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

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.

Figures (5)

• Figure 1: For $\alpha = 10^{-3}$ and discretization parameter $n=64$, nominal critical point, that is, a critical point of \ref{['eq:nom']}(left), and a reference critical point of \ref{['eq:ocp']}, that is, a critical point of \ref{['eq:saasob']}(right).
• Figure 2: For $n = 64$, empirical estimate $\widehat{\mathrm{E}}[\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})]$ of $\cE{\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})}$ over $N$ with $\alpha = 10^{-3}$(left) and empirical estimate $\widehat{\mathrm{E}}[\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})]$ of $\cE{\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})}$ over $\alpha$ with $N = 256$(right). Here, $\bar{u}_{N,\alpha,n}$ are SAA critical points of \ref{['eq:saafinitedim']} computed with sample size $N$, regularization parameter $\alpha$, and discretization parameter $n$. The approximated criticality measure $\widetilde{\chi}_n$ is defined in \ref{['eq:chiapprox']}. For both plots, $48$ independent realizations of the criticality measure $\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})$ are depicted. The convergence rates were computed using least squares. For each plot, the first four empirical means were excluded for the least squares computations.
• Figure 3: For $n = 64$ and $N=256$, empirical mean $\widehat{\mathrm{E}}[\widetilde{\Psi}_{n}(\bar{u}_{N,\alpha,n})]$ of $\mathbb{E}[\widetilde{\Psi}_{n}(\bar{u}_{N,\alpha,n})]$ over $\alpha$(left) and empirical mean $\widehat{\mathrm{E}}[\hat{\Psi}_{N,n}(\bar{u}_{N,\alpha,n})]$ of $\mathbb{E}[\hat{\Psi}_{N,n}(\bar{u}_{N,\alpha,n})]$ over $\alpha$(right). Here, $\bar{u}_{N,\alpha,n}$ are the SAA critical points of \ref{['eq:saafinitedim']} used in \ref{['fig:errors']}(right). The criticality measures $\widetilde{\Psi}_{n}$ and $\hat{\Psi}_{N,n}$ are defined in \ref{['eq:tikhonovapprox']}.
• Figure 4: For multiple values of the discretization parameter $n$, empirical estimates $\widehat{\mathrm{E}}[\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})]$ of $\cE{\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})}$ over $N$ with $\alpha = 10^{-1}$(left) and with $\alpha = 10^{-3}$(right). Here, $\bar{u}_{N,\alpha,n}$ are SAA critical points of \ref{['eq:saafinitedim']} computed with sample size $N$, regularization parameter $\alpha$, and discretization parameter $n$. The approximated criticality measure $\widetilde{\chi}_n$ is defined in \ref{['eq:chiapprox']}. After averaging the empirical estimates for each $N$, the convergence rates were computed using least squares.
• Figure 5: Empirical estimate $\widehat{\mathrm{E}}[\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})]$ of $\cE{\widetilde{\chi}_n(\bar{u}_{N,\alpha, n})}$ over the discretization parameter $n$ with $N = 256$ and $\alpha = 10^{-1}$(left) and with $N = 256$ and $\alpha = 10^{-3}$(right). Here, $\bar{u}_{N,\alpha,n}$ are SAA critical points of \ref{['eq:saafinitedim']} computed with sample size $N$, regularization parameter $\alpha$, and discretization parameter $n$. The approximated criticality measure $\widetilde{\chi}_n$ is defined in \ref{['eq:chiapprox']}. For both plots, $48$ independent realizations of the criticality measure $\widetilde{\chi}_n(\bar{u}_{N,\alpha,n})$ are depicted. The convergence rates were computed using least squares.

Theorems & Definitions (44)

• Lemma 3.3
• Lemma 3.4
• Proof 1
• Proof 2: Proof of \ref{['lem:properties_B']}
• Lemma 3.5
• Proof 3
• Proposition 3.6
• Lemma 3.7
• Proof 4
• Lemma 3.8