21 pages. 0 figuresSubjects: Numerical Analysis (math.NA)
In this article a theoretical framework for the Galerkin finite element approximation to the timedependent tempered fractional problem is presented, which does not require for the fractional regularity assumption [V. J. Ervin and J. P. Roop, {\em Numer. Meth. Part. D. E.}, 22 (2005), pp. 558576]. Because the timedependent problems should become easier to solve as the time step $\tau \rightarrow 0$, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, {\em Math. Comp.}, 153 (1981), pp. 3551]. As far as we know, the convergence rate of the Vcycle multigrid finite element method has not been consider with $\tau\rightarrow 0$. Based on the introduced and analysis of the fractional $\tau$norm, the uniform convergence estimates of the Vcycle multigrid method (MGM) with the timedependent fractional problem is strictly proved, which means that the convergence rates of the Vcycle MGM is independent of the mesh size $h$ and the time step $\tau$. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform method, where $N$ is the number of the grid points.
 [5] arXiv:1711.08235 [pdf, other]

Title: A closedform update for orthogonal matrix decompositions under arbitrary rankone modificationsAuthors: Ralf ZimmermannComments: 11 pages, 1 figureSubjects: Numerical Analysis (math.NA)
We consider rankone adaptations $X_{new} = X+ab^T$ of a given matrix $X\in \mathbb{R}^{n\times p}$ with known matrix factorization $X = UW$, where $U\in\mathbb{R}^{n\times p}$ is columnorthogonal, i.e. $U^TU=I$.
Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U\Sigma V^T$, and the QRdecomposition, where $X = UW = QR$.
By using a geometric approach, we derive a closedform expression for a columnorthogonal matrix $U_{new}$ whose columns span the same subspace as the columns of the rankone modified $X_{new} = X +ab^T$.
This may be interpreted as a rankone adaptation of the $U$factor in the SVD or a rankone adaptation of the $Q$factor in the QRdecomposition, respectively.
As a consequence, we obtain a decomposition for the adapted matrix $X_{new} = U_{new}W_{new}$.
Moreover, the formula for $U_{new}$ allows us to determine the subspace distance between the subspaces colspan$(X) =\mathcal{S}$ and colspan$(X_{new}) =\mathcal{S}_{new}$ without additional computational effort.
In contrast to the existing approaches, the method does not require a numerical recomputation of the SVD or the QRdecomposition of an auxiliary matrix as an intermediate step.  [6] arXiv:1711.08335 [pdf, other]

Title: Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal smallscales and isogeometric analysis. I: The convectivediffusive contextSubjects: Numerical Analysis (math.NA)
This paper presents the construction of novel stabilized finite element methods in the convectivediffusive context that exhibit correctenergy behavior. Classical stabilized formulations can create unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal smallscales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large and smallscales should be $H_0^1$orthogonal. Using this orthogonality the VMS method can be converted into the streamlineupwind PetrovGalerkin (SUPG) or the Galerkin/leastsquares (GLS) method. Incorporating both large and smallscales in the energy definition asks for dynamic behavior of the smallscales. Therefore, the large and smallscales are treated as separate equations.
Two consistent variational formulations which depict correctenergy behavior are proposed: (i) the Galerkin/leastsquares method with dynamic smallscales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energydecaying generalized$\alpha$ timeintegrator. Numerical verification shows that dissipation due to the smallscales in classical stabilized methods can become negative, both on a local and global scale. The results show that without loss of accuracy the correctenergy behavior can be recovered by the proposed methods. The computations employ NURBSbased isogeometric analysis for the spatial discretization.  [7] arXiv:1711.08340 [pdf, ps, other]

Title: A fully discrete approximation of the onedimensional stochastic heat equationSubjects: Numerical Analysis (math.NA)
A fully discrete approximation of the onedimensional stochastic heat equation driven by multiplicative spacetime white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFLtype step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in $L^q(\Omega)$, for all $q\geq2$, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.
 [8] arXiv:1711.08343 [pdf, other]

Title: Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal smallscales and isogeometric analysis. II: The incompressible NavierStokes equationsSubjects: Numerical Analysis (math.NA)
This paper presents the construction of a correctenergy stabilized finite element method for the incompressible NavierStokes equations. The framework of the methodology and the correctenergy concept have been developed in the convectivediffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal smallscales and isogeometric analysis. The convectivediffusive context, CMAME, Accepted 2018]. This work extends ideas of this paper to build a stabilized method within the variational multiscale (VMS) setting which displays correctenergy behavior. Similar to the convectiondiffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the smallscales. This is demanded for correct energy behavior and links the VMS framework to the streamlineupwind PetrovGalerkin (SUPG) and the Galerkin/leastsquares method (GLS).
The presented method is a Galerkin/leastsquares formulation with dynamic divergencefree smallscales (GLSDD). It is locally massconservative for both the large and smallscales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBSbased isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.  [9] arXiv:1711.08390 [pdf, ps, other]

