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A Probabilistic Approach to Trajectory-Based Optimal Experimental Design

Ahmed Attia

TL;DR

A novel probabilistic approach to optimal path experimental design that enables exploration of the utility function's distribution tail and treats the utility function of the design as a black box, making it applicable to linear and nonlinear inverse problems and beyond experimental design.

Abstract

We present a novel probabilistic approach for optimal path experimental design. In this approach a discrete path optimization problem is defined on a static navigation mesh, and trajectories are modeled as random variables governed by a parametric Markov policy. The discrete path optimization problem is then replaced with an equivalent stochastic optimization problem over the policy parameters, resulting in an optimal probability model that samples estimates of the optimal discrete path. This approach enables exploration of the utility function's distribution tail and treats the utility function of the design as a black box, making it applicable to linear and nonlinear inverse problems and beyond experimental design. Numerical verification and analysis are carried out by using a parameter identification problem widely used in model-based optimal experimental design.

A Probabilistic Approach to Trajectory-Based Optimal Experimental Design

TL;DR

A novel probabilistic approach to optimal path experimental design that enables exploration of the utility function's distribution tail and treats the utility function of the design as a black box, making it applicable to linear and nonlinear inverse problems and beyond experimental design.

Abstract

We present a novel probabilistic approach for optimal path experimental design. In this approach a discrete path optimization problem is defined on a static navigation mesh, and trajectories are modeled as random variables governed by a parametric Markov policy. The discrete path optimization problem is then replaced with an equivalent stochastic optimization problem over the policy parameters, resulting in an optimal probability model that samples estimates of the optimal discrete path. This approach enables exploration of the utility function's distribution tail and treats the utility function of the design as a black box, making it applicable to linear and nonlinear inverse problems and beyond experimental design. Numerical verification and analysis are carried out by using a parameter identification problem widely used in model-based optimal experimental design.
Paper Structure (57 sections, 3 theorems, 62 equations, 42 figures, 3 algorithms)

This paper contains 57 sections, 3 theorems, 62 equations, 42 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background: Model-Based OED
  3. Probabilistic Optimization Approach
  4. Probabilistic Approach to Optimal Path OED
  5. Probabilistic Policies for the Experimental Path
  6. First-Order Policy
  7. Modeling the initial distribution
  8. Modeling the transition distribution
  9. Higher-Order Path Policies
  10. Path policy based on the Raftery model
  11. Path policy based on a generalized Raftery model
  12. Fixing the Lag Weights
  13. Sampling Trajectories
  14. Probabilistic Path OED Problem
  15. Complete Algorithmic Statement
  16. ...and 42 more sections

Key Result

Proposition 3.1

The policy eqn:first_order_trajectory_distribution_full has the following gradient of log-probability:

Figures (42)

  • Figure 1: Wall times against mesh cardinality (number of nodes) for sampling (top) and log-policy gradient (bottom) across the three policy models, averaged over $32$ runs on an Apple M1 Mac laptop without parallelization. First-order, higher-order, and generalized higher-order models correspond to \ref{['defn:first_order_path_model']}, \ref{['defn:higher_order_path_model']}, and \ref{['defn:generalized_higher_order_path_model']}, respectively. Results of the first-order policy are repeated for comparison.
  • Figure 1: Advection-diffusion model \ref{['eqn:advection_diffusion']}. Panels from left to right are (1) the constant velocity field; (2) the prior variance field, that is, the diagonal of the prior covariance matrix; (3) ground truth of the inference parameter; that is, the true initial condition; and (4) the model state at the final simulation time instance $T=3.6$.
  • Figure 1: Directed graph describing a simple navigation mesh. The graph consists of $N=5$ nodes $v_1, \ldots, v_5$ with transition parameters $p^{i}_{j}$ describing the probability of moving from node $v_i$ to node $v_j$ displayed on the respective arcs $(v_i, v_j)$.
  • Figure 1: Results of \ref{['alg:probabilistic_path_optimization']} with the first-order policy (\ref{['defn:first_order_path_model']}) applied to the fine navigation mesh (\ref{['fig:navigation_meshes']}, right) with trajectory length of $n=19$ and a group of $s=7$ moving sensors.
  • Figure 2: Navigation meshes used. Left: a coarse mesh with $75$ candidate locations (graph vertices/nodes). Right: a fine navigation mesh with $332$ nodes.
  • ...and 37 more figures

Theorems & Definitions (9)

  • Proposition 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • Proof 4: Proof of \ref{['proposition:first_order_gradient']}
  • Proof 5: Proof of \ref{['proposition:higher_order_gradient']}
  • Proof 6: Proof of \ref{['proposition:generalized_higher_order_gradient']}