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Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs

Matthieu Cadiot, Jonathan Jaquette, Jean-Philippe Lessard, Akitoshi Takayasu

TL;DR

A validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, is introduced, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors.

Abstract

In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors. Second, we examine the Zernike polynomials, a spectral basis well-suited for problems on the disk, and highlight their essential properties. We further demonstrate how the MMT approach can be effectively employed to compute the product of truncated Zernike series, ensuring both accuracy and efficiency. Finally, we combine the MMT framework and Zernike series to construct computer-assisted proofs that establish the existence of solutions to two distinct nonlinear elliptic PDEs on the disk.

Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs

TL;DR

A validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, is introduced, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors.

Abstract

In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors. Second, we examine the Zernike polynomials, a spectral basis well-suited for problems on the disk, and highlight their essential properties. We further demonstrate how the MMT approach can be effectively employed to compute the product of truncated Zernike series, ensuring both accuracy and efficiency. Finally, we combine the MMT framework and Zernike series to construct computer-assisted proofs that establish the existence of solutions to two distinct nonlinear elliptic PDEs on the disk.

Paper Structure

This paper contains 24 sections, 11 theorems, 124 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials and the MMT approach
  3. Background on orthogonal polynomials and Gaussian quadrature
  4. Computing the entries of the MMT and iMMT matrices
  5. Computing the nodes $x_j$
  6. Computing the weights $\omega_j$
  7. Evaluating $p_n(x_j)$: the orthogonal polynomials at the grid points
  8. Computing the scaling factors $W_n$
  9. Validated numerics and comparisons
  10. Comparison: Evaluation of the Jacobi Polynomials
  11. Zernike Polynomials
  12. Definition of the Zernike polynomials
  13. Banach Algebra
  14. The MMT operator for Zernike polynomials and evaluating nonlinearities
  15. Pseudo-differential Operators
  16. ...and 9 more sections

Key Result

Theorem 2.2

Let $\{p_n(x)\}_{n=0}^\infty$ be a sequence of orthogonal polynomials with respect to a weight function $w(x)$ on the interval $[a, b]$ and let $\{x_j\}_{j=0}^N$ be the $N+1$ distinct roots of the orthogonal polynomial $p_{N+1}(x)$. Then, any polynomial $f(x)$ of degree at most $N$ can be uniquely r Hence, each entry of the iMMT matrix $M^{-1}$ is defined as where $n$ represents the row index and

Figures (11)

  • Figure 1: Diagram of evaluating the polynomial nonlinearity $\mathcal{G}$ via the MMT approach.
  • Figure 2: Real part of a numerically computed approximate solution to \ref{['eq : zero finding original 1/z']}.
  • Figure 3: The plot illustrates the runtime (in seconds) as a function of the polynomial degree $N$ for evaluating $P^{1,1}_N(x)$ at a random point $x \in [-1,1]$ using interval arithmetic. It is evident that the Forsythe algorithm is significantly faster than the linear system approach.
  • Figure 4: The relationship between the polynomial degree $N$ and the maximum radius of the output interval inclusions for evaluating $P^{1,1}_N$ at 50 random points within $[-1,1]$ is shown. We observe that the error bounds for the Forsythe algorithm increase rapidly, primarily due to the wrapping effect that arises when using interval arithmetic for the recursion formula.
  • Figure 6: In the plot, we illustrate the precision of the arithmetic (as a function of $N$ required to ensure that the output radius of the interval inclusion of $P_N^{1,1}(x)$ is below the machine precision $\epsilon$ for 64-bit floating-point numbers. It is observed that both algorithms require more than $N$-precision (which corresponds to high-precision arithmetic when $N > 64$ to achieve the machine precision $\epsilon$.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Definition 1.1: Matrix Multiplication Transform
  • Remark 1.2: Clebsch-Gordon vs MMT
  • Definition 2.1: Jacobi polynomials
  • Theorem 2.2: iMMT matrix via Gaussian quadrature
  • Remark 2.3: Computing the weights for Jacobi polynomials
  • Theorem 3.1: Newton-Kantorovich Theorem
  • Definition 4.1: Zernike polynomials
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 21 more