Wong--Zakai Approximations of Stochastic Allen-Cahn Equation

We establish a unconditional and optimal strong convergence rate of Wong--Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen--Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the fully use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation, in combination with the factorization method and stochastic calculus in martingale type 2 Banach spaces applied to deduce sharp error estimation between the exact and approximate Ornstein--Uhlenbeck processes, in Banach space norm

Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises

In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the well-posedness of the problem, the linear operator of the equations is only need to be a generator of a \CC_0-semigroup and the proposed noises are quite general, which include space-time white noise and rougher noises. When the linear operator generates an analytic \CC_0-semigroup, we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method

Approximating Stochastic Evolution Equations with Additive White and Rough Noises

In this paper, we analyze Galerkin approximations for stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to $1/2$. First we regularize the noise by the Wong-Zakai approximation and obtain its optimal order of convergence. Then we apply the Galerkin method to discretize the stochastic evolution equations with regularized noises. Optimal error estimates are obtained for the Galerkin approximations. In particular, our error estimates remove an infinitesimal factor which appears in the error estimates of various numerical methods for stochastic evolution equations in existing literatures.Comment: 32 page

Finite element approximations for second order stochastic differential equation driven by fractional Brownian motion

We consider finite element approximations for a one dimensional second order stochastic differential equation of boundary value type driven by a fractional Brownian motion with Hurst index $H\le 1/2$. We make use of a sequence of approximate solutions with the fractional noise replaced by its piecewise con- stant approximations to construct the finite element approximations for the equation. The error estimate of the approximations is derived through rigorous convergence analysis.Comment: To appear in IMA Journal of Numerical Analysis; the time-dependent case such as stochastic heat equation and stochastic wave equation driven by fractional Brownian sheet with temporal Hurst index $1/2$ and spatial Hurst index $H\le 1/2$ has been considered by arXiv:1601.02085 for spatially Galerkin approximations and a forthcoming paper for fully discrete approximation

Optimal Regularity of Stochastic Evolution Equations in M-type 2 Banach Spaces

In this paper, we prove the well-posedness and op- timal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [HL] dealing with Hilbert setting and a version of Burkholder inequality in M-type 2 Banach space. Applying our main results to the stochastic heat equation gives a positive an- swer to an open problem proposed in [JR12].Comment: 17 page

Deep Scene Text Detection with Connected Component Proposals

A growing demand for natural-scene text detection has been witnessed by the computer vision community since text information plays a significant role in scene understanding and image indexing. Deep neural networks are being used due to their strong capabilities of pixel-wise classification or word localization, similar to being used in common vision problems. In this paper, we present a novel two-task network with integrating bottom and top cues. The first task aims to predict a pixel-by-pixel labeling and based on which, word proposals are generated with a canonical connected component analysis. The second task aims to output a bundle of character candidates used later to verify the word proposals. The two sub-networks share base convolutional features and moreover, we present a new loss to strengthen the interaction between them. We evaluate the proposed network on public benchmark datasets and show it can detect arbitrary-orientation scene text with a finer output boundary. In ICDAR 2013 text localization task, we achieve the state-of-the-art performance with an F-score of 0.919 and a much better recall of 0.915.Comment: 10 pages, 5 figure

Instrumental variable estimation of early treatment effect in randomized screening trials

The primary analysis of randomized screening trials for cancer typically adheres to the intention-to-screen principle, measuring cancer-specific mortality reductions between screening and control arms. These mortality reductions result from a combination of the screening regimen, screening technology and the effect of the early, screening-induced, treatment. This motivates addressing these different aspects separately. Here we are interested in the causal effect of early versus delayed treatments on cancer mortality among the screening-detectable subgroup, which under certain assumptions is estimable from conventional randomized screening trial using instrumental variable type methods. To define the causal effect of interest, we formulate a simplified structural multi-state model for screening trials, based on a hypothetical intervention trial where screening detected individuals would be randomized into early versus delayed treatments. The cancer-specific mortality reductions after screening detection are quantified by a cause-specific hazard ratio. For this, we propose two estimators, based on an estimating equation and a likelihood expression. The methods extend existing instrumental variable methods for time-to-event and competing risks outcomes to time-dependent intermediate variables. Using the multi-state model as the basis of a data generating mechanism, we investigate the performance of the new estimators through simulation studies. In addition, we illustrate the proposed method in the context of CT screening for lung cancer using the US National Lung Screening Trial (NLST) data.Comment: Lifetime Data Anal (2021

Well-posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator. Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems

Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise

We derive an asymptotic log-Harnack inequality for nonlinear monotone SPDE driven by possibly degenerate multiplicative noise. Our main tool is the asymptotic coupling by the change of measure. As an application, we show that, under certain monotone and coercive conditions on the coefficients, the corresponding Markov semigroup is asymptotically strong Feller, asymptotic irreducibility, and possesses a unique and thus ergodic invariant measure. The results are applied to highly degenerate finite-dimensional or infinite-dimensional diffusion processes

Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in LpL^p

We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive rough noise in the $L^p$-space, for any $p \ge 2$. These inequalities are used to study the ergodicity of the corresponding Markov semigroup $(P_t)_{t \ge 0}$. The main ingredient of our method is a coupling by the change of measure. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having a negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)_{t \ge 0}$ possesses a unique and thus ergodic invariant measure in $L^p$ for all $p \ge 2$, which is independent of the Lipschitz term