Title: Multiplicative Updates for Polynomial Root FindingAuthors: Nicolas GillisComments: 9 pages, 2 figuresSubjects: Numerical Analysis (math.NA)
Let $f(x)=p(x)q(x)$ be a polynomial with real coefficients whose roots have nonnegative real part, where $p$ and $q$ are polynomials with nonnegative coefficients. In this paper, we prove the following: Given an initial point $x_0 > 0$, the multiplicative update $x_{t+1} = x_t \, p(x_t)/q(x_t)$ ($t=0,1,\dots$) monotonically and linearly converges to the largest (resp. smallest) real roots of $f$ smaller (resp. larger) than $x_0$ if $p(x_0) < q(x_0)$ (resp. $q(x_0) < p(x_0)$). The motivation to study this algorithm comes from the multiplicative updates proposed in the literature to solve optimization problems with nonnegativity constraints; in particular many variants of nonnegative matrix factorization.
Crosslists for Thu, 23 Nov 17
 [10] arXiv:1711.08448 (crosslist from cs.SI) [pdf, other]

Title: Node and layer eigenvector centralities for multiplex networksSubjects: Social and Information Networks (cs.SI); Numerical Analysis (math.NA); Physics and Society (physics.socph)
Eigenvectorbased centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, monolayer networks. Moreover, several efficient computational tools are available for their computation. Moving up in dimensionality, several efforts have been made in the past to describe an eigenvectorbased centrality measure that generalizes Bonacich index to the case of multiplex networks. In this work, we propose a new definition of eigenvector centrality that relies on the Perron eigenvector of a multihomogeneous map defined in terms of the tensor describing the network. We prove that existence and uniqueness of such centrality are guaranteed under very mild assumptions on the multiplex network. Extensive numerical studies are proposed to test the newly introduced centrality measure and to compare it to other existing eigenvectorbased centralities.
Replacements for Thu, 23 Nov 17
 [11] arXiv:1612.08077 (replaced) [pdf, other]

Title: Optimaltransportbased mesh adaptivity on the plane and sphere using finite elementsComments: Updated following reviews, 36 pagesSubjects: Numerical Analysis (math.NA)
 [12] arXiv:1701.07620 (replaced) [pdf, other]

Title: A fully discretised filtered polynomial approximation on spherical shellsAuthors: Yoshihito KazashiSubjects: Numerical Analysis (math.NA)
 [13] arXiv:1707.03765 (replaced) [pdf, other]

Title: Computing Singularly Perturbed Differential EquationsComments: This paper has appeared in Journal of Computational Physics, Volume 354 (pages 417446)Journalref: Journal of Computational Physics 354 (2018) 417446Subjects: Numerical Analysis (math.NA); Materials Science (condmat.mtrlsci); Computational Physics (physics.compph)
 [14] arXiv:1707.08459 (replaced) [pdf, other]

Title: Highorder numerical methods for 2D parabolic problems in single and composite domainsAuthors: Gustav Ludvigsson, Kyle R. Steffen, Simon Sticko, Siyang Wang, Qing Xia, Yekaterina Epshteyn, Gunilla KreissComments: 45 pages, 12 figures, in revision for Journal of Scientific ComputingSubjects: Numerical Analysis (math.NA)
 [15] arXiv:1707.08603 (replaced) [pdf, ps, other]

Title: Some Extensions of the CrouzeixPalencia ResultSubjects: Numerical Analysis (math.NA)
 [16] arXiv:1711.06874 (replaced) [pdf, ps, other]

Title: Tucker Tensor analysis of Matern functions in spatial statisticsAuthors: Alexander Litvinenko, David Keyes, Venera Khoromskaia, Boris N. Khoromskij, Hermann G. MatthiesComments: 22 pages, 2 diagrams, 2 tables, 9 figuresSubjects: Numerical Analysis (math.NA)
 [17] arXiv:1705.10299 (replaced) [pdf, ps, other]

Title: Robustness to unknown error in sparse regularizationComments: To appear in IEEE Transactions on Information TheorySubjects: Information Theory (cs.IT); Numerical Analysis (math.NA)
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