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Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations Math. Comp. (IF 2.0) Pub Date : 2022-08-11 Elisenda Feliu, AmirHosein Sadeghimanesh
Abstract:Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration
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Finite element/holomorphic operator function method for the transmission eigenvalue problem Math. Comp. (IF 2.0) Pub Date : 2022-08-11 Bo Gong, Jiguang Sun, Tiara Turner, Chunxiong Zheng
Abstract:The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. It plays a key role in the unique determination of inhomogeneous media. Furthermore, transmission eigenvalues can be reconstructed from the scattering data and used to estimate the material properties of the unknown object. The problem is posted as a system of two second order partial differential
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Explicit Tamagawa numbers for certain algebraic tori over number fields Math. Comp. (IF 2.0) Pub Date : 2022-08-09 Thomas Rüd
Abstract:Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of $\operatorname {Gal}(K/k)$ of prime order $p$, there exists an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ by the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a
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Recovery of Sobolev functions restricted to iid sampling Math. Comp. (IF 2.0) Pub Date : 2022-08-09 David Krieg, Erich Novak, Mathias Sonnleitner
Abstract:We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\Omega )$ and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use identically distributed (iid) sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption
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Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate Math. Comp. (IF 2.0) Pub Date : 2022-08-05 Josef Dick, Takashi Goda, Kosuke Suzuki
Abstract:We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted
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ℚ-Curves, Hecke characters and some Diophantine equations Math. Comp. (IF 2.0) Pub Date : 2022-08-04 Ariel Pacetti, Lucas Villagra Torcomian
Abstract:In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb {Q}(\sqrt {-d})$ is attached to each primitive solution, which happens to be a $\mathbb {Q}$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $Gal_\mathbb
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Improved computation of fundamental domains for arithmetic Fuchsian groups Math. Comp. (IF 2.0) Pub Date : 2022-08-03 James Rickards
Abstract:A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the
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Anti-Gaussian quadrature formulae of Chebyshev type Math. Comp. (IF 2.0) Pub Date : 2022-08-03 Sotirios Notaris
Abstract:We prove that there is no positive measure $d\sigma$ on the interval $[a,b]$ such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure $d\sigma (t)=d\sigma (-t)$ on the symmetric interval $[-a,a]$, for which the anti-Gaussian formula has the form $\int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{
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An algorithm to recognize regular singular Mahler systems Math. Comp. (IF 2.0) Pub Date : 2022-08-01 Colin Faverjon, Marina Poulet
Abstract:This paper is devoted to the study of the analytic properties of Mahler systems at $0$. We give an effective characterisation of Mahler systems that are regular singular at $0$, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and ($q$-)difference systems but they do not apply in the Mahler case. This work fills in the gap by
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Delay-dependent elliptic reconstruction and optimal 𝐿^{∞}(𝐿²) a posteriori error estimates for fully discrete delay parabolic problems Math. Comp. (IF 2.0) Pub Date : 2022-07-29 Wansheng Wang, Lijun Yi
Abstract:We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the $L^\infty (L^2)$-norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced
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Why large time-stepping methods for the Cahn-Hilliard equation is stable Math. Comp. (IF 2.0) Pub Date : 2022-07-28 Dong Li
Abstract:We consider the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford
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An algorithm for Hodge ideals Math. Comp. (IF 2.0) Pub Date : 2022-07-27 Guillem Blanco
Abstract:We present an algorithm to compute the Hodge ideals (see M. Mustaţă and M. Popa [Mem. Amer. Math. Soc. 262 (2019), pp. v + 80; J. Éc. polytech. Math. 6 (2019), pp. 283–328]) of $\mathbb {Q}$-divisors associated to any reduced effective divisor $D$. The computation of the Hodge ideals is based on an algorithm to compute parts of the $V$-filtration of Kashiwara and Malgrange on $\iota _{+}\mathscr
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Sampling and homology via bottlenecks Math. Comp. (IF 2.0) Pub Date : 2022-07-22 Sandra Di Rocco, David Eklund, Oliver Gäfvert
