Control constrained parabolic optimal control problems are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can be directly applied to such problems. An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations. At each iteration of the ADMM, the main computation is for solving an unconstrained parabolic optimal control subproblem. Because of its inevitably high dimensionality after space-time discretization, the parabolic optimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required. It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and that can be performed automatically with no need to set empirically perceived constant accuracy a priori. The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly, yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by some numerical results. Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations, and fractional parabolic equations.

In this paper, we are concerned about the stability analysis for a Perfectly Matched Layer (PML) recently developed by Bécache et al. [5] for simulating wave propagation in the Drude metamaterial. This PML is proved to be stable originally in [6] through a modal analysis. Here we establish its stability by the energy method. A FDTD scheme is developed and analyzed. Numerical simulations illustrate the stability of the PML model and its effectiveness in absorbing outgoing waves in the Drude medium.

We introduce a novel algorithm for gradient-based optimization of stochastic objective functions. The method may be seen as a variant of SGD with momentum equipped with an adaptive learning rate automatically adjusted by an ‘energy’ variable. The method is simple to implement, computationally efficient, and well suited for large-scale machine learning problems. The method exhibits unconditional energy stability for any size of the base learning rate. We provide a regret bound on the convergence rate under the online convex optimization framework. We also establish the energy-dependent convergence rate of the algorithm to a stationary point in the stochastic non-convex setting. In addition, a sufficient condition is provided to guarantee a positive lower threshold for the energy variable. Our experiments demonstrate that the algorithm converges fast while generalizing better than or as well as SGD with momentum in training deep neural networks, and compares also favorably to Adam.

We introduce a population-age-time (PAT) model which describes the temporal evolution of the population distribution in age. The surprising result is that the qualitative nature of the population distribution dynamics is robust with respect to the birth rate and death rate distributions in age, and initial conditions. When the number of children born per woman is 2, the population distribution approaches an asymptotically steady state of a kink shape; thus the total population approaches a constant. When the number of children born per woman is greater than 2, the total population increases without bound; and when the number of children born per woman is less than 2, the total population decreases to zero.

We study the homogenization of a boundary obstacle problem on a $C^{1,α}$-domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $\gamma.$ For any $\epsilon \in \mathbb{R}_+,$ $∂D=\Gamma ∪Σ,$ $\Gamma ∩Σ=∅$ and $S_{\epsilon}\subset Σ$ with suitable assumptions, we prove that as $\epsilon$ tends to zero, the energy minimizer $u^{\epsilon}$ of $\int_D |\gamma ∇u|^2dx,$ subject to $u≥\varphi$ on $S_{\epsilon},$ up to a subsequence, converges weakly in $H^1 (D)$ to $\tilde{u},$ which minimizes the energy functional $$\int_D |\gamma∇u|^2+ \int_Σ (u−\varphi)^2\_\mu (x)dS_x,$$ where $\mu (x)$ depends on the structure of $S_{\epsilon}$ and $\varphi$ is any given function in $C^∞(\overline{D}).$