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Polytope Division Method: A Scalable Sampling Method for Problems with High-dimensional Parameters

Evie Nielen, Oliver Tse, Karen Veroy

TL;DR

The Polytope Division Method is introduced, a scalable greedy-type approach that adaptively partitions the parameter space and targets regions of high loss, overcoming the limitations of traditional methods.

Abstract

Configuration Optimization Problems (COPs), which involve minimizing a loss function over a set of discrete points $\boldsymbolγ \subset P$, are common in areas like Model Order Reduction, Active Learning, and Optimal Experimental Design. While exact solutions are often infeasible, heuristic methods such as the Greedy Sampling Method (GSM) provide practical alternatives, particularly for low-dimensional cases. GSM recursively updates $\boldsymbolγ$ by solving a continuous optimization problem, which is typically approximated by a search over a discrete sample set $S \subset P$. However, as the dimensionality grows, the sample size suffers from the curse of dimensionality. To address this, we introduce the Polytope Division Method (PDM), a scalable greedy-type approach that adaptively partitions the parameter space and targets regions of high loss. PDM achieves linear scaling with problem dimensionality and offers an efficient solution approach for high-dimensional COPs, overcoming the limitations of traditional methods.

Polytope Division Method: A Scalable Sampling Method for Problems with High-dimensional Parameters

TL;DR

The Polytope Division Method is introduced, a scalable greedy-type approach that adaptively partitions the parameter space and targets regions of high loss, overcoming the limitations of traditional methods.

Abstract

Configuration Optimization Problems (COPs), which involve minimizing a loss function over a set of discrete points , are common in areas like Model Order Reduction, Active Learning, and Optimal Experimental Design. While exact solutions are often infeasible, heuristic methods such as the Greedy Sampling Method (GSM) provide practical alternatives, particularly for low-dimensional cases. GSM recursively updates by solving a continuous optimization problem, which is typically approximated by a search over a discrete sample set . However, as the dimensionality grows, the sample size suffers from the curse of dimensionality. To address this, we introduce the Polytope Division Method (PDM), a scalable greedy-type approach that adaptively partitions the parameter space and targets regions of high loss. PDM achieves linear scaling with problem dimensionality and offers an efficient solution approach for high-dimensional COPs, overcoming the limitations of traditional methods.

Paper Structure

This paper contains 11 sections, 6 theorems, 56 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Greedy Sampling Method
  3. Polytope Division Method
  4. COPs in Model Order Reduction
  5. Reduced Basis Method
  6. Empirical Interpolation Method
  7. Numerical Tests
  8. Thermal block example
  9. Heat conduction model with Gaussian heat source
  10. Outlook
  11. Proof of Lemma \ref{['lemma: facets p-boundary polytopes']}

Key Result

Theorem 3.10

\newlabelthm: main theorem polytope division0 Let $P$ be a $d$-dimensional hyperrectangle and let $\mathscr{D}^N$ be the polytope division of $P$ constructed by Algorithm pdm at step $N \ge 1$. Then where $\mathscr{S} \subset \mathbb{S}$ is a family of simplices and $\mathscr{B} \subset \mathbb{B}_{P}$ a family of $P$-boundary polytopes.

Figures (13)

  • Figure 1: Depiction of the steps in PDM for the $2$-dimensional parameter case. (1) Sample first parameter. (2) Compute barycenters. (3) Select the barycenter with the largest objective function value. (4) Update polytope division. (5) Compute the new barycenters.
  • Figure 1: Thermal Block with $6$ parameters
  • Figure 2: Example of a $2$-dimensional polytope along with its facets and vertices. (a) Example of a polytope, (b) facets of a polytope, and (c) vertices of a polytope.
  • Figure 3: Example of the convex hull of a point and a facet. The dashed line in (a) represents the polytope, the red dot a point $p$, and the thick red line the facet $F$. The filled area in (b) represents $\mathrm{Conv}(p \cup F)$.
  • Figure 3: Verification stage: Maximum squared residual error for the reduced bases of the thermal block problem for PDM, GSM-LHS, and GSM-Random
  • ...and 8 more figures

Theorems & Definitions (19)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: Polytopes, faces, facets, and vertices
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Definition 3.9
  • Theorem 3.10
  • ...and 9 more