Abstract:In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact affine variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered
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Explicit stabilized multirate method for stiff differential equations Math. Comp. (IF 2.0) Pub Date : 2022-07-19 Assyr Abdulle, Marcus Grote, Giacomo Rosilho de Souza
Abstract:Stabilized Runge–Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge–Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced
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A Trefftz method with reconstruction of the normal derivative applied to elliptic equations Math. Comp. (IF 2.0) Pub Date : 2022-07-15 Bruno Després, Maria El Ghaoui, Toni Sayah
Abstract:This article deals with the application of the Trefftz method to the Laplace problem. We introduce a new discrete variational formulation using a penalisation of the continuity of the solution on the edges which is compatible with the discontinuity of the Trefftz basis functions in the cells. We prove the existence and uniqueness of the discrete solution. A high order error estimate is established
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Algorithms for fundamental invariants and equivariants of finite groups Math. Comp. (IF 2.0) Pub Date : 2022-07-08 Evelyne Hubert, Erick Rodriguez Bazan
Abstract:For a finite group, we present three algorithms to compute a generating set of invariants simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants. Its symmetry adapted basis delivers the fundamental equivariants
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Density function of numerical solution of splitting AVF scheme for stochastic Langevin equation Math. Comp. (IF 2.0) Pub Date : 2022-07-07 Jianbo Cui, Jialin Hong, Derui Sheng
Abstract:In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. We first show the existence of the density function of the numerical solution by proving its exponential integrability property, Malliavin differentiability and the almost surely non-degeneracy of the associated Malliavin covariance
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Inf-sup stability implies quasi-orthogonality Math. Comp. (IF 2.0) Pub Date : 2022-07-08 Michael Feischl
Abstract:We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms
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Exact sequences on Worsey–Farin splits Math. Comp. (IF 2.0) Pub Date : 2022-07-12 Johnny Guzmán, Anna Lischke, Michael Neilan
Abstract:We construct several smooth finite element spaces defined on three-dimensional Worsey–Farin splits. In particular, we construct $C^1$, $H^1(\operatorname {curl})$, and $H^1$-conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at subsimplices of the mesh
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Error estimates for discrete generalized FEMs with locally optimal spectral approximations Math. Comp. (IF 2.0) Pub Date : 2022-07-12 Chupeng Ma, Robert Scheichl
Abstract:This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation spaces are constructed using eigenvectors of local eigenvalue problems solved by the finite element method on some sufficiently fine mesh with mesh size $h$
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A symmetric low-regularity integrator for nonlinear Klein-Gordon equation Math. Comp. (IF 2.0) Pub Date : 2022-07-05 Yan Wang, Xiaofei Zhao
Abstract:In this work, we propose a symmetric exponential-type low- regularity integrator for solving the nonlinear Klein-Gordon equation under rough data. The scheme is explicit in the physical space, and it is efficient under the Fourier pseudospectral discretization. Moreover, it achieves the second-order accuracy in time without loss of regularity of the solution, and its time-reversal symmetry
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Toric eigenvalue methods for solving sparse polynomial systems Math. Comp. (IF 2.0) Pub Date : 2022-06-30 Matías Bender, Simon Telen
Abstract:We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical
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Explicit Vologodsky integration for hyperelliptic curves Math. Comp. (IF 2.0) Pub Date : 2022-06-30 Enis Kaya
Abstract:Vologodsky’s theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry [Mosc. Math. J. 3 (2003), pp. 205–247, 260]. In contrast with the theory developed by Coleman [Invent. Math. 69 (1982), pp. 171–208; Duke Math. J. 52 (1985), pp. 765–770; Ann. of Math. (2) 121 (1985), pp. 111–168; Invent. Math. 93 (1988), pp. 239-266], it has
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On the 𝑞-analogue of the Pair Correlation Conjecture via Fourier optimization Math. Comp. (IF 2.0) Pub Date : 2022-06-14 Oscar Quesada-Herrera
Abstract:We study the $q$-analogue of the average of Montgomery’s function $F(\alpha ,\, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of $1$. To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee
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Computing equilibrium measures with power law kernels Math. Comp. (IF 2.0) Pub Date : 2022-06-14 Timon Gutleb, José Carrillo, Sheehan Olver
Abstract:We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form $K(x-y) = \frac {|x-y|^\alpha }{\alpha }-\frac {|x-y|^\beta }{\beta }$ using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases. The proposed method reduces what is naïvely a difficult to
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A finite element elasticity complex in three dimensions Math. Comp. (IF 2.0) Pub Date : 2022-06-14 Long Chen, Xuehai Huang
Abstract:A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The $H^1$ conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(\operatorname {inc})$-conforming finite
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Avoiding squares over words with lists of size three amongst four symbols Math. Comp. (IF 2.0) Pub Date : 2022-06-08 Matthieu Rosenfeld
Abstract:In 2007, Grytczuk conjectured that for any sequence $(\ell _i)_{i\ge 1}$ of alphabets of size $3$ there exists a square-free infinite word $w$ such that for all $i$, the $i$-th letter of $w$ belongs to $\ell _i$. The result of Thue from 1906 implies that there is an infinite square-free word if all the $\ell _i$ are identical. On the other hand, Grytczuk, Przybyło and Zhu showed in 2011 that
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Crouzeix-Raviart triangular elements are inf-sup stable Math. Comp. (IF 2.0) Pub Date : 2022-06-08 Carsten Carstensen, Stefan Sauter
Abstract:The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a conjecture of Crouzeix-Falk from 1989 for $p=3$. Our proof applies to any odd degree $p\ge 3$ and concludes the overall stability analysis: Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the
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Norm one Tori and Hasse norm principle Math. Comp. (IF 2.0) Pub Date : 2022-06-07 Akinari Hoshi, Kazuki Kanai, Aiichi Yamasaki
Abstract:Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskiǐ proved that there exists an exact sequence $0\to A(T)\to H^1(k,\operatorname {Pic}\overline {X})^\vee \to \Sha (T)\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $\Sha (T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\overline {X}=X\times
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Time integrators for dispersive equations in the long wave regime Math. Comp. (IF 2.0) Pub Date : 2022-06-07 María Cabrera Calvo, Frédéric Rousset, Katharina Schratz
Abstract:We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $\varepsilon = 1$ up to long wave limit regime $\varepsilon \ll 1$ on the natural time scale of the PDE $t= \mathcal {O}(\frac {1}{\varepsilon })$. Most notably the global error of our new schemes is of order $\tau \varepsilon$ (for the first-order
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Time domain boundary integral equations and convolution quadrature for scattering by composite media Math. Comp. (IF 2.0) Pub Date : 2022-06-07 Alexander Rieder, Francisco–Javier Sayas, Jens Melenk
Abstract:We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and provide a priori estimates for the convergence in space and time.
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Approximating viscosity solutions of the Euler system Math. Comp. (IF 2.0) Pub Date : 2022-06-01 Eduard Feireisl, Mária Lukáčová-Medvid’ová, Simon Schneider, Bangwei She
Abstract:Applying the concept of S-convergence, based on averaging in the spirit of Strong Law of Large Numbers, the vanishing viscosity solutions of the Euler system are studied. We show how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the viscosity finite volume method. Theoretical results are illustrated by numerical simulations
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Series reversion in Calderón’s problem Math. Comp. (IF 2.0) Pub Date : 2022-05-31 Henrik Garde, Nuutti Hyvönen
Abstract:This work derives explicit series reversions for the solution of Calderón’s problem. The governing elliptic partial differential equation is $\nabla \cdot (A\nabla u)=0$ in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends $A$ to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and
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Error estimates for a class of continuous Bonse-type inequalities Math. Comp. (IF 2.0) Pub Date : 2022-05-23 Diego Marques, Pavel Trojovský
Abstract:Let $p_n$ be the $n$th prime number. In 2000, Papaitopol proved that the inequality $p_1\cdots p_n>p_{n+1}^{n-\pi (n)}$ holds, for all $n\geq 2$, where $\pi (x)$ is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing $n-\pi (n)$ by $n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n))$, however there is a small mistake in their argument. In this paper
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Analysis of finite element methods for surface vector-Laplace eigenproblems Math. Comp. (IF 2.0) Pub Date : 2022-05-16 Arnold Reusken
Abstract:In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce
