Numerical Analysis
New submissions
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New submissions for Thu, 22 Feb 18
 [1] arXiv:1802.07322 [pdf, ps, other]

Title: Broyden's method for nonlinear eigenproblemsAuthors: Elias JarlebringSubjects: Numerical Analysis (math.NA)
Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE is used to illustrate the approach. The simulations are done in the julia programming language, and are provided as publicly available module for reproducability.
 [2] arXiv:1802.07341 [pdf, other]

Title: An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and Numerical VerificationAuthors: Marvin Bohm, Andrew R. Winters, Gregor J. Gassner, Dominik Derigs, Florian Hindenlang, Joachim SaurComments: arXiv admin note: substantial text overlap with arXiv:1711.05576Subjects: Numerical Analysis (math.NA)
The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on threedimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible NavierStokes equations, the resistive MHD equations need special considerations because of the divergencefree constraint on the magnetic field. For instance, it is well known that for the symmetrization of the ideal MHD system as well as the continuous entropy analysis a nonconservative term proportional to the divergence of the magnetic field, typically referred to as the Powell term, must be included. As a consequence, the mimicry of the continuous entropy analysis in the discrete sense demands a suitable DG approximation of the nonconservative terms in addition to the ideal MHD terms.
This paper focuses on the resistive MHD equations: Our first contribution is a proof that the resistive terms are symmetric and positivedefinite when formulated in entropy space as gradients of the entropy variables. This enables us to show that the entropy inequality holds for the resistive MHD equations. This continuous analysis is the key for our DG discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality, typically termed entropy stability. Our second contribution is a detailed derivation and analysis of the discretization on threedimensional curvilinear meshes. The discrete analysis relies on the summationbyparts property, which is satisfied by the DG spectral element method (DGSEM) with LegendreGaussLobatto (LGL) nodes. Although the divergencefree constraint is included in the nonconservative terms, the resulting method has no particular treatment of the magnetic field divergence errors...  [3] arXiv:1802.07386 [pdf, other]

Title: Subspace Methods for 3Parameter Eigenvalue ProblemsComments: 26 pages, 6 figures, 7 tablesSubjects: Numerical Analysis (math.NA)
We propose subspace methods for 3parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for 2parameter eigenvalue problems exist, extensions to 3parameter setting have not been worked out thoroughly. An inherent difficulty is that, while for 2parameter eigenvalue problems we can exploit a relation to Sylvester equations to obtain a fast Arnoldi type method, this relation does not seem to extend to three or more parameters in a straightforward way. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a JacobiDavidson type method for three or more parameters, which we generalize from its 2parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The JacobiDavidson approach is devised to locate eigenvalues close to a prescribed target, yet it often performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. Matlab implementations of both methods are made available in package MultiParEig and we present extensive numerical experiments which indicate that both approaches are effective in locating eigenvalues in the exterior of the spectrum.
 [4] arXiv:1802.07395 [pdf, other]

Title: A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed SphereSubjects: Numerical Analysis (math.NA)
In a previous article [\emph{J. Comp. Phys.} $\mathbf{357}$ (2018) 282304], hereafter LPG18, the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a nonaffine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in $H(\mathrm{rot})$, $H(\mathrm{div})$ and $L_2$. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the $H(\mathrm{div})$ form of the Piola transformation used to construct the metric terms for the normal velocities.
 [5] arXiv:1802.07484 [pdf, other]

Title: A Godunov type scheme for a class of scalar conservation laws with nonlocal fluxSubjects: Numerical Analysis (math.NA)
We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used LaxFriedrichs type scheme. In contrast to other approaches, we consider a nonlocal mean velocity instead of a mean density and provide $L^\infty$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the wellposedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to LaxFriedrichs is proved by a variety of numerical examples.
 [6] arXiv:1802.07493 [pdf, other]

Title: The real polynomial eigenvalue problem is well conditioned on the averageSubjects: Numerical Analysis (math.NA); Probability (math.PR)
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries are very wellconditioned on the average.
 [7] arXiv:1802.07539 [pdf, other]

Title: Continuous Level Monte Carlo and SampleAdaptive Model HierarchiesComments: 22 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
In this paper, we present a generalisation of the Multilevel Monte Carlo (MLMC) method to a setting where the level parameter is a continuous variable. This Continuous Level Monte Carlo (CLMC) estimator provides a natural framework in PDE applications to adapt the model hierarchy to each sample. In addition, it can be made unbiased with respect to the expected value of the true quantity of interest provided the quantity of interest converges sufficiently fast. The practical implementation of the CLMC estimator is based on interpolating actual evaluations of the quantity of interest at a finite number of resolutions. As our new level parameter, we use the logarithm of a goaloriented finite element error estimator for the accuracy of the quantity of interest. We prove the unbiasedness, as well as a complexity theorem that shows the same rate of complexity for CLMC as for MLMC. Finally, we provide some numerical evidence to support our theoretical results, by successfully testing CLMC on a standard PDE test problem. The numerical experiments demonstrate clear gains for samplewise adaptive refinement strategies over uniform refinements.
 [8] arXiv:1802.07540 [pdf, other]

