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Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and…
In this paper, a local-global model reduction method is presented to solve stochastic optimal control problems governed by partial differential equations (PDEs). If the optimal control problems involve uncertainty, we need to use a few…
We present a new splitting method for time-dependent convection-dominated diffusion problems. The original convection diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a…
One of the beauties of the projected gradient descent method lies in its rather simple mechanism and yet stable behavior with inexact, stochastic gradients, which has led to its wide-spread use in many machine learning applications.…
In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the…
This article presents a new numerical scheme for the discretization of dissipative particle dynamics with conserved energy. The key idea is to reduce elementary pairwise stochastic dynamics (either fluctuation/dissipation or thermal…
In this work, we consider a distributed multi-agent stochastic optimization problem, where each agent holds a local objective function that is smooth and convex, and that is subject to a stochastic process. The goal is for all agents to…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle…
This paper considers spectral-difference methods of a high-order of accuracy for solving the one-way wave equation using the Laguerre integral transform with respect to time as the base. In order to provide a high spatial accuracy and…
In this paper, a high-order exponential scheme is developed to solve the 1D unsteady convection-diffusion equation with Neumann boundary conditions. The present method applies fourth-order compact exponential difference scheme in spatial…
A new analytic approximate technique for addressing nonlinear problems, namely the optimal perturbation iteration method, is introduced and implemented to singular initial value Lane-Emden type problems to test the effectiveness and…
In this work, we study a single-machine scheduling problem that aims at minimizing the total cost of a schedule subject to start-time dependent costs. This framework naturally captures scenarios where costs fluctuate throughout the day,…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
Time distributed optimization is an implementation strategy that can significantly reduce the computational burden of model predictive control by exploiting its robustness to incomplete optimization. When using this strategy, optimization…
Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for…
Optimization via simulation has been well established to find optimal solutions and designs in complex systems. However, it still faces modeling and computational challenges when extended to the multi-stage setting. This survey reviews the…
In this paper we introduce adaptive time step control for simulation of evolution of ice sheets. The discretization error in the approximations is estimated using "Milne's device" by comparing the result from two different methods in a…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…