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On enforcing non-negativity in polynomial approximations in high dimensions

Yuan Chen, Dongbin Xiu, Xiangxiong Zhang

TL;DR

This paper first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems.

Abstract

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range, due to constraints posed by the underlying physical problems. Efficient numerical methods are thus needed to enforce such conditions. In this paper, we discuss effective numerical algorithms for polynomial approximation under non-negativity constraints. We first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems. Numerical examples are provided, for up to $200$ dimensions, to demonstrate the effectiveness and scalability of the methods.

On enforcing non-negativity in polynomial approximations in high dimensions

TL;DR

This paper first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems.

Abstract

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range, due to constraints posed by the underlying physical problems. Efficient numerical methods are thus needed to enforce such conditions. In this paper, we discuss effective numerical algorithms for polynomial approximation under non-negativity constraints. We first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems. Numerical examples are provided, for up to dimensions, to demonstrate the effectiveness and scalability of the methods.
Paper Structure (24 sections, 2 theorems, 44 equations, 17 figures, 1 table, 2 algorithms)

This paper contains 24 sections, 2 theorems, 44 equations, 17 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Least squares polynomial approximation
  4. Constrained Polynomial Approximation
  5. Efficient Iterative Methods
  6. The primal and dual problems
  7. Closed convex proper functions
  8. First order splitting methods
  9. FISTA with restart on the dual
  10. The convergence rate of FISTA methods
  11. Numerical Examples
  12. One-dimensional Examples
  13. Runge function
  14. Truncated sine function
  15. A step function
  16. ...and 9 more sections

Key Result

Theorem 3.2

\newlabelthm:FISTAn0 Let $\left\{\mathbf{u}_k\right\}_{k \geq 0}$ be the sequence generated by FISTA equ:FISTA for solving problem equ:probd2. Then for any $k \geq 1$, where $L$ is the maximum eigenvalue of the matrix $\mathbf{B}\mathbf{K}^\dagger\mathbf{B}^T$, $F(\mathbf{u}) := G^*(\mathbf{u}) + h^\star(\mathbf{u})$ is the objective function defined in equ:probd2 with optimal solution $\mathbf{

Figures (17)

  • Figure 1: The approximated polynomials for $f_1(x)$\ref{['equ:f1']} with positivity constraints for $n=10$ (Left), $n=20$ (Right). The positive approximation is found via solving \ref{['primal']} by restarted FISTA on \ref{['equ:probd2']}.
  • Figure 2: Comparasion for unconstrained approximation ($L^2$ projection) and non-negativity constrained approximation for $f_1(x)$\ref{['equ:f1']}. The convergence with respect to polynomial order (left) and percentage of negative points in $201$ non-negativity enforced sample points (right).
  • Figure 3: The approximated polynomials for $f_1(x)$\ref{['equ:f1']} with positivity constraints for $n=20$ with $M=31$ equispaced (left), Chebshev (middle) and random (right) constraint points. The areas enclosed by the dotted line are zoomed in for detailed presentation.
  • Figure 4: The approximated polynomials for $f_2(x)$\ref{['equ:tsin']} with positivity constraints for $n=5$ (Left), $n=20$ (Right). The positive approximation is found via solving \ref{['primal']} by restarted FISTA on \ref{['equ:probd2']}.
  • Figure 5: The comparison of convergence curve of several methods for approximating $f_2(x)$\ref{['equ:tsin']} with positivity constraints when $n=20$. The reference minimizer $\mathbf{c}^\star$ is obtained numerically via restarted FISTA with $5,000$ steps. For simplicity, Douglas-Rachford splitting uses the same step size as the FISTA method. We emphasize that Douglas-Rachford splitting could be much faster if tuning parameters.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Remark 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4