Related papers: A superconvergent stencil-adaptive SBP-SAT finite …
Stochastic approximation is a foundation for many algorithms found in machine learning and optimization. It is in general slow to converge: the mean square error vanishes as $O(n^{-1})$. A deterministic counterpart known as quasi-stochastic…
Recently-derived high-order splitting schemes with complex coefficients are shown to exhibit reduced convergence rates for certain parabolic evolution equations. When applied to semilinear reaction-diffusion equations with periodic boundary…
We construct a high-order adaptive time stepping scheme for vesicle suspensions with viscosity contrast. The high-order accuracy is achieved using a spectral deferred correction (SDC) method, and adaptivity is achieved by estimating the…
We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion are non-globally Lipschitz continuous. This stiffness may…
Stochastic gradient descent (SGD) for strongly convex functions converges at the rate $\bO(1/k)$. However, achieving good results in practice requires tuning the parameters (for example the learning rate) of the algorithm. In this paper we…
We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt + dL_t, $$ with irregular $\beta$-H\"older drift, $\beta > 0$, driven by a L\'evy…
We describe an asynchronous parallel stochastic coordinate descent algorithm for minimizing smooth unconstrained or separably constrained functions. The method achieves a linear convergence rate on functions that satisfy an essential strong…
In this paper, we develop a self-adaptive ADMM that updates the penalty parameter adaptively. When one part of the objective function is strongly convex i.e., the problem is semi-strongly convex, our algorithm can update the penalty…
In this paper, we analyze the convergence rate of the Jacobi-Proximal Alternating Direction Method of Multipliers (ADMM) initially introduced by Deng et al. for the block-structured optimization problem with linear constraint. The algorithm…
We consider energy stable summation by parts finite difference methods (SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam equation (DBE). Previously the constant coefficient problem has been solved with SBP-FD together with…
A new adaptive weighted essentially non-oscillatory WENO-$\theta$ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter…
We present an adaptive approximation scheme for jump-diffusion SDEs with discontinuous drift and (possibly) degenerate diffusion. This transformation-based doubly-adaptive quasi-Milstein scheme is the first scheme that has strong…
Large-scale optimization problems require algorithms both effective and efficient. One such popular and proven algorithm is Stochastic Gradient Descent which uses first-order gradient information to solve these problems. This paper studies…
In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{\sigma |X_s|^\alpha dW_s}, \;x_0>0,\;\sigma>0,\;…
We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a…
We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Gr\"unwald-Letnikov type central difference stencil. The filter…
The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite of its success in practice, the convergence properties of the standard ADMM for minimizing the sum of $N$…
The push-sum based subgradient is an important method for distributed convex optimization over unbalanced directed graphs, which is known to converge at a rate of $O(\ln t/\sqrt{t})$. This paper shows that the subgradient-push algorithm…
This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-by-parts (SBP) finite-difference operators. More precisely, the focus is on…
In this paper, we propose and analyze an adaptive time-stepping fully discrete scheme which possesses the optimal strong convergence order for the stochastic nonlinear Schr\"odinger equation with multiplicative noise. Based on the splitting…