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  • Pointwise error estimates for 𝐶⁰ interior penalty approximation of biharmonic problems
    Math. Comp. (IF 2.07) Pub Date : 2020-10-08
    D. Leykekhman

    Abstract:The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green's function estimates for the continuous problem which

    更新日期:2020-11-27
  • The spectrum of the abelian sandpile model
    Math. Comp. (IF 2.07) Pub Date : 2020-08-26
    Robert Hough; Hyojeong Son

    Abstract:In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral parameters which govern the asymptotic spectral gap and asymptotic mixing time. This paper gives a general method of determining the spectral parameters either computationally or asymptotically

    更新日期:2020-10-27
  • A numerical toolkit for multiprojective varieties
    Math. Comp. (IF 2.07) Pub Date : 2020-10-02
    Jonathan Hauenstein; Anton Leykin; Jose Rodriguez; Frank Sottile

    Abstract:A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation

    更新日期:2020-10-27
  • Upper bounds for the usual measures of totally positive algebraic integers with house less than 5.8
    Math. Comp. (IF 2.07) Pub Date : 2020-09-08
    V. Flammang

    Abstract:Previously, we established lower bounds for the usual measures (trace, length, Mahler measure) of totally positive algebraic integers, i.e., all of whose conjugates are positive real numbers. We used the method of explicit auxiliary functions and we noticed that the house of most of the totally positive polynomials involved in our functions are bounded by 5.8. Thanks to this observation, we

    更新日期:2020-10-27
  • Multiplicative series, modular forms, and Mandelbrot polynomials
    Math. Comp. (IF 2.07) Pub Date : 2020-09-09
    Michael Larsen

    Abstract:We say a power series is multiplicative if the sequence is so. In this paper, we consider multiplicative power series such that is also multiplicative. We find a number of examples for which is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over . The precise determination of this variety turns out to

    更新日期:2020-10-27
  • Geometry of error amplification in solving the Prony system with near-colliding nodes
    Math. Comp. (IF 2.07) Pub Date : 2020-09-09
    Andrey Akinshin; Gil Goldman; Yosef Yomdin

    Abstract:We consider a reconstruction problem for ``spike-train'' signals of an a priori known form from their moments We assume that the moments , , are known with an absolute error not exceeding . This problem is essentially equivalent to solving the Prony system We study the ``geometry of error amplification'' in reconstruction of from in situations where the nodes near-collide, i.e., form a cluster

    更新日期:2020-10-27
  • On the continuous time limit of the ensemble Kalman filter
    Math. Comp. (IF 2.07) Pub Date : 2020-10-06
    Theresa Lange; Wilhelm Stannat

    Abstract:We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719-1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J

    更新日期:2020-10-27
  • Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation
    Math. Comp. (IF 2.07) Pub Date : 2020-09-14
    Xiao Li; Zhonghua Qiao; Cheng Wang

    Abstract:In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete norm for the error function to establish the convergence result

    更新日期:2020-10-27
  • Consistency of finite volume approximations to nonlinear hyperbolic balance laws
    Math. Comp. (IF 2.07) Pub Date : 2020-10-06
    Matania Ben-Artzi; Jiequan Li

    Abstract:This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of ``balance laws''. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and

    更新日期:2020-10-27
  • A second-order numerical method for the aggregation equations
    Math. Comp. (IF 2.07) Pub Date : 2020-08-18
    José Carrillo; Ulrik Fjordholm; Susanne Solem

    Abstract:Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, -convex potentials with a possible Lipschitz singularity at the origin

    更新日期:2020-10-27
  • New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media
    Math. Comp. (IF 2.07) Pub Date : 2020-09-08
    Weiwei Sun; Chengda Wu

    Abstract:Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure

    更新日期:2020-10-27
  • A note on devising HDG+ projections on polyhedral elements
    Math. Comp. (IF 2.07) Pub Date : 2020-09-23
    Shukai Du; Francisco-Javier Sayas

    Abstract:In this paper, we propose a simple way of constructing HDG+ projections on polyhedral elements. The projections enable us to analyze the Lehrenfeld-Schöberl HDG (HDG+) methods in a very concise manner, and make many existing analysis techniques of standard HDG methods reusable for HDG+. The novelty here is an alternative way of constructing the projections without using -decompositions as

    更新日期:2020-10-27
  • Implicitization of tensor product surfaces via virtual projective resolutions
    Math. Comp. (IF 2.07) Pub Date : 2020-06-29
    Eliana Duarte; Alexandra Seceleanu

    Abstract:We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.

