Sample Size Estimates for Risk-Neutral Semilinear PDE-Constrained Optimization

TL;DR

This paper develops nonasymptotic sample size estimates for risk-neutral PDE-constrained optimization problems using the SAA approach, providing bounds on sample requirements for accurate solutions.

Contribution

It introduces a novel method to derive explicit sample size bounds for SAA critical points in risk-neutral PDE optimization, with quantifiable accuracy guarantees.

Findings

Derived upper bounds on sample size for accurate critical points

Quantified accuracy using expectation and tail bounds

Provided numerical illustrations validating the approach

Abstract

The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented.