Mathematics of Computation最新文献

Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter ε varepsilon . In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of ε varepsilon and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS.

Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.

The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the L 2 L^2 - and H 1 H^1 -norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted N × N Ntimes N matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the N → ∞ Nto infty behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.

This is a brief report on some recent large-scale Monte Carlo simulations for approximating the density of zeros in character tables of large symmetric groups. Previous computations suggested that a large fraction of zeros cannot be explained by classical vanishing results. Our computations eclipse previous ones and suggest that the opposite is true. In fact, we find empirically that almost all of the zeros are of a single classical type.

The blocky optimization has gained a significant amount of attention in far-reaching practical applications. Following the recent work (M. Nikolova and P. Tan [SIAM J. Optim. 29 (2019), pp. 2053–2078]) on solving a class of nonconvex nonsmooth optimization, we develop a stochastic alternating structure-adapted proximal (s-ASAP) gradient descent method for solving blocky optimization problems. By deploying some state-of-the-art variance reduced gradient estimators (rather than full gradient) in stochastic optimization, the s-ASAP method is applicable to nonconvex optimization whose objective is the sum of a nonsmooth data-fitting term and a finite number of differentiable functions. The sublinear convergence rate of s-ASAP is built upon the proximal point algorithmic framework, whilst the linear convergence rate of s-ASAP is achieved under the error bound condition. Furthermore, the convergence of the sequence produced by s-ASAP is established under the Kurdyka-Łojasiewicz property. Preliminary numerical simulations on some image processing applications demonstrate the compelling performance of the proposed method.

We present a complete numerical analysis for a general discretization of a coupled flow–mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix–fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix–fracture interfaces in order to cover a wide range of normal fracture conductivities.

The numerical analysis is carried out in the Gradient Discretization framework (see J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin [The gradient discretisation method, Springer, Cham, 2018]), encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ( P 2 mathbb {P}_2 ) for the mechanical displacement coupled with face-wise constant ( P 0 mathbb P_0 ) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over a hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done by adapting the strategy we have adopted to study the totally real case, but for this new setting several new computational issues had to be overcome. As a consequence, we detect the four number fields of signature ( r 1 , r 2 ) = ( 6 , 1 ) (r_1,r_2)=(6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures ( 3 , 1 ) (3,1) , ( 4 , 1 ) (4,1) and ( 5 , 1 ) (5,1) .

A new H ( div ⁡ div ) H(operatorname {div}operatorname {div}) -conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and C 0 C^0 discontinuous Galerkin methods for the biharmonic equation are derived.

In this paper, we study the ultraweak-local discontinuous Galerkin (UWLDG) method for time-dependent linear fourth-order problems with four types of boundary conditions. In one dimension and two dimensions, stability and optimal error estimates of order k + 1 k+1 are derived for the UWLDG scheme with polynomials of degree at most k k ( k ≥ 1 kge 1 ) for solving initial-boundary value problems. The main difficulties are the design of suitable penalty terms at the boundary for numerical fluxes and the construction of projections. More precisely, in two dimensions with the Dirichlet boundary condition, an elaborate projection of the exact boundary condition is proposed as the boundary flux, which, in combination with some proper penalty terms, leads to the stability and optimal error estimates. For other three types of boundary conditions, optimal error estimates can also be proved for fluxes without any penalty terms when special projections are designed to match different boundary conditions. Numerical experiments are presented to confirm the sharpness of theoretical results.