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Hybrid Surrogate Models: Circumventing Gibbs Phenomenon for Partial Differential Equations with Finite Shock-Type Discontinuities

Juan-Esteban Suarez Cardona, Shashank Reddy, Michael Hecht

Abstract

We introduce the concept of Hybrid Surrogate Models (HSMs) -- combining multivariate polynomials with Heavyside functions -- as approximates of functions with finitely many jump discontinuities. We exploit the HSMs for formulating a variational optimization approach, solving non-regular partial differential equations (PDEs) with non-continuous shock-type solutions. The HSM technique simultaneously obtains a parametrization of the position and the height of the shocks as well as the solution of the PDE. We show that the HSM technique circumvents the notorious Gibbs phenomenon, which limits the accuracy that classic numerical methods reach. Numerical experiments, addressing linear and non-linearly propagating shocks, demonstrate the strong approximation power of the HSM technique.

Hybrid Surrogate Models: Circumventing Gibbs Phenomenon for Partial Differential Equations with Finite Shock-Type Discontinuities

Abstract

We introduce the concept of Hybrid Surrogate Models (HSMs) -- combining multivariate polynomials with Heavyside functions -- as approximates of functions with finitely many jump discontinuities. We exploit the HSMs for formulating a variational optimization approach, solving non-regular partial differential equations (PDEs) with non-continuous shock-type solutions. The HSM technique simultaneously obtains a parametrization of the position and the height of the shocks as well as the solution of the PDE. We show that the HSM technique circumvents the notorious Gibbs phenomenon, which limits the accuracy that classic numerical methods reach. Numerical experiments, addressing linear and non-linearly propagating shocks, demonstrate the strong approximation power of the HSM technique.
Paper Structure (11 sections, 4 theorems, 21 equations, 3 figures)

This paper contains 11 sections, 4 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notation
  3. Sobolev Cubatures
  4. Contribution
  5. Hybrid Surrogate Models HSMs
  6. Distributional Derivative for $H_s$
  7. Reconstruction problem and discretized loss
  8. Reconstruction loss discretization
  9. Transport equation in 1D with discontinuous solution
  10. Numerical Experiments
  11. Conclusions

Key Result

lemma 1

Let $x\in \Omega$ be an arbitrary point, and let $F: \mathcal{D}'((0,T)) \rightarrow \mathbb{R}$ be a distribution defined as $F[\phi] := \int\limits_{0}^{T}H_s(x,t)\phi(x,t)dt$ for $\phi(x,\cdot)\in D((0,T))$ and $H_s\in BV((0,T)))$ the Heaviside function, then its distributional derivative $F'\in and is given by $F'[\phi] = \phi(x,s) = \int\limits_{0}^T \phi(x,t)\delta_s(t)dt$, where $\delta_s(

Figures (3)

  • Figure 1: Ground truth for the numerical experiments
  • Figure 2: Approximation errors for the reconstruction problem
  • Figure 3: Approximation of the 1D transport equation

Theorems & Definitions (8)

  • lemma 1
  • proof
  • theorem 1
  • theorem 2
  • lemma 2
  • proof
  • proof
  • remark 1