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Optimal bounds for POD approximations of infinite horizon control problems based on time derivatives

Javier de Frutos, Bosco Garcia-Archilla, Julia Novo

TL;DR

This paper considers the numerical approximation of infinite horizon problems via the dynamic programming approach using a new proper orthogonal decomposition (POD) method based on time derivatives that bound some terms of the error that could not be bounded in a standard POD approach.

Abstract

In this paper we consider the numerical approximation of infinite horizon problems via the dynamic programming approach. The value function of the problem solves a Hamilton-Jacobi-Bellman (HJB) equation that is approximated by a fully discrete method. It is known that the numerical problem is difficult to handle by the so called curse of dimensionality. To mitigate this issue we apply a reduction of the order by means of a new proper orthogonal decomposition (POD) method based on time derivatives. We carry out the error analysis of the method using recently proved optimal bounds for the fully discrete approximations. Moreover, the use of snapshots based on time derivatives allow us to bound some terms of the error that could not be bounded in a standard POD approach. Some numerical experiments show the good performance of the method in practice.

Optimal bounds for POD approximations of infinite horizon control problems based on time derivatives

TL;DR

This paper considers the numerical approximation of infinite horizon problems via the dynamic programming approach using a new proper orthogonal decomposition (POD) method based on time derivatives that bound some terms of the error that could not be bounded in a standard POD approach.

Abstract

In this paper we consider the numerical approximation of infinite horizon problems via the dynamic programming approach. The value function of the problem solves a Hamilton-Jacobi-Bellman (HJB) equation that is approximated by a fully discrete method. It is known that the numerical problem is difficult to handle by the so called curse of dimensionality. To mitigate this issue we apply a reduction of the order by means of a new proper orthogonal decomposition (POD) method based on time derivatives. We carry out the error analysis of the method using recently proved optimal bounds for the fully discrete approximations. Moreover, the use of snapshots based on time derivatives allow us to bound some terms of the error that could not be bounded in a standard POD approach. Some numerical experiments show the good performance of the method in practice.
Paper Structure (11 sections, 11 theorems, 107 equations, 10 figures, 1 table)

This paper contains 11 sections, 11 theorems, 107 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model problem and standard numerical approximation
  3. POD approximation of the optimal control problem based on time derivatives
  4. POD approximation based on time derivatives
  5. The POD control problem
  6. Error analysis of the method
  7. Numerical Experiments
  8. Test 1: Semilinear equation
  9. Test 2: Advection-diffusion equation
  10. Test 3: A two-dimensional reaction-diffusion equation.
  11. Conclusions

Key Result

Theorem 1

Let assumptions lip_f, infty_f, invariance, lip_g, cota_g, semi_con_f and semi_con_g hold and let $\lambda>\max(2L_g,L_f)$. Let $v$ and $v_h$ be the solutions of HJB and discrete_HJB, respectively. Then, there exists a constant $C\ge 0$, that can be bounded explicitly, such that the following bound

Figures (10)

  • Figure 1: Test 1: Optimal HJB states computed with r=4 POD basis functions (top-left), difference between optimal solution with 4 and 2 POD basis functions (top-middle), difference between optimal solution with 4 and 3 POD bases (top-right). Optimal HJB controls with $r=4,3,2$ (bottom). The red crosses correspond to the values of the controls that we have joined by a blue line.
  • Figure 2: Test 1: Value of the cost functional \ref{['cost_f']} on the optimal HJB states for $r=2,3,4$. The red crosses correspond to the values of the cost values that are joined by a pdf line.
  • Figure 3: Test 1: Relative error between the optimal HJB states with $r=4$ corresponding to solving \ref{['fully_discrete_pod']} by fixed point iteration with tolerances ${\rm TOL}_v=5\times 10^{-4}$ and ${\rm TOL}_v=1\times 10^{-4}$ (left), ${\rm TOL}_v=1\times 10^{-4}$ and ${\rm TOL}_v=2\times 10^{-5}$ (centre), and optimal HJB controls (right).
  • Figure 4: Test 1: Results for different values of $k_r$; relative errors between HJB states (top left) and controls (bottom left) with respect to to $k_r=0.005$; HBJ controls (centre) and values of the cost functional \ref{['cost_f']} (right).
  • Figure 5: Test 1: Results for POD basis extracted from snapshots ($\Delta x =1/50$, $r=3$); Relative errors of HJB state (left) and control (centre) with respect to the case where POD basis is taken from time derivatives; HBJ control (right).
  • ...and 5 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Proof 1
  • Lemma 3
  • Proof 2
  • Lemma 4
  • Proof 3
  • ...and 8 more