Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

2021 ◽  
Vol 19 (1) ◽  
pp. 147-174
Author(s):  
Wei Liu ◽  
Ziqing Xie ◽  
Wenfan Yi
Keyword(s):  
Elliptic Pdes ◽  
Minimax Method ◽  
Semilinear Elliptic ◽  
Unstable Solutions ◽  
Semilinear Elliptic Pdes

Combinatorial optimal control of semilinear elliptic PDEs

2018 ◽  
Vol 70 (3) ◽  
pp. 641-675 ◽  
Author(s):  
Christoph Buchheim ◽  
Renke Kuhlmann ◽  
Christian Meyer
Keyword(s):  
Optimal Control ◽  
Elliptic Pdes ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Pdes

scholarly journals Risk-averse optimal control of semilinear elliptic PDEs

2020 ◽  
Vol 26 ◽  
pp. 53 ◽  
Author(s):  
D.P. Kouri ◽  
T.M. Surowiec
Keyword(s):  
Optimal Control ◽  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Measures ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Continuity Properties ◽  
Semilinear Elliptic ◽  
Solution Regularity ◽  
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.

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