13 research outputs found
Optimality Conditions for Convex Stochastic Optimization Problems in Banach Spaces with Almost Sure State Constraints
We analyze a convex stochastic optimization problem where the state is
assumed to belong to the Bochner space of essentially bounded random variables
with images in a reflexive and separable Banach space. For this problem, we
obtain optimality conditions that are, with an appropriate model, necessary and
sufficient. Additionally, the Lagrange multipliers associated with optimality
conditions are integrable vector-valued functions and not only measures. A
model problem is given demonstrating the application to PDE-constrained
optimization under uncertainty with an outlook for further applications
Optimality conditions for convex stochastic optimization problems in Banach spaces with almost sure state constraint
We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty
Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo
In this work we develop a new hierarchical multilevel approach to generate
Gaussian random field realizations in an algorithmically scalable manner that
is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC)
algorithms. This approach builds off of other partial differential equation
(PDE) approaches for generating Gaussian random field realizations; in
particular, a single field realization may be formed by solving a
reaction-diffusion PDE with a spatial white noise source function as the
righthand side. While these approaches have been explored to accelerate forward
uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous
constructions are not directly applicable to multilevel MCMC frameworks which
build fine scale random fields in a hierarchical fashion from coarse scale
random fields. Our new hierarchical multilevel method relies on a hierarchical
decomposition of the white noise source function in which allows us to
form Gaussian random field realizations across multiple levels of
discretization in a way that fits into multilevel MCMC algorithmic frameworks.
After presenting our main theoretical results and numerical scaling results to
showcase the utility of this new hierarchical PDE method for generating
Gaussian random field realizations, this method is tested on a four-level MCMC
algorithm to explore its feasibility