Abstract. The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity pro...

We characterize the long time behaviour of a discrete-in-time approximation volume preserving fractional mean curvature flow. In particular, we prove that discrete flow starting from any bounded set finite perimeter converges exponentially fast to single ball if dimension N≤7 and esponent s≈1, or for s∈(0,1) when N=2. As an intermediate result establish quantitative Alexandrov type estimate nor...

Let Sn = 1 nXnX∗ n where Xn = {Xij } is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t2, σ ) = √ p(xn(t1)∗(Sn + σI)−1xn(t2)− xn(t1)∗xn(t2)mn(σ )) in which σ > 0 and mn(σ)= ∫ dFyn (x) x+σ where Fyn(x) is the Marčenko–Pastur law with parameter yn = p/n; which converges to a positive constant as n→∞, and xn(t1) and xn(t2) are unit vectors in Cp ,...

We prove that the implicit time Euler scheme coupled with finite elements space discretization for 2D Navier–Stokes equations on torus subject to a random perturbation converges in $$L^2(\varOmega )$$ , and describe rate of convergence an $$H^1$$ -valued initial condition. This refines previous results which only established probability these numerical approximations. Using exponential moment e...

in this paper, a new numerical method for solving time-fractional diffusion equations is introduced. for this purpose, finite difference scheme for discretization in time and chebyshev collocation method is applied. also, to simplify application of the method, the matrix form of the suggested method is obtained. illustrative examples show that the proposed method is very efficient and accurate.

This paper is devoted to proposing a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations. To further improve its convergence rate, we derive preconditioned based on Kronecker product approximations A theoretical analysis shows that proposed converges an exact solution any initial at most finite steps in absence round-off errors. Compared with gradie...

Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge solution $x...

We study error estimates for a finite volume discretization of an elliptic equation. We prove that, for s ∈ [0,1], if the exact solution belongs to H1+s and the right-hand side is f + div(G) with f ∈ L2 and G ∈ (Hs)N , then the solution of the finite volume scheme converges in discrete H1norm to the exact solution, with a rate of convergence of order hs (where h is the size of the mesh).

We consider latent structural versions of probit loss and ramp loss. We show that these surrogate loss functions are consistent in the strong sense that for any feature map (finite or infinite dimensional) they yield predictors approaching the infimum task loss achievable by any linear predictor over the given features. We also give finite sample generalization bounds (convergence rates) for th...