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A combination technique for optimal control problems constrained by random PDEs

Fabio Nobile, Tommaso Vanzan

TL;DR

A combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems constrained by random partial differential equations and states that the asymptotic complexity is exclusively determined by the spatial solver.

Abstract

We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the (possible) convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.

A combination technique for optimal control problems constrained by random PDEs

TL;DR

A combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems constrained by random partial differential equations and states that the asymptotic complexity is exclusively determined by the spatial solver.

Abstract

We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the (possible) convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.
Paper Structure (13 sections, 9 theorems, 75 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 9 theorems, 75 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Combination technique
  4. Construction of the multi-index set $\mathcal{I}$
  5. An a-priori construction
  6. Complexity analysis
  7. Numerical section
  8. Discussion on the validity of the assumptions
  9. Numerical tests
  10. Conclusions
  11. Appendix
  12. Proof of Theorem \ref{['thm:stoc']}
  13. Proof of Theorem \ref{['thm:stoc_det']}

Key Result

Theorem 1

\newlabelthm:stoc0 There exist constants $C_1$ and $C_2$ such that for any $W_{\max}$ satisfying $W_{\max}\geq \frac{|\widetilde{\bm{ g}}|^{2N}C_1}{(2N)!}$, and choosing $\widehat{L}=\sqrt[2N]{\frac{W_{\max}(2N)!}{C_1}}-|\widetilde{\bm{ g}}|$,

Figures (4)

  • Figure 1: Numerical validation of Assumption \ref{['eq:ass:2']}, \ref{['eq:ass:4']} and \ref{['ass:produc_structure_E']} on the decay of the error contributions. Spatial error contributions with $\bar{{\bm \beta}}=\bm{1}$ (left), stochastic error contributions with $\bar{{\bm \alpha}}=(3,3)$ (center) for the first four random variables, and mixed spatial and stochastic error contributions (right). The solid lines are based on computed values, the dashed lines are the fitted ansatzes.
  • Figure 2: Convergence behaviour of the different methods for $N=2,6,10$ random variables.
  • Figure 3: The left and center panel show the sparsity pattern of $(\beta_1,\beta_2,\beta_3)$ and $(\beta_1,\beta_9,\beta_{10})$ of Alg. 2 for $d=2$ and $N=10$. The right panel verifies numerically the simplified convergence estimate \ref{['eq:decay_simplified']}.
  • Figure 4: Convergence of the combined $({\bm \alpha},{\bm \beta})$ CT for a one-dimensional physical problem (top row) and a two-dimensional one (bottom row)

Theorems & Definitions (18)

  • Example 1
  • Remark 1: On the convergence of the infinite series
  • Remark 2: Sparse grids vs Combination technique
  • Remark 3: Multilevel/multi-index (Quasi-)Monte Carlo methods
  • Theorem 1
  • Theorem 2
  • Remark 4: Infinite dimensional setting
  • Lemma 1: Lemma 4 in haji2016multi
  • Lemma 2
  • Proof 1
  • ...and 8 more