# Numerical Analysis

## New submissions

[ total of 17 entries: 1-17 ]
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### New submissions for Thu, 23 Nov 17

[1]
Title: C2 continuous time dependent feedrate scheduling with configurable kinematic constraints
Subjects: Numerical Analysis (math.NA)

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent 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]
Title: Mathematical Analysis of the 1D Model and Reconstruction Schemes for Magnetic Particle Imaging
Comments: This is joint work of the members of the scientific network MathMPI (DFG project ER777/1-1)
Subjects: Numerical Analysis (math.NA)

Magnetic particle imaging (MPI) is a promising new in-vivo medical imaging modality in which distributions of super-paramagnetic 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 ill-posedness 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]
Title: Adomian decomposition method for solving derivative-dependent doubly singular boundary value problems
Authors: Randhir Singh
Subjects: Numerical Analysis (math.NA)

In this work, we apply Adomian decomposition method for solving nonlinear derivative-dependent doubly singular boundary value problems: $(py')'= qf(x,y,y')$. This method is based on the modification of ADM and new two-fold 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]
Title: A fast multigrid finite element method for the time-dependent tempered fractional problem
Subjects: Numerical Analysis (math.NA)

In this article a theoretical framework for the Galerkin finite element approximation to the time-dependent 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. 558-576]. Because the time-dependent 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. 35--51]. As far as we know, the convergence rate of the V-cycle 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 V-cycle multigrid method (MGM) with the time-dependent fractional problem is strictly proved, which means that the convergence rates of the V-cycle 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]
Title: A closed-form update for orthogonal matrix decompositions under arbitrary rank-one modifications
Authors: Ralf Zimmermann
Subjects: Numerical Analysis (math.NA)

We consider rank-one 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 column-orthogonal, 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 QR-decomposition, where $X = UW = QR$.
By using a geometric approach, we derive a closed-form expression for a column-orthogonal matrix $U_{new}$ whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$.
This may be interpreted as a rank-one adaptation of the $U$-factor in the SVD or a rank-one adaptation of the $Q$-factor in the QR-decomposition, 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 QR-decomposition of an auxiliary matrix as an intermediate step.

[6]
Title: Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context
Subjects: Numerical Analysis (math.NA)

This paper presents the construction of novel stabilized finite element methods in the convective-diffusive context that exhibit correct-energy 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 small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be $H_0^1$-orthogonal. Using this orthogonality the VMS method can be converted into the streamline-upwind Petrov-Galerkin (SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large- and small-scales in the energy definition asks for dynamic behavior of the small-scales. Therefore, the large- and small-scales are treated as separate equations.
Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized-$\alpha$ time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, both on a local and global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization.

[7]
Title: A fully discrete approximation of the one-dimensional stochastic heat equation
Subjects: Numerical Analysis (math.NA)

A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time 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 CFL-type 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 non-globally 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]
Title: Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
Subjects: Numerical Analysis (math.NA)

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective-diffusive 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 small-scales and isogeometric analysis. The convective-diffusive context, CMAME, Accepted 2018]. This work extends ideas of this paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection-diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.

[9]
Title: Multiplicative Updates for Polynomial Root Finding
Authors: Nicolas Gillis
Subjects: 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.

### Cross-lists for Thu, 23 Nov 17

[10]  arXiv:1711.08448 (cross-list from cs.SI) [pdf, other]
Title: Node and layer eigenvector centralities for multiplex networks
Subjects: Social and Information Networks (cs.SI); Numerical Analysis (math.NA); Physics and Society (physics.soc-ph)

Eigenvector-based 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, mono-layer 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 eigenvector-based 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 multi-homogeneous 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 eigenvector-based centralities.

### Replacements for Thu, 23 Nov 17

[11]  arXiv:1612.08077 (replaced) [pdf, other]
Title: Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements
Comments: Updated following reviews, 36 pages
Subjects: Numerical Analysis (math.NA)
[12]  arXiv:1701.07620 (replaced) [pdf, other]
Title: A fully discretised filtered polynomial approximation on spherical shells
Subjects: Numerical Analysis (math.NA)
[13]  arXiv:1707.03765 (replaced) [pdf, other]
Title: Computing Singularly Perturbed Differential Equations
Comments: This paper has appeared in Journal of Computational Physics, Volume 354 (pages 417-446)
Journal-ref: Journal of Computational Physics 354 (2018) 417-446
Subjects: Numerical Analysis (math.NA); Materials Science (cond-mat.mtrl-sci); Computational Physics (physics.comp-ph)
[14]  arXiv:1707.08459 (replaced) [pdf, other]
Title: High-order numerical methods for 2D parabolic problems in single and composite domains
Comments: 45 pages, 12 figures, in revision for Journal of Scientific Computing
Subjects: Numerical Analysis (math.NA)
[15]  arXiv:1707.08603 (replaced) [pdf, ps, other]
Title: Some Extensions of the Crouzeix-Palencia Result
Subjects: Numerical Analysis (math.NA)
[16]  arXiv:1711.06874 (replaced) [pdf, ps, other]
Title: Tucker Tensor analysis of Matern functions in spatial statistics
Comments: 22 pages, 2 diagrams, 2 tables, 9 figures
Subjects: Numerical Analysis (math.NA)
[17]  arXiv:1705.10299 (replaced) [pdf, ps, other]
Title: Robustness to unknown error in sparse regularization
Comments: To appear in IEEE Transactions on Information Theory
Subjects: Information Theory (cs.IT); Numerical Analysis (math.NA)
[ total of 17 entries: 1-17 ]
[ showing up to 2000 entries per page: fewer | more ]

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