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Certified dimension reduction in nonlinear Bayesian inverse problems Math. Comp. (IF 2.0) Pub Date : 2022-04-27 Olivier Zahm, Tiangang Cui, Kody Law, Alessio Spantini, Youssef Marzouk
Abstract:We propose a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. The likelihood function is approximated by a ridge function, i.e., a map which depends nontrivially only on a few linear combinations of the parameters. We build this ridge approximation by minimizing an upper bound on the Kullback–Leibler
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Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh Math. Comp. (IF 2.0) Pub Date : 2022-04-26 Buyang Li
Abstract:The Galerkin finite element solution $u_h$ of the Poisson equation $-\Delta u=f$ under the Neumann boundary condition in a possibly nonconvex polygon $\varOmega$, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: \begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*}
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Full discretization error analysis of exponential integrators for semilinear wave equations Math. Comp. (IF 2.0) Pub Date : 2022-04-05 Benjamin Dörich, Jan Leibold
Abstract:In this article we prove full discretization error bounds for semilinear second-order evolution equations. We consider exponential integrators in time applied to an abstract nonconforming semidiscretization in space. Since the fully discrete schemes involve the spatially discretized semigroup, a crucial point in the error analysis is to eliminate the continuous semigroup in the representation
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The equilateral small octagon of maximal width Math. Comp. (IF 2.0) Pub Date : 2022-03-30 Christian Bingane, Charles Audet
Abstract:A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24%$ larger than the width of the regular octagon: $\cos (\pi /8)$. In addition, the paper proposes a family of equilateral small
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A sharp discrepancy bound for jittered sampling Math. Comp. (IF 2.0) Pub Date : 2022-03-24 Benjamin Doerr
Abstract:For $m, d \in {\mathbb N}$, a jittered (or stratified) sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c > 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected
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A discontinuous Galerkin pressure correction scheme for the incompressible Navier–Stokes equations: Stability and convergence Math. Comp. (IF 2.0) Pub Date : 2022-03-24 Rami Masri, Chen Liu, Beatrice Riviere
Abstract:A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier–Stokes equations is formulated and analyzed. We prove unconditional stability of the proposed scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.
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A posteriori error analysis for approximations of time-fractional subdiffusion problems Math. Comp. (IF 2.0) Pub Date : 2022-03-14 Lehel Banjai, Charalambos Makridakis
Abstract:In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability
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Time discretizations of Wasserstein–Hamiltonian flows Math. Comp. (IF 2.0) Pub Date : 2022-03-14 Jianbo Cui, Luca Dieci, Haomin Zhou
Abstract:We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on a graph (lattice) with different weights are derived, which can be viewed as spatial discretizations of the original Hamiltonian systems. We prove consistency of these discretizations. Furthermore
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Fast and stable augmented Levin methods for highly oscillatory and singular integrals Math. Comp. (IF 2.0) Pub Date : 2022-02-15 Yinkun Wang, Shuhuang Xiang
Abstract:In this paper, augmented Levin methods are proposed for the computation of oscillatory integrals with stationary points and an algebraically or logarithmically singular kernel. Different from the conventional Levin method, to overcome the difficulties caused by singular and stationary points, the original Levin ordinary differential equation (Levin-ODE) is converted into an augmented ODE system
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Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis Math. Comp. (IF 2.0) Pub Date : 2022-02-15 Anne-Maria Ernvall-Hytönen, Neea Palojärvi
Abstract:We prove an explicit error term for the $\psi (x,\chi )$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.
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Multiscale scattering in nonlinear Kerr-type media Math. Comp. (IF 2.0) Pub Date : 2022-02-07 Roland Maier, Barbara Verfürth
Abstract:We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously
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Explicit interval estimates for prime numbers Math. Comp. (IF 2.0) Pub Date : 2022-01-25 Michaela Cully-Hugill, Ethan Lee
Abstract:Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta ,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1-\Delta ^{-1}),x]$.
