This paper has introduced a novel fully discrete hybridizable discontinuous Galerkin (HDG) ensemble Monte Carlo method (FEMC-HDG) tailored for solving the heat equation with random diffusion and Robin coefficients. The FEMC-HDG method solves a single linear system with multiple right-hand side vectors per time step. We established stability analysis and error estimates that are optimal in the spatial and first-order accuracy in time for the $L^{\mathrm{\infty }}(0,T,L^2(D))$-norm error estimate. Numerical experiments were included to confirm the theoretical convergence and showcase the method's efficiency.

A new ensemble Monte Carlo (EMC) method is proposed and applied to numerically simulate a parabolic optimal control problem with random coefficients. The state equation is discretized by the EMC method, which shares a common coefficient matrix with multiple right-hand vectors. It saves the computational cost compared with the Monte Carlo (MC) method. For this new EMC method, it is unconditionally stable and does not need to subgroup the samples in the simulation. Under natural regularity condition, some error estimates are obtained for the EMC approximation of the optimal control problem. Two numerical examples are presented to test the theoretical results.

The keys to constructing numerical schemes for nonlinear partial differential equations are accuracy, handling of the nonlinear terms, and physical properties (energy dissipation or conservation). In this paper, we employ the exponential scalar auxiliary variable (E-SAV) method to solve a semi-linear wave equation. By defining two different variables and combining the Crank−Nicolson scheme, two semi-discrete schemes are proposed, both of which are second-order and maintain Hamiltonian conservation. Two numerical experiments are presented to verify the reliability of the theory.

Deep operator networks is a popular machine learning approach. Some problems require multiple inputs and outputs. In this work, a multi-input and multi-output operator neural network (MIMOONet) for solving optimal control problems was proposed. To improve the accuracy of the numerical solution, a physics-informed MIMOONet was also proposed. To test the performance of the MIMOONet and the physics-informed MIMOONet, three examples, including elliptic (linear and semi-linear) and parabolic problems, were presented. The numerical results show that both methods are effective in solving these types of problems, and the physics-informed MIMOONet achieves higher accuracy due to its incorporation of physical laws.

A variational MAX ensemble-based time-stepping numerical method is proposed to simulate a transient heat equation with uncertain Robin boundary and diffusion coefficients. Instead of employing ensemble means for Robin coefficients as well as diffusion coefficients, the maximums of these coefficients are utilized at per time step. This is a new variational ensemble Monte Carlo (MC) numerical method, which we call the variational MAX ensemble Monte Carlo (VMEMC) method. In contrast with related methodologies, the novelty of this algorithm is that it is unconditionally stable. And also, the error estimates are proved. Numerical tests illustrate the theoretical properties for the VMEMC method.

This paper focused on approximating a second-order nonlinear hyperbolic optimal control problem. By introducing a new variable, the hyperbolic equation was converted into two parabolic equations. A second-order fully discrete scheme was obtained by combining the Crank-Nicolson formula with the finite element method. The error estimation for this scheme was derived utilizing the second-order sufficient optimality condition and auxiliary problems. To validate the effectiveness of the fully discrete scheme, a numerical example was presented.