Stochastic subgradient methods play an important role in machine learning. We introduced the concepts of subgradient methods and stochastic subgradient methods in this project, discussed their convergence conditions as well as the strong and weak points against their competitors. We demonstrated the application of (stochastic) subgradient methods to machine learning with a running example of tr...

in the present article, we focus on the numerical approximation of stochastic partial differential equations of itˆo type with space-time white noise process, in particular, parabolic equations. for each case of additive andmultiplicative noise, the numerical solution of stochastic diffusion equations is approximated using two stochastic finite difference schemes and the stability and consisten...

Many Bayesian models involve continuous but non-differentiable log-posteriors, including the sparse Bayesian methods with a Laplace prior and the regularized Bayesian methods with maxmargin posterior regularization that acts like a likelihood term. In analogy to the popular stochastic subgradient methods for deterministic optimization, we present the stochastic subgradient MCMC for efficient po...

In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving differential equations with conserved quantity. The order conditions the are given based on theory B-series and multicolored rooted tree. Sufficient preserving quantity derived in terms coefficients. Conservative mean square convergence 1 general equations, as well high single integ...

We give a short introduction to the stochastic calculus for ItôLévy processes and review briefly the two main methods of optimal control of systems described by such processes: (i) Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation (ii) The stochastic maximum principle and its associated backward stochastic differential equation (BSDE). The two methods are illustrated by applica...

The immersed boundary method is a numerical approach for simulating elastic structures which interact with a fluid flow. In many physical systems thermal fluctuations become significant at small scales and play a fundamental role. In this paper stochastic numerical methods are developed which extend the immersed boundary approach to account for thermal fluctuations by including appropriate stoc...