Projection Onto A Simplex

This mini-paper presents a fast and simple algorithm to compute the projection onto the canonical simplex $\triangle^n$. Utilizing the Moreau's identity, we show that the problem is essentially a univariate minimization and the objective function is strictly convex and continuously differentiable. Moreover, it is shown that there are at most n candidates which can be computed explicitly, and the minimizer is the only one that falls into the correct interval

Approximation of Solution Operators for High-dimensional PDEs

We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method.Comment: 14 pages, 4 page appendix, 4 figure

On Optimal Control at the Onset of a New Viral Outbreak

An optimal control problem for the early stage of an infectious disease outbreak is considered. At that stage, control is often limited to non-medical interventions like social distancing and other behavioral changes. We show that the running cost of control satisfying mild, problem-specific, conditions generates an optimal control strategy that stays inside its admissible set for the entire duration of the study period $[0 ,T]$. For the optimal control problem, restricted by SIR compartmental model of disease transmission, we prove that the optimal control strategy, $u(t)$, may be growing until some moment $\bar{t} \in [0 ,T)$. However, for any $t \in [\bar{t}, T]$, the function $u(t)$ will decline as $t$ approaches $T$, which may cause the number of newly infected people to increase. So, the window from $0$ to $\bar{t}$ is the time for public health officials to prepare alternative mitigation measures, such as vaccines, testing, antiviral medications, and others. Our theoretical findings are illustrated with numerical examples showing optimal control strategies for various cost functions and weights. Simulation results provide a comprehensive demonstration of the effects of control on the epidemic spread and mitigation expenses, which can serve as invaluable references for public health officials