In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second-order time-stepping for the numerical solution of the “good” Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, a ∞(0, T ∗;H 2) convergence for the solution and ∞(0, T ∗; 2)...

It is well known that the optimal convergence rate for stochastic optimization of smooth functions is O(1/ √ T ), which is same as stochastic optimization of Lipschitz continuous convex functions. This is in contrast to optimizing smooth functions using full gradients, which yields a convergence rate of O(1/T ). In this work, we consider a new setup for optimizing smooth functions, termed as Mi...

In this paper, we propose an inexact alternating direction method with square quadratic proximal  (SQP) regularization for  the structured variational inequalities. The predictor is obtained via solving SQP system  approximately  under significantly  relaxed accuracy criterion  and the new iterate is computed directly by an explicit formula derived from the original SQP method. Under appropriat...

We consider the differential sum rule for the effective scattering rate 1/t(v) and optical conductivity s1(v) in a dirty BCS superconductor, for arbitrary ratio of the superconducting gap D and the normal state constant damping rate 1/t . We show that if t is independent of T, the area under 1/t(v) does not change between the normal and the superconducting states, i.e., there exists an exact di...

We study the homogenization of 2D linear transport equations, ut + ~a(~x/ε) · ∇~xu = 0, where ~a is a non-vanishing vector field with integral invariance on the torus T . When the underlying flow on T 2 is ergodic, we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate. This result unifies and illuminates the ...

In a Hilbert setting, we introduce a new dynamical system and associated algorithms for solving monotone inclusions by rapid methods. Given a maximal monotone operator A, the evolution is governed by the time dependent operator I− (I+λ(t)A)−1, where the positive control parameter λ(t) tends to infinity as t→ +∞. The tuning of λ(·) is done in a closed-loop way, by resolution of the algebraic equ...

Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: Cρn , where ρ ...

Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...

Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...

In this short note, a direct proof of L2 convergence of an Euler–Maruyama approximation of a Zakai equation driven by a square integrable martingale is shown. The order of convergence is as known for real-valued stochastic differential equations and for less general driving noises O( √ ∆t) for a time discretization step size ∆t.