Title: Operator splitting technique using streamline projection for twophase flow in highly heterogeneous and anisotropic porous mediaSubjects: Numerical Analysis (math.NA); Computational Physics (physics.compph)
In this paper, we present a fast streamlinebased numerical method for the twophase flow equations in highrate flooding scenarios for incompressible fluids in heterogeneous and anisotropic porous media. A fractional flow formulation is adopted and a discontinuous Galerkin method (DG) is employed to solve the pressure equation. Capillary effects can be neglected in highrate flooding scenarios. This allows us to present an improved streamline approach in combination with the onedimensional front tracking method to solve the transport equation. To handle the high computational costs of the DG approximation, domain decomposition is applied combined with an algebraic multigrid preconditioner to solve the linear system. Special care at the interior interfaces is required and the streamline tracer has to include a dynamic communication strategy. The method is validated in various two and threedimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fullyimplicit finite volume methods are provided.
 [9] arXiv:1802.07682 [pdf, ps, other]

Title: Stability and error analysis of an implicit Milstein finite difference scheme for a twodimensional Zakai SPDEComments: 24 pagesSubjects: Numerical Analysis (math.NA)
In this article, we propose an implicit finite difference scheme for a twodimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. The scheme is based on a Milstein approximation to the stochastic integral and an alternating direction implicit (ADI) discretisation of the elliptic term. We prove its meansquare stability and convergence in $L_2$ of first order in time and second order in space, by Fourier analysis, in the presence of Dirac initial data. Numerical tests confirm these findings empirically.
 [10] arXiv:1802.07684 [pdf, other]

Title: Multiscale finite elements through advectioninduced coordinates for transient advectiondiffusion equationsComments: 26 pages, 13 figures, 6 tablesSubjects: Numerical Analysis (math.NA); Data Structures and Algorithms (cs.DS); Computational Physics (physics.compph)
Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable long term simulations. This is typically done via parametrizations but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel upscaling approach inspired by multiscale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advectiondominated problems arising in climate simulations the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general and we demonstrate it as a proofofconcept on a onedimensional passive advectiondiffusion equation with oscillatory background velocity and diffusion.
Crosslists for Thu, 22 Feb 18
 [11] arXiv:1802.07716 (crosslist from math.AT) [pdf, other]

Title: Sampling real algebraic varieties for topological data analysisSubjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); Numerical Analysis (math.NA)
Topological data analysis (TDA) provides a growing body of tools for computing geometric and topological information about spaces from a finite sample of points. We present a new adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to minimize the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We demonstrate our algorithm on several examples.
Replacements for Thu, 22 Feb 18
 [12] arXiv:1607.05210 (replaced) [pdf, other]

Title: Hierarchical Approximate Proper Orthogonal DecompositionSubjects: Numerical Analysis (math.NA)
 [13] arXiv:1610.02270 (replaced) [pdf, other]

Title: A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz MethodsComments: 71 pages, final version, accepted by SIAM ReviewSubjects: Numerical Analysis (math.NA)
 [14] arXiv:1705.08138 (replaced) [pdf, ps, other]

Title: A twolevel domaindecomposition preconditioner for the timeharmonic Maxwell's equationsAuthors: Marcella Bonazzoli (JAD), Victorita Dolean (JAD), Ivan Graham, Euan Spence, PierreHenri Tournier (LJLL, ALPINES)Subjects: Numerical Analysis (math.NA)
 [15] arXiv:1705.08139 (replaced) [pdf, ps, other]

Title: Twolevel preconditioners for the Helmholtz equationAuthors: Marcella Bonazzoli (JAD), Victorita Dolean (JAD), Ivan Graham, Euan Spence, PierreHenri Tournier (LJLL, ALPINES)Subjects: Numerical Analysis (math.NA)
 [16] arXiv:1708.07743 (replaced) [pdf, other]

Title: Bézier $\bar{B}$ ProjectionComments: 26 pagesSubjects: Numerical Analysis (math.NA)
 [17] arXiv:1710.08898 (replaced) [pdf, other]

Title: Overview of the Incompressible NavierStokes simulation capabilities in the MOOSE FrameworkComments: 54 pages, 16 figures, includes peer reviewer revisionsSubjects: Numerical Analysis (math.NA)
 [18] arXiv:1802.05759 (replaced) [pdf, other]

Title: A Krylov subspace method for the approximation of bivariate matrix functionsAuthors: Daniel KressnerComments: Revised version contains polynomial approximation results for phi function in appendixSubjects: Numerical Analysis (math.NA); Operator Algebras (math.OA)
 [19] arXiv:1802.07014 (replaced) [pdf, ps, other]

Title: Electromechanical coupling of waves in nerve fibresSubjects: Biological Physics (physics.bioph); Numerical Analysis (math.NA)
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