• Abstract

In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods possibly with variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deterministic or stochastic ADMM. This new stochastic optimization algorithm is shown to have ergodic converge in expectation with $\mathcal{O}(1/T)$ convergence rate, where $T$ denotes the number of outer iterations. Our preliminary experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications. Finally, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are discussed in the appendix.

• Abstract

We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables, which is the crucial step to derive macroscopic free energy. It is at the level of kernel expansion, or equivalently the free energy, that the symmetries of the interacting rigid molecules can be fully recognized. Thus, writing down the form of expansion consistent with the symmetries is significant. Symmetries of two types are considered. First, we examine the symmetry of an interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, with the linearly independent terms identified. Then, we study the molecular symmetry characterized by a point group in $O(3).$ The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.

• Abstract

In this paper, we develop a fully discrete scheme to solve the confined ternary blended polymers (TBP) model with four order parameters based on the stabilized-scalar auxiliary variable (S-SAV) approach in time and the Fourier spectral method in space. Then, theoretical analysis is given for the scheme based on the backward differentiation formula. The unconditional energy stability and mass conservation are derived. Rigorous error analysis is carried out to show that the fully discrete scheme converges with order $\mathcal{O}(\tau^2+h^m)$ in the sense of the $L^2$ norm, where $\tau$ is the time step, $h$ is the spatial step, and $m$ is the regularity of the exact solution. Finally, some numerical results are given to demonstrate the theoretical analysis. Moreover, the phase separation of two kinds of polymer particles, namely, Ashura and Janus core-shell particles, is presented to show the morphological structures.

• Abstract

In this paper, we propose and develop a saturation value total variation (SV-TV) regularization model for simultaneously image denoising and luminance adjustment. The idea is to propose a variational approach containing an energy functional to adjust the luminance between image patches, and the noise of the image can be removed. In the proposed model, we establish the adjustment term based on the concept of structure, luminance, and contrast similarity, and we make use of the SV-TV regularization to remove the noise simultaneously. We present an efficient and effective algorithm with convergence guaranteed to solve the proposed minimization model. Experimental results are presented to show the effectiveness of the proposed model compared with existing methods.