    更新日期:2020-08-20
  • An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives
    Math. Comp. (IF 2.07) Pub Date : 2020-08-04
    Qi Tao; Yan Xu; Chi-Wang Shu

    Abstract:In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply

    更新日期:2020-08-20
  • Dörfler marking with minimal cardinality is a linear complexity problem
    Math. Comp. (IF 2.07) Pub Date : 2020-06-24
    Carl-Martin Pfeiler; Dirk Praetorius

    Abstract:Most adaptive finite element strategies employ the Dörfler marking strategy to single out certain elements of a triangulation for refinement. In the literature, different algorithms have been proposed to construct , where usually two goals compete. On the one hand, should contain a minimal number of elements. On the other hand, one aims for linear costs with respect to the cardinality of

    更新日期:2020-08-20
  • Adaptive iterative linearization Galerkin methods for nonlinear problems
    Math. Comp. (IF 2.07) Pub Date : 2020-07-07
    Pascal Heid; Thomas Wihler

    Abstract:A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract

    更新日期:2020-08-20
  • A note on the Monge–Ampère type equations with general source terms
    Math. Comp. (IF 2.07) Pub Date : 2020-06-19
    Weifeng Qiu; Lan Tang

    Abstract:In this paper we consider numerical approximation to the generalised solutions to the Monge-Ampère type equations with general source terms. We first give some important propositions for the border of generalised solutions. Then, for both the classical and weak Dirichlet boundary conditions, we present well-posed numerical methods for the generalised solutions with general source terms. Finally

    更新日期:2020-08-20
  • Second order splitting of a class of fourth order PDEs with point constraints
    Math. Comp. (IF 2.07) Pub Date : 2020-07-27
    Charles Elliott; Philip Herbert

    Abstract:We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the

    更新日期:2020-08-20
  • Guaranteed a posteriori bounds for eigenvalues and eigenvectors: Multiplicities and clusters
    Math. Comp. (IF 2.07) Pub Date : 2020-07-30
    Eric Cancès; Geneviève Dusson; Yvon Maday; Benjamin Stamm; Martin Vohralík

    Abstract:This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the

    更新日期:2020-08-20
  • Quadratic points on modular curves with infinite Mordell–Weil group
    Math. Comp. (IF 2.07) Pub Date : 2020-08-13
    Josha Box

    Abstract:Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578-602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461-2484] have recently determined the quadratic points on each modular curve of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of are 37, 43, 53, 61, 57, 65

    更新日期:2020-08-13
  • Low-regularity integrators for nonlinear Dirac equations
    Math. Comp. (IF 2.07) Pub Date : 2020-08-07
    Katharina Schratz; Yan Wang; Xiaofei Zhao

    Abstract:In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in for solutions in , i.e., without requiring any additional regularity on the solution. In contrast to classical methods

    更新日期:2020-08-07
  • A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications
    Math. Comp. (IF 2.07) Pub Date : 2020-08-04
    Liping Zhang; Chiyu Chen

    Abstract:We focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP), and we have two contributions in this paper. First, a smoothing Newton-type algorithm is proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is

    更新日期:2020-08-04
  • Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix
    Math. Comp. (IF 2.07) Pub Date : 2020-08-01
    Bartosz Makuracki; Andrzej Mróz

    Abstract:Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix is -equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of . We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of , of the

    更新日期:2020-08-01
  • Weak discrete maximum principle of finite element methods in convex polyhedra
    Math. Comp. (IF 2.07) Pub Date : 2020-07-27
    Dmitriy Leykekhman; Buyang Li

    Abstract:We prove that the Galerkin finite element solution of the Laplace equation in a convex polyhedron , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree , satisfies the following weak maximum principle: with a constant independent of the mesh size . By using this result, we show that the Ritz projection operator is stable in norm uniformly

    更新日期:2020-07-27
  • Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall
    Math. Comp. (IF 2.07) Pub Date : 2020-07-20
    Zhijian He; Xiaoqun Wang

    Abstract:Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of the quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions

    更新日期:2020-07-20
  • Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem
    Math. Comp. (IF 2.07) Pub Date : 2020-07-14
    Natalia Kopteva

