Risk-averse optimal control of semilinear elliptic PDEs

In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurable multifunctions. We illustrate our theoretical results with two numerical examples motivated by the optimal doping of semiconductor devices.

Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics.

On Solution Regularity of Linear Hyperbolic Stochastic PDE Using the Method of Characteristics

The generalized Polynomial Chaos (gPC) method is one of the most widely used numerical methods for solving stochastic differential equations. Recently, attempts have been made to extend the the gPC to solve hyperbolic stochastic partial differential equations (SPDE). The convergence rate of the gPC depends on the regularity of the solution. It is shown that the characteristics technique can be used to derive general conditions for regularity of linear hyperbolic PDE, in a detailed case study of a linear wave equation with a random variable coefficient and random initial and boundary data.

Study on Alkaline Mass Transfer Regularity with Dissolution in Porous Media

Inducting dissolution speed constant and multistage reaction series to the alkaline solution transmission equation, this article established the alkaline solution transmission equation with multistage reaction dynamics in the porous media of stratum mineral and calculated one-dimensional alkaline solution concentration distribution. Experimental results verified the correctness of transmission equation, moreover, it further analyzed the alkaline solution regularity in the porous media. The model will be used to predict alkali loss, optimize alkaline solution concentration and slug size, thereby alkaline waterflooding or combination drive can obtain better displacement characteristics and improve the oil recovery.