Efficient estimation of coherent risk measures for risk-averse optimization problems governed by partial differential equations with random inputs

The scope of this thesis is the assessment and design of structure-exploiting methods for the efficient estimation of risk measures of quantities of interest in the context of optimization of partial differential equations (PDEs) with random inputs. Risk measures of the quantities of interest arise as objective functions or as constraints in the PDE-constrained optimization problems under uncertainty. A single evaluation of a risk measure requires numerical integration in a high-dimensional parameter space, which requires the solution of the PDE at many parameter samples. When the integrand is smooth in the random parameters, efficient methods, such as sparse grids, exist that substantially reduce the sample size. Unfortunately, many risk-averse formulations, such as semideviation and Conditional Value-at-Risk, introduce a non-smoothness in integrand. This work demonstrates that naive application of sparse grids and other smoothness-exploiting approaches is not beneficial in the risk-averse case. For the widely used class of coherent risk measures, this thesis proposes a new method for evaluating risk-averse objectives based on the biconjugate representation of coherent risk functions and importance sampling. The method is further enhanced by utilizing reduced order models of the PDEs under consideration. The proposed method leads to substantial reduction in the number of PDE solutions required to accurately estimate coherent risk measures. The performance of existing and of the new methods for the estimation of risk measures is demonstrated on examples of risk-averse PDE-constrained optimization problems. The resulting method can substantially reduce the number of PDE solutions required to solve optimization problems, and, therefore, enlarge the applicability of important risk measures for PDE-constrained optimization problems under uncertainty.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Engineering

Grantor

Rice University

Year dc:date.issued

2017

Author dc:creator

• Takhtaganov, Timur

Advisor dc:contributor.advisor

• Heinkenschloss, Matthias

• optimization under uncertainty
• PDE-constrained optimization
• risk-averse optimization
• risk measures
• importance sampling
• reduced order models
• conditional value-at-risk

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/105454

OAI identifier oai:identifier

oai:repository.rice.edu:1911/105454