Mathematics of Computation最新文献

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.

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at t = 0 t = 0 which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H 1 ( H − 1 ) H^1(H^{-1}) , L 2 ( L 2 ) L^2(L^2) , and the L 2 ( H 1 ) L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H 1 ( H −