Numerical Analysis

New submissions

[ total of 19 entries: 1-19 ]
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New submissions for Thu, 22 Feb 18

[1]
Title: Broyden's method for nonlinear eigenproblems
Authors: Elias Jarlebring
Subjects: 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]
Title: An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and Numerical Verification
Subjects: 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 three-dimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible Navier-Stokes equations, the resistive MHD equations need special considerations because of the divergence-free 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 non-conservative 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 non-conservative 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 positive-definite 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 three-dimensional curvilinear meshes. The discrete analysis relies on the summation-by-parts property, which is satisfied by the DG spectral element method (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. Although the divergence-free constraint is included in the non-conservative terms, the resulting method has no particular treatment of the magnetic field divergence errors...

[3]
Title: Subspace Methods for 3-Parameter Eigenvalue Problems
Comments: 26 pages, 6 figures, 7 tables
Subjects: Numerical Analysis (math.NA)

We propose subspace methods for 3-parameter 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 2-parameter eigenvalue problems exist, extensions to 3-parameter setting have not been worked out thoroughly. An inherent difficulty is that, while for 2-parameter 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 Jacobi-Davidson type method for three or more parameters, which we generalize from its 2-parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi-Davidson 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]
Title: A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere
Subjects: Numerical Analysis (math.NA)

In a previous article [\emph{J. Comp. Phys.} $\mathbf{357}$ (2018) 282--304], 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 non-affine, 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]
Title: A Godunov type scheme for a class of scalar conservation laws with non-local flux
Subjects: Numerical Analysis (math.NA)

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local 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 well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

[6]
Title: The real polynomial eigenvalue problem is well conditioned on the average
Subjects: 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 well-conditioned on the average.

[7]
Title: Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies
Subjects: 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 goal-oriented 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 sample-wise adaptive refinement strategies over uniform refinements.

[8]
Title: Operator splitting technique using streamline projection for two-phase flow in highly heterogeneous and anisotropic porous media
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)

In this paper, we present a fast streamline-based numerical method for the two-phase flow equations in high-rate 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 high-rate flooding scenarios. This allows us to present an improved streamline approach in combination with the one-dimensional 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 three-dimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fully-implicit finite volume methods are provided.

[9]
Title: Stability and error analysis of an implicit Milstein finite difference scheme for a two-dimensional Zakai SPDE
Subjects: Numerical Analysis (math.NA)

In this article, we propose an implicit finite difference scheme for a two-dimensional 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 mean-square 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]
Comments: 26 pages, 13 figures, 6 tables
Subjects: Numerical Analysis (math.NA); Data Structures and Algorithms (cs.DS); Computational Physics (physics.comp-ph)

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 up-scaling approach inspired by multi-scale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advection-dominated 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 proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion.

Cross-lists for Thu, 22 Feb 18

[11]  arXiv:1802.07716 (cross-list from math.AT) [pdf, other]
Title: Sampling real algebraic varieties for topological data analysis
Subjects: 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 Decomposition
Subjects: 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 Methods
Comments: 71 pages, final version, accepted by SIAM Review
Subjects: Numerical Analysis (math.NA)
[14]  arXiv:1705.08138 (replaced) [pdf, ps, other]
Title: A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations
Subjects: Numerical Analysis (math.NA)
[15]  arXiv:1705.08139 (replaced) [pdf, ps, other]
Title: Two-level preconditioners for the Helmholtz equation
Subjects: Numerical Analysis (math.NA)
[16]  arXiv:1708.07743 (replaced) [pdf, other]
Title: Bézier $\bar{B}$ Projection
Subjects: Numerical Analysis (math.NA)
[17]  arXiv:1710.08898 (replaced) [pdf, other]
Title: Overview of the Incompressible Navier-Stokes simulation capabilities in the MOOSE Framework
Comments: 54 pages, 16 figures, includes peer reviewer revisions
Subjects: Numerical Analysis (math.NA)
[18]  arXiv:1802.05759 (replaced) [pdf, other]
Title: A Krylov subspace method for the approximation of bivariate matrix functions
Authors: Daniel Kressner
Comments: Revised version contains polynomial approximation results for phi function in appendix
Subjects: Numerical Analysis (math.NA); Operator Algebras (math.OA)
[19]  arXiv:1802.07014 (replaced) [pdf, ps, other]
Title: Electromechanical coupling of waves in nerve fibres
Subjects: Biological Physics (physics.bio-ph); Numerical Analysis (math.NA)
[ total of 19 entries: 1-19 ]
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