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Strictly convex entropy and entropy stable schemes for reactive Euler equations Math. Comp. (IF 2.0) Pub Date : 2022-01-25 Weifeng Zhao
Abstract:This paper presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity
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Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on ℝ^{𝕕} Math. Comp. (IF 2.0) Pub Date : 2022-01-14 Alexander Gilbert, Frances Kuo, Ian Sloan
Abstract:We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathbb {R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence
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Enumeration of set-theoretic solutions to the Yang–Baxter equation Math. Comp. (IF 2.0) Pub Date : 2022-01-14 Ö. Akgün, M. Mereb, L. Vendramin
Abstract:We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang–Baxter equation of small size. We show that there are 321,931 involutive solutions of size nine, 4,895,272 involutive solutions of size ten and 422,449,480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles.
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Abel Maps for nodal curves via tropical geometry Math. Comp. (IF 2.0) Pub Date : 2022-01-11 Alex Abreu, Sally Andria, Marco Pacini
Abstract:We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an
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Boosted optimal weighted least-squares Math. Comp. (IF 2.0) Pub Date : 2022-01-05 Cécile Haberstich, Anthony Nouy, Guillaume Perrin
Abstract:This paper is concerned with the approximation of a function $u$ in a given subspace $V_m$ of dimension $m$ from evaluations of the function at $n$ suitably chosen points. The aim is to construct an approximation of $u$ in $V_m$ which yields an error close to the best approximation error in $V_m$ and using as few evaluations as possible. Classical least-squares regression, which defines a
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Greenberg’s conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions Math. Comp. (IF 2.0) Pub Date : 2021-12-30 Lorenzo Pagani
Abstract:Let $\mathcal {A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb {Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal {A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal {A}_n$’s stabilizes
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An extended Galerkin analysis in finite element exterior calculus Math. Comp. (IF 2.0) Pub Date : 2021-12-30 Qingguo Hong, Yuwen Li, Jinchao Xu
Abstract:For the Hodge–Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contractible domains, this framework utilizes seven fields and provides a unifying inf-sup analysis with respect to all discretization and penalty parameters. It is shown that the proposed methods can be hybridized as a
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Four consecutive primitive elementsin a finite field Math. Comp. (IF 2.0) Pub Date : 2021-12-28 Tamiru Jarso, Tim Trudgian
Abstract:For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $\mathbb {F}_{q}$ when $q> 2401$.
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Stochastic gradient descent for linear inverse problems in Hilbert spaces Math. Comp. (IF 2.0) Pub Date : 2021-12-22 Shuai Lu, Peter Mathé
Abstract:We investigate stochastic gradient decent (SGD) for solving full infinite dimensional ill-posed problems in Hilbert spaces. We allow for batch-size versions of SGD where the randomly chosen batches incur noise fluctuations. Based on the corresponding bias-variance decomposition we provide bounds for the root mean squared error. These bounds take into account the discretization levels, the
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On symmetric-conjugate composition methods in the numerical integration of differential equations Math. Comp. (IF 2.0) Pub Date : 2021-12-22 S. Blanes, F. Casas, P. Chartier, A. Escorihuela-Tomàs
Abstract:We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and
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A log-log speedup for exponent one-fifth deterministic integer factorisation Math. Comp. (IF 2.0) Pub Date : 2021-12-15 David Harvey, Markus Hittmeir
Abstract:Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most \[ O\left ( \frac {N^{1/5} \log ^{16/5} N}{(\log \log N)^{3/5}}\right ) \] bit operations. This improves on the previous best known result by a factor of $(\log \log N)^{3/5}$
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A first-order Fourier integrator for the nonlinear Schrödinger equation on 𝕋 without loss of regularity Math. Comp. (IF 2.0) Pub Date : 2021-12-14 Yifei Wu, Fangyan Yao
Abstract:In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first-order accuracy in $H^\gamma$ for any initial data belonging to $H^\gamma$, for any $\gamma >\frac 32$. That is
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An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative Math. Comp. (IF 2.0) Pub Date : 2021-10-21 Jay Jorgenson, Lejla Smajlović, Holger Then
Abstract:Let $N$ be one of the $38$ distinct square-free integers such that the arithmetic group $\Gamma _0(N)^+$ has genus one. We constructed canonical generators $x_N$ and $y_N$ for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of $x_N$, which we express as a polynomial in $y_N$ with