    Abstract:An initial-boundary value problem with a Caputo time derivative of fractional order is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative operator of order is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-monotonicity of this operator are established, which yields sharp pointwise-in-time

    更新日期:2020-07-14
  • Computing GIT-fans with symmetry and the Mori chamber decomposition of \overline{𝑀}_{0,6}
    Math. Comp. (IF 2.07) Pub Date : 2020-06-30
    Janko Böhm; Simon Keicher; Yue Ren

    Abstract:We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry, and group theory. We have implemented our algorithm in the SINGULAR library GITFAN.LIB. Using our implementation, we compute the Mori chamber decomposition of  .

    更新日期:2020-06-30
  • Generalized matrix spectral factorization and quasi-tight framelets with a minimum number of generators
    Math. Comp. (IF 2.07) Pub Date : 2020-06-05
    Chenzhe Diao; Bin Han

    Abstract:As a generalization of orthonormal wavelets in , tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large

    更新日期:2020-06-05
  • Orthogonal polynomials in and on a quadratic surface of revolution
    Math. Comp. (IF 2.07) Pub Date : 2020-06-05
    Sheehan Olver; Yuan Xu

    Abstract:We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation

    更新日期:2020-06-05
  • On computing the eventual behavior of an 𝐹𝐼-module over the rational numbers
    Math. Comp. (IF 2.07) Pub Date : 2020-06-01
    John Wiltshire-Gordon

    Abstract:We give a formula for the eventual multiplicities of irreducible representations appearing in a finitely presented -module over the rational numbers. The result relies on structure theory due to Sam-Snowden [Trans. Amer. Math. Soc. 146 (2018), no. 10, pp. 4117-4126].

    更新日期:2020-06-01
  • Computing isomorphisms between lattices
    Math. Comp. (IF 2.07) Pub Date : 2020-06-01
    Tommy Hofmann; Henri Johnston

    Abstract:Let be a number field, let be a finite-dimensional semisimple -algebra, and let be an -order in . It was shown in previous work that, under certain hypotheses on , there exists an algorithm that for a given (left) -lattice either computes a free basis of over or shows that is not free over . In the present article, we generalize this by showing that, under weaker hypotheses on , there exists

    更新日期:2020-06-01
  • Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs
    Math. Comp. (IF 2.07) Pub Date : 2020-05-26
    Kristin Kirchner

    Abstract:Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive

    更新日期:2020-05-26
  • The Magnus expansion and post-Lie algebras
    Math. Comp. (IF 2.07) Pub Date : 2020-05-26
    Charles Curry; Kurusch Ebrahimi-Fard; Brynjulf Owren

    Abstract:We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows us to place the classical Magnus expansion in the context of Lie group integrators.

    更新日期:2020-05-26
  • Explicit Coleman integration for curves
    Math. Comp. (IF 2.07) Pub Date : 2020-05-22
    Jennifer Balakrishnan; Jan Tuitman

    Abstract:The Coleman integral is a -adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author [Math. Comp. 85 (2016), pp. 961-981] and [Finite Fields Appl. 45 (2019), pp. 301-322] to compute the action of Frobenius on -adic cohomology. We present

    更新日期:2020-05-22
  • On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm
    Math. Comp. (IF 2.07) Pub Date : 2020-05-19
    Nadir Murru; Lea Terracini

    Abstract:Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field of -adic numbers . MCFs have also been recently introduced in , including, in particular, a -adic Jacobi-Perron algorithm

    更新日期:2020-05-19
  • How far away must forced letters be so that squares are still avoidable?
    Math. Comp. (IF 2.07) Pub Date : 2020-04-28
    Matthieu Rosenfeld

    Abstract:We describe a new nonconstructive technique to show that squares are avoidable by an infinite word even if we force some letters from the alphabet to appear at certain occurrences. We show that as long as forced positions are at a distance at least 19 (resp., 3, resp., 2) from each other, then we can avoid squares over 3 letters (resp., 4 letters, resp., 6 or more letters). We can also deduce

    更新日期:2020-04-28
  • Computational high frequency scattering from high-contrast heterogeneous media
    Math. Comp. (IF 2.07) Pub Date : 2020-03-09
    Daniel Peterseim; Barbara Verfürth

    Abstract:This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We

    更新日期:2